Dynamics and Bifurcations-Hale
Dynamics and Bifurcations-Hale
Dynamics and Bifurcations-Hale
Editors
J.E. Marsden
L. Sirovich
M. Golubitsky
W.Jiger
F.John (deceased)
Advisors
G.Iooss
P. Holmes
D. Barkley
M. Dellnitz
P. Newton
Springer
New York
Berlin
Heidelberg
Barcelona
Hong Kong
London
Milan
Paris
Singapore
Tokyo
Texts in Applied Mathematics
Dynamics and
Bifurcations
Springer
Jack K. Hale Huseyin ~ak
School of Mathematics Department of Mathematics and
Georgia Institute of Technology Computer Science
Atlanta, GA 30332 Uni versity of Miam i
USA Coral Gables. FL 33 124
hale@math.gatech.edu USA
hk @math.miami.edu
Editors
lE. Marsden L. Sirovich
Control and Dynamical Systems 107-8 1 Division of Applied Mathematics
Cali fornia Institute of Technology Brown University
Pasadena. CA 9 1125 Providence. RI 02912
USA USA
M. GolubilSky W. Higer
Department of Mathematics Depanment of Applied Mathematics
Uni versity of Houston Universitlit Heidelberg
Houston, TX 77204·3476 1m Neuenheimer Feld 294
USA 69 120 Heidelberg. Gennany
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9 8 7 6 5 4
ISB N-13: 978- 1-461 2·8765-0 e- ISBN- 13: 978·1-46 12-4426-4
DOl: 10. 10071978- 1·4612-4426-4
To Students:
Thank you for opening our book. Inside you will find ideas
and examples about the geometry of dynamics and bifur-
cations of ordinary differential and difference equations.
As it is an unusual book in both content and style, let
us explain how it evolved from our courses in the Divi-
sion of Applied Mathematics at Brown University during
a three-year period, and came into being.
The subject of differential and difference equations, alias dynamical
systems, is an old and much-honored chapter in science, one which germi-
nated in applied fields such as celestial mechanics, nonlinear oscillations,
and fluid dynamics. Over the centuries, as a result of the efforts of scien-
tists and mathematicians alike, an attractive and far-reaching theory has
emerged. In recent years, due primarily to the proliferation of computers,
dynamical systems has once more turned to its roots in applications with
perhaps a more mature outlook. Currently, the level of excitement and
activity, not only on the mathematical front but in almost all allied fields
of learning, is unique. It is the aim of our book to provide a modest foun-
dation for taking part in certain theoretical and practical facets of these
exciting developments.
The subject of dynamical systems is a vast one not easily accessible to
undergraduate and beginning graduate students in mathematics or science
and engineering. Many of the available books and expository narratives
either require extensive mathematical preparation, or are not designed to
be used as textbooks. It is with the desire to fill this void that we have
written the present book.
It is both our conviction and our experience that many of the fun-
damental ideas of dynamics and bifurcations can be explained in a sim-
ple setting, one that is mathematically insightful yet devoid of extensive
viii Greeting
FAREWELL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537
ApPENDIX: A Catalogue of Fundamental Theorems . . . . . . . . . . 539
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545
INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559
1
Scalar
Autonomous
Equations
1t'i11
•
In thi, opening chapt"" we p"",mt ""lected bM" oon-
cepts about the geometry of solutions of ordinary differ-
ential equations. To keep the ideas free from technical
~ complications, the setting is one-dimensional-the scalar
autonomous differential equations. Despite their simplic-
====.=~ ity, these concepts are central to our subject and reap-
pear in various incarnations throughout the book. Following a collec-
tion of examples, we first state a theorem on the existence and unique-
ness of solutions. Then we explain what a differential equation is ge-
ometrically. To facilitate qualitative analysis, geometric concepts such
as vector field, orbit, equilibrium point, and limit set are included in
this discussion. The next topic is the notion of stability of an equilib-
rium point and the role of linear approximation in determining stability.
We conclude the chapter with an example of a scalar differential equa-
tion defined on a one-dimensional space other than the real line-a circle.
4 Chapter 1: Scalar Autonomous Equations
(1.3)
1.1. Existence and Uniqueness 5
when the integral is defined. One obtains x(t) by finding the inverse of
the function on the left-hand side of this equation. Occasionally, we will
use this formula to exhibit solutions of special differential equations for the
purposes of illustrations. However, in general, it is impossible to perform
these integrations and one should not expect to obtain explicit formulas for
solutions. It is important to realize this fact from the beginning. In fact,
our objective in this book is to understand as much as possible about the
behavior of solutions of differential equations without the knowledge of an
explicit formula for the solutions.
Let us now give several examples of differential equations and their
solutions in order to realize some of the difficulties that arise in laying the
foundations for the theory, that is, the existence and the uniqueness of
solutions of Eq. (1.2).
Example 1.1. The first example: Consider the differential equation
X= -x. (1.4)
(ii) If, in addition, f E e 1 (IR, IR), then 'P(t, xo) is unique on Ixo and
'P(t, xo) is continuous in (t, xo) together with its first partial deriva-
tives, that is, rp(t, xo) is a e 1 function. <)
The largest possible interval Ixo in part (i) of the theorem above is
called the maximal interval of existence of the solution 'P(t, xo). The maxi-
mal interval of existence of a solution of Example 1.2 is shown in Figure 1.0.
1.1. Existence and Uniqueness 7
~
Ii
~ I
0 1 t
.0.0
on JR. There are also other ways of obtaining dynamical systems and we
will see one such important case in Chapter 3.
Exercises _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
1.1. Verifying hypotheses: Show that the initial-value problem:i; = 1+x2 , x(O) =
o has a unique solution by verifying the hypotheses of Theorem 1.4. Find
the solution. What is the maximal interval of existence of the solution?
1.2. Multiple solutions and numerics: Reexamine Example 1.3, :i; = ,.;x, with
the initial value x(O) = 0 in light of Theorem 1.4. Which hypothesis of the
theorem is not met by the example to prevent the uniqueness of solutions?
Solve this initial value problem numerically on the computer. If you have not
studied numerical solutions of differential equations before, you may want
to return to this task after reading Chapter 3 or consult the reference below.
Which solution do you obtain? Try several different numerical algorithms.
Do you succeed in obtaining the nonzero solution?
Help: In the sequel, some of the exercises will include numerical experiments
with differential equations. To eliminate the burden of programming, we
suggest the computer program PHASER: An Animator/Simulator for Dy-
namical Systems which accompanies one of our earlier books, Koc;ak [1989].
1.3. Infinitely many solutions: Show that the differential equation :i; = X 2 / 3 has
infinitely many solutions satisfying x(O) = 0 on every interval [0, a].
1.4. No solution: Consider the function
f(x) = {I2 ~fx ~
If x>
0o.
Show that the differential equation :i; = f(x) has no solution satisfying
x(O) = 0 on any open interval about to = o.
x x
\
\
\
\
\
\
\
\
\
\
l
~
"--. --. --.
~
" "--. "--. "--. ~
0
---- ---- ---- ---- ---- ---- t
/ I / /
I I I I I t
(a)
I I / I I /
(b)
t
Figure 1.1. (a) Direction field along with several trajectories, and (b) vec-
tor field of x = -a;.
x x
t
I I I I I t
-- -- -- -- -- 0
t
I I I I I t
I I I I I t
(a) (b)
t
Figure 1.2. (a) Direction field along with several trajectories, and (b) vec-
tor field of x = a;2.
x x
o o
(a) (b)
Figure 1.3. (a) Positive orbit, negative orbit, orbit through Xo, and (b)
phase portrait of x = -x.
x x
'>I"}{ ")'(xo)
xo
"),-(xo) {
0 0
(a) (b)
Figure 1.4. (a) Positive orbit, negative orbit, orbit through Xo, and (b)
phase portrait of x = x 2 •
There are some orbits which are especially simple, but they play a
central role in the qualitative study of differential equations, as well as in
applications.
Definition 1.6. A point x E ill. is called an equilibrium point (also critical
point, steady state solution, etc.) ofi; = f(x), if f(x) = 0.
When x is an equilibrium point, the constant function x(t) = x for all
t is a solution, and thus the orbit ')'(x) is x itself.
It is very easy to draw orbits of Eq. (1.1) from the graph of f(x).
In fact, the sign of f determines the direction of the motion along an or-
bit. If f(xo) < 0, then the solution is decreasing in t, and cp(t, xo) either
approaches an equilibrium point or tends to -00 as t --+ (3xo. Similarly,
if f(xo) > 0, then the solution is increasing in t, and cp(t, xo) either ap-
proaches an equilibrium point or tends to +00 as t --+ (3xo; see Figures 1.5a
and 1.6a. Furthermore, if solutions ofthe initial-value problem for Eq. (1.1)
are unique, then the solutions through two different initial conditions with
Xo < Yo satisfy cp(t, xo) < cp(t, yo). Thus we have the following lemma:
Lemma 1.7. Suppose that the solution cp(t, xo) of the initial-value prob-
lem is unique for every Xo. Then
(i) cp(t, xo) is a monotone function in t;
(ii) cp(t, xo) < cp(t, Yo) for all t if Xo < Yo;
(iii) if ')'+(xo) [respectively, ,),-(xo)] is bounded, then (3xo = +00 [respec-
tively, Q xo = -00] and cp(t, xo) --+ x as t --+ +00 [respectively, t --+
-00], where x is an equilibrium point. <>
Let us now illustrate the approach to drawing phase portraits discussed
above. We note, however, that unlike the following example, it may not
12 Chapter 1: Scalar Autonomous Equations
(a)
(b)
x= x - x3 . (1.6)
The equilibrium points of this equation are -1, 0, and 1, and the function
f(x) = x - x 3 is positive on the interval (-00, -1), negative on (-1,0),
1.2. Geometry of Flows 13
(a)
(b)
Figure 1.6. Determining the phase portrait of x = x 2 from (a) the func-
tion f(x) = x 2 , and (b) from the potential function F(x) = -x 3 /3.
positive on (0, 1), and negative on (1, +00). Therefore, its phase portrait
can easily be drawn as in Figure 1. 7a. The orbits are the open intervals
(-00, -1), (-1,0), (0, 1), (1, +00), and the points {-1}, {O}, and {1}. <)
Determining the origins and ultimate destinations of orbits will be one
of our primary concerns. Therefore, we introduce the following important
concepts:
Definition 1.9. If ,- (xo) is bounded, then the set
14 Chapter 1: Scalar Autonomous Equations
f(x) = x - X 3
(a)
(b)
is called the a-limit set of Xo. Similarly, if ,+ (xo) is bounded, then the set
that a and ware, respectively, the first and the last letters of the Greek
alphabet.
We now present another method that is particularly useful in deter-
mining the flows of certain specific differential equations. Equation (1.1)
can be rewritten in the form
where
F(x) == -fox f(s) ds.
Equation (1.7) in this form is a special case of gmdient systems which we
will study later in a more general setting. At this time, it is sufficient to
note that, if x(t) is a solution of Eq. (1.7), then
d d d
dt F (x(t)) = dx F (x(t)) . dt x(t) = - [f (x(t))]2 ::; o.
Thus F is always decreasing along the solution curves, and hence can be
thought of as a "potential" function of Eq. (1.1). It is evident that the
equilibrium points of the differential equation (1.7) are extreme points of
the potential function F.
Let us now reconsider the previous examples from this new viewpoint.
Example 1.1 revisited. Equation (1.4) can be written as a gradient
system (1.7) with the potential function F(x) = x 2/2. The orbits of
± = -x = - d~ (~2)
can be drawn by thinking of the motion of a particle on the graph of the
potential function F(x). As shown in Figure 1.5b, a particle at any point Xo
goes downhill with (ever decreasing) velocity f (x) towards the equilibrium
point O. Therefore, the orbits for this equation are the intervals (-00, 0),
(0, +00), and the equilibrium point O. <>
Example 1.2 revisited. The differential equation (1.5) can be written as
with the potential function F(x) = -x3 /3. The graph of F(x) and the flow
are shown in Figure 1.6b. Here the orbits are again the same intervals as
in the previous example, however, the two flows are different. <>
16 Chapter 1: Scalar Autonomous Equations
±= x_ X3 = _ d~ (_ x; + ~4)
with the potential function F( x) = -x 2/2 + x4 /4. The graph of F( x)
and the flow are shown in Figure 1. 7b. Here the orbits are the intervals
(-00, -1), (-1,0), (0, 1), and (1, +00), and the equilibrium points -1, 0,
and 1. ¢
Exercises _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ .. C/. ¢
1.5. Many examples: Describe all of the orbits and sketch the phase portrait
of each of the following scalar differential equations in two ways: first, by
determining the intervals on which the vector field is of constant sign; second,
by using a potential function:
(a) x = -2x; (b) x = 1 + x; (c) X = x(l- x);
(d) x = x - x 3 + 1; (e) x = x - x 3 + 0.2;
(f) x = -x - x 3 + 1; (g) x = -x - x 3 + A, where A is a constant.
(h) x = 2sinx; (i) x = 1- 2sinx; (j) x = 1- sinx;
' 2 '
(k) x = - sm x; (I) . h ( ). {0 if x = 0
x = tan x; m x = x In Ixl if x =1= O.
Suggestion: You may wish to determine the phase portraits of some of these
examples numerically using PRASER. In this case, to compute a negative
orbit you should use a negative step size and a negative "end time."
1.6. Show that if f E C 1 (lR, lR) and the positive orbit ,+(xo) is bounded, then
/3"'0 = +00. State and prove a similar fact for negative orbits.
Hint: Use uniqueness.
1.7. Show that if f E C 1 (lR, lR) and the positive orbit ,+(xo) is bounded, then
w(xo) is an equilibrium point. State and prove a similar fact for negative
orbits.
Hint: Use the Mean Value Theorem to show that t.jJ(t, xo) ---> 0 as t ---> +00.
Figure 1.8. Phase portrait of:i; = _x 3 sinx- 1 . The origin is stable, but
not asymptotically stable.
iJ = f'(x)y + g(y),
which can be considered as a perturbation of the linear differential equation
iJ = f'(x)y. In fact, the function g(y) satisfies g(O) = 0 and g'(O) = O.
Since g'(O) = 0, for any E> 0, there is a 8 > 0 such that 19'(y)1 < Eif
Iyl < 8. Using the formula g(y) = J~ g'(s) ds, it follows that Ig(y)1 ::; EIYI if
Iyl < 8. Now suppose that f'(x) =f. 0 and E< 1f'(x)l. Then Iyl < 8 implies
that the sign of the function f(x + y) = f'(x)y + g(y) is determined by
the sign of f'(x)y. Therefore, the conclusion of the theorem follows from
Lemma 1.12. <>
The linear differential equation x = f'(x)x is called the linear varia-
tional equation or the linearization of the vector field x = f(x) about its
equilibrium point x. Theorem 1.14 asserts that, when f'(x) =f. 0, the stabil-
ity type of the equilibrium point x of x = f(x) is the same as the stability
type of the equilibrium point at the origin of its linearized vector field.
We introduce the following common terminology for an equilibrium
point satisfying the hypothesis of the theorem above.
Definition 1.15. An equilibrium point x ofx = f(x) is called a hyperbolic
equilibrium if f'(x) =f. O.
If f'(x) = 0, then x is called a nonhyperbolic or degenerate equilib-
rium point. Unlike near a hyperbolic equilibrium where the linear term
of the vector field determines the flow locally, the stability properties of a
nonhyperbolic equilibrium x depends on higher-order terms in the Taylor
expansion of the function f(x + y). For instance, while x = 0 is an unsta-
ble equilibrium for x = x 2 , it is asymptotically stable for x = -x 3 . There
are other complications associated with nonhyperbolic equilibria; infinitely
many equilibria are present in any open neighborhood of the nonhyperbolic
equilibrium x = 0 of the differential equation x = -x 3 sin x- 1, as we saw
in Example 1.13. These examples point to the realization that a study of
nonhyperbolic equilibria will not be trivial. Despite the difficulties associ-
ated with them, however, nonhyperbolic equilibria playa prominent role
in our subject, as we shall soon see.
N=-)..N,
substance for r, that is, the necessary time for the substance to decay to
half of its original size. Show that>. = (In2)/r, where r is the half-life.
Once >. is determined, N (t) can readily be found for any t by evaluating
the exponential function on a pocket calculator. Have you ever wondered
how your calculator or computer determines the values of the exponential
or logarithm function?
The half-life of the naturally occurring radioactive element 14e, carbon-14,
is known to be 5568 years. Compute the length of time it takes for a mass
of 14e to reduce to 20 percent of its original weight.
Radiocarbon Dating: An effective method of estimating the ages of archeo-
logical finds of organic origin is the method of 14e dating discovered by W.
Libby in 1949. The key idea of the method is remarkably simple: 14e is
in equilibrium in living plants-the amount absorbed from the atmosphere
balances the anlOunt that radiates. Once the plant dies, it ceases to absorb
any more 14e but the radiation continues. One basic cosmological premise
is that the concentration of 14e in the atmosphere has been constant over
millennia. Suppose that at t = 0 a tree dies. Let R(t) be the rate of
disintegration of 14e in the dead wood at time t. Derive the formula
1 R(O)
t = ~ In R(t)"
Now, we can measure R(t) at the present time. R(O) is also measurable using
a piece of living plant. So, the age of the dead wood is easy to compute.
Here is an example.
Agn Dagmda: In 1956, a piece of old wood excavated at Mount Ararat gave
a count of 5.96 disintegrations per minute per gram of He while living wood
gave 6.68. Did the piece of old wood come from the Ark? Well, 14e dating
is not always reliable over relatively short time spans.
Using the methods of Section 1.2, it is easy to determine the flow of Eq. (1.9)
on IR. The corresponding flow on 81 can be obtained by identifying the end
points of any interval of length 211"; see Figure 1.9. <>
I I I
,
·O~·
I I
I I
•
I I x
4 •
• • I
Figure 1.9. Phase portrait of x = sinx on the line and on the circle.
Exercises _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ .. Q. <>
1.14. Sketch the phase portraits on the circle and analyze the stability of equilibria
of the following scalar differential equations:
(a) x = 2sinx; (b) x = 1- 2sinx; (c) x = 1- sinx;
(d) x = 1- 2sin(x + 1); (e) x = cos(2x) - cos x + 1.
Bibliographical Notes - - - - - - - - - - - - - - - - - - I f @
There is a vast literature on the fundamental theorem of existence and
uniqueness of solutions of ordinary differential equations. Variations on the
statements and methods of proofs of such theorems abound; see, for exam-
ple, the Appendix, Coddington and Levinson [1955], Hale [1980], Hartman
[1964], and Robbin [1968]. We will have no need to resort to "pathological"
functions as vector fields; the dynamics of polynomial vector fields, with
a few trigonometric functions thrown in, provide us with more complexity
than anyone is able to understand.
It is not our intent in this book to dwell on specialized results that are
manifestations of low dimensionality. Indeed, many of the concepts and
1.4. Equations on a Circle 23
c<O
c=O------~--~~--+-----
x
c>O
F(O, x) = -x
F(O, x) = X2
c<O
c=O--------~~~~------
x
c>O II
I
I
I
For a scalar differential equation :i; = f(x), the equilibrium points and
the sign of the function f(x) between the equilibria determine the number
of orbits and the direction of the flow on the orbits. We refer to the number
of orbits and the direction of the flow on the orbits as the orbit structure
of the differential equation or the qualitative structure of the flow.
The study of changes in the qualitative structure of the flow of a dif-
ferential equation as parameters are varied is called bifurcation theory. At
a given parameter value, a differential equation is said to have stable orbit
structure if the qualitative structure of the flow does not change for suf-
ficiently small variations of the parameter. A parameter value for which
the flow does not have stable orbit structure is called a bifurcation value,
and the equation is said to be at a bifurcation point. It is evident from
the analysis above that Eq. (2.1) has stable orbit structure for all values of
28 Chapter 2: Elementary Bifurcations
c, and that Eq. (2.2) has stable orbit structure for any c =I- 0, but is at a
bifurcation point for c = O. The particular bifurcation behavior of Eq. (2.2)
described above is called saddle-node bifurcation. The choice of terminol-
ogy saddle-node will become apparent when we discuss two-dimensional
systems in Part III of our book.
There is another very useful graphical method for depicting some of
the important dynamical features in equations ± = F(c, x) depending on
a parameter c. This method consists of drawing curves on the (c, x)-plane,
where the curves depict the equilibrium points for each value of the param-
eter. More specifically, a point (co, xo) lies on one of these curves if and
only if F(eo, xo) = o. Also, to represent the stability types of these equi-
libria, we label stable equilibria with solid curves and unstable equilibria
with dotted curves. The resulting picture is called a bifurcation diagmm.
For instance, the bifurcation diagram of the saddle-node bifurcation in Ex-
ample 2.2, ± = c + x 2 , is the parabola c = -x 2 labeled as in Figure 2.3.
Example 2.3. Transcritical bifurcation: Consider the differential equation
containing a real parameter c:
(2.3)
which is another perturbation of Eq. (1.5). Unlike the previous example,
this perturbation is not a translation of the unperturbed vector field. Nev-
ertheless, it is still easy to determine the phase portrait of Eq. (2.3) from
the graph of the function F(c, x) = cx + x 2 as shown in Figure 2.4. Notice
that the origin is an equilibrium point for all values of the parameter c.
For c < 0, the origin is asymptotically stable and there is another equi-
librium point x = -c which is unstable. The parameter value c = 0 is a
2.1. Dependence on Parameters - Examples 29
F(e, xl F(e, xl
c<o C"'O
c>o
,,
,,
,,
,,
,
------->k- - - - - - - c
bifurcation value at which the two equilibria coalesce at the origin, which
is a nonhyperbolic unstable equilibrium point. For c > 0, the origin be-
comes unstable by transferring its stability to another equilibrium point,
x = -c. For this reason, the bifurcation that Eq. (2.3) undergoes is called
trans critical bifurcation; see Figure 2.5. <)
Example 2.4. Hysteresis: Consider the cubic differential equation con-
taining a real parameter c:
j; = c+ x - x 3 . (2.4)
2
C=-3~-----+--4r--~---+~~~---+--------
C=O--------__--~----~----~r_---------- x
,,
\
,
,, c
\
\
\
\
\ ,
c
---
Figure 2.8. Hysteresis loop.
a very large positive value, the system will follow the equilibria on the right
leg of the cubic until c = -Cl, at which point it will jump to the left leg.
The solid lines in Figure 2.8 indicate the equilibria the system will follow
as C is decreased from a very large positive value to a very large negative
value. The important observation about this experiment is that the system
experiences a jump at two different values of the parameter; moreover,
the parameter value at which the jump takes place is determined by the
direction in which the physical parameter is varied! This phenomenon
is referred to as hysteresis and the part in Figure 2.8 that resembles a
parallelogram is called the hysteresis loop. (;
The perturbation of the cubic differential equation given in Eq. (2.4) is,
in a way, the simplest one since it is equivalent to a translation of the x-axis.
In the next two examples, we will study the effects of other perturbations
of Eq. (1.6).
Example 2.5. Pitchfork bifurcation: Consider the differential equation
x= dx - x 3 , (2.5)
where d is a real parameter. The effect of varying d is equivalent to changing
the slope of the cubic at the origin while keeping the x-axis the same.
Reasoning as before, it is easy to see that Eq. (2.5) has three equilibria and
stable orbit structure for all d > O. At d = 0, the equilibria come together
at the origin and the system is at a bifurcation point. For all d < 0, the
equation again has stable orbit structure, with one asymptotically stable
equilibrium point; see Figure 2.9.
2.1. Dependence on Parameters - Examples 33
F(d, xl = dx _x 3
d<O d=O
d>O
depending on two real parameters c and d. The vector field (2.6) is the
most general perturbation of the function -x3 with lower order terms be-
cause any term involving x 2 can always be eliminated by an appropriate
translation of the variable. In fact, for a general cubic _x 3 + ex 2 + dx + c,
use the change of variable x t--t x + e/3 and determine the new coefficients
c and d in terms of e, c, and d.
We begin the analysis of Example 2.6 by first finding the bifurcation
values of the parameters. As we have seen in the previous examples, at
bifurcation points, a differential equation must have a nonhyperbolic equi-
2.1. Dependence on Parameters - Examples 35
-................ ....,
....
,
'\
----------+---------d
c + dx - x 3 = 0 and d - 3x 2 = o.
Our objective is to determine all values of c and d for which these two
equations can have some common solution x. Therefore, we can consider
these equations as defining c and d parametrically in terms of x. Solving
the second equation for d and then substituting the result into the first
equation yields
(2.7)
If we now eliminate x from these two equations, we obtain the following
equation for a cusp:
(2.8)
In Figure 2.12 we have drawn the graph of Eq. (2.8) in the (c, d)-plane;
this graph is a cusp. In each appropriate region of that (c, d)-plane we
have sketched a graph for the function F(c, d, x). In each such sketch we
have indicated the flow determined by Eq. (2.6).
There is a wealth of information packed in Figure 2.12, including the
dynamics of Eqs. (2.4) and (2.5). To extract the dynamics of Eq. (2.4), we
fix d at a positive value, say d = 1, and then obtain the bifurcation diagram
of hysteresis shown in Figure 2.7. To extract the dynamics of Eq. (2.5),
we fix c = 0 and obtain the pitchfork bifurcation diagram in Figure 2.10.
Equation (2.6) also contains a supercritical saddle-node bifurcation in dis-
guise: fix c i- 0, say c = 1, and vary d, as illustrated in Figure 2.13.
36 Chapter 2: Elementary Bifurcations
d
4d 3 = 27c 2
~
4 c
Figure 2.12. Pictures of the cusp 4d3 = 27c 2 in the (c, d)-plane and some
representative phase portraits of x = c + dx - x 3 .
•---------- d
"'----
Figure 2.13. Supercritical saddle-node bifurcation in x = 1 + dx - x3 •
~, d
\ A \~'
Figure 2.14. The bifurcation diagram of the cubic differential equation
:i; = c + dx -
x 3 in the (c, d, x )-space. The cusp below is the set of points in
the (c, d)-plane for which the folding surface above has vertical tangency;
over a point inside the cusp, the surface above has multiple values.
\-,d
(a)
~,
d
\ X \ \'
(b)
t-'
d
(c)
\ X \ \'
Figure 2.15. Slices of the bifurcation diagram of:i; = c + dx - x 3 : (a) hys-
teresis for d = 1, (b) supercritical pitchfork for c = 0, and (c) supercritical
saddle-node for c = 1.
2.1. Dependence on Parameters - Examples 39
c=~
\
d = m~ + 1 - - - - - t - - - - - '-
c
---+-=~---
" .....
---
d
/
c ~2 ~
I
I
I
I
I I I
I~ ............... I~·....•...•...•..• ...• .....·--
I I I
Exercises _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ .. C7. 0
2.1. Identify the groups of examples in Exercise 1.5 with the same orbit structure.
2.2. Provide the details of the computation of the bifurcation diagrams in Ex-
ample 2.7.
40 Chapter 2: Elementary Bifurcations
X= kX-=2 -h,
where all the coefficients are positive; k and c reflect the intrinsic growth rate
of the population and h is the rate of harvesting. Notice that the population
grows at a rate proportional to its size when the population density is small.
However, when the population gets large, growth is impaired because of, for
example, overcrowding; the x 2 term reflects this behavior.
Now, the problem is, for fixed k and c, to determine the effect of harvesting
on the population. Since the population density cannot be negative, we are
interested in the solutions of this equation for x ~ O. For a positive initial
population density, the population is exterminated if there is a finite value of
t such that cp(t, xo) = O. Without finding explicit solutions of the differential
equation, show the following:
(a) If the harvesting rate h satisfies 0 < h ~ k 2 / ( 4c), then there is threshold
value of the initial size of the population such that if the initial size
is below the threshold value, then the population is exterminated. On
the other hand, if the initial size is above the threshold value, then the
population approaches an equilibrium point.
(b) If the harvesting rate h satisfies h > k 2 / ( 4c), then the population is
exterminated regardless of its initial size.
2.4. Proportional harvesting: Suppose that a population grows according to the
"logistic model" as in the previous exercise, but is harvested at a rate pro-
portional to the size of the population:
x = kx - =2 - hx,
where k, c, and h are positive constants. Show that if k < h, then, regardless
of the initial density Xo > 0, such a population tends toward extermination
as t ----> +00, but is not exterminated in finite time. Also, analyze the fate of
the population in the cases k = hand k > h.
2.5. Hydroplane: The rectilinear motion of a hydroplane, ignoring pitching and
rolling, is determined by a scalar differential equation of the form
mv = T(v) - W(v),
where v is the velocity of the hydroplane, m is its mass, T is the thrust of the
driving mechanism, and W is the resistance. It is reasonable to assume, for
simplicity, that the thrust is approximately constant. The resistance, on the
other hand, should increase with small and large v, but can be negative for
intermediate values of the velocity due to rising of the hydroplane and the
decrease of the wetted area. Discuss the possible motions of the hydroplane
for various values of the constant thrust.
2.2. The Implicit Function Theorem 41
8F
F(O, 0) = 0 and 8x (0, 0) =f. O.
Then there are constants 8 > 0 and 'rJ > 0, and a C 1 function
F(A,X)=O
quite arbitrary, but small, perturbations. Thus, our results are valid only
in a sufficiently small neighborhood around the equilibrium point. For
simplicity, we shall assume that :i; = I(x) has an equilibrium point at OJ if
not, we always can translate to a new coordinate system, as in the proof
of Theorem 1.14.
Case I: Hyperbolic equilibria. Suppose that 1 is a 0 1 function
with 1(0) = 0 and 1'(0) -:f. O. We saw in Theorem 1.14 that the stability
properties ofthe equilibrium point 0 of the differential equation:i; = I(x) is
determined by the linear approximation of the vector field near 0, that is,
the higher order perturbations in the Taylor expansion of the vector field
do not effect the qualitative structure of the flow near zero. However, the
question remains: what happens if we make perturbations that influence
the constant and the linear terms? We will show below that the situation
in Example 2.1 also prevails in the general case.
To be precise, consider the perturbed differential equation
F(O, 0) = 0 and
of
ox (0, 0) = I' (0) -:f. O.
Hence, the Implicit Function Theorem implies that there are constants
8 > 0 and TJ > 0, and a 0 1 function 'IjJ(A) defined for IIAII < 8 with 'IjJ(0) = 0
such that
F(A, 'IjJ(A)) = o.
Moreover, every (A, x) with IIAII < 8 and Ixl < TJ satisfying F(A, x) = 0 is
given by (A, 'IjJ(A)). Therefore, for each IIAII < 8, and Ixl < TJ, there is a
unique equilibrium x = 'IjJ(A) of Eq. (2.12) satisfying Ixl < TJ·
The stability behavior of the equilibrium 'IjJ(A) can easily be determined
from Theorem 1.14. To do so, we need to compute the sign ofthe derivative
of
ox (A, 'IjJ(A)). (2.14)
From Eq. (2.13) and the fact that 'IjJ(0) = 0, we have ~~ (0, 'IjJ(0)) = I' (0) -:f.
O. Thus, there is a 8 > 0 such that, for IIAII < 8, the sign of Eq. (2.14) is
44 Chapter 2: Elementary Bifurcations
the same as that of 1'(0). Therefore, the stability type of the equilibrium
'l/J(A) of the perturbed equation (2.12) is the same as the stability type of
the equilibrium 0 of the unperturbed equation x = f(x).
We can summarize the discussion above by saying that the flow near
a hyperbolic equilibrium point is insensitive to small perturbations of the
vector field.
Case II: Equilibria with quadratic degeneracy. Suppose that f
is a C 2 function with f(O) = 0, 1'(0) = 0, but 1"(0) =1= O. This is the next
order of complication that occurs when we cannot make a decision about
the stability of an equilibrium point based on the linearization.
Let us consider the perturbed differential equation
of
F(O, x) = f(x), ax (0, 0) = 0, ox
0 2 F (0, 0)
2 = f "()
0 =1= O. (2.16)
These conditions together with f(O) = 0 imply that the Taylor expan-
sion of F about the origin has the following form:
x2
F(A, x) = a(A) + b(A)X + c(A)2 + G(A, x),
with a(O) = 0, b(O) = 0, c(O) = 1"(0) =1= 0, and, for any f > 0, there are
8 > 0 and T/ > 0 such that the function G satisfies IG(A, x)1 < flxl 2 for
IIAII < 8 and Ixl < T/.
As an instance of Case II, let us recall the differential equation x =
f(x) = x 2 and its perturbation (2.2) depending on one parameter (k = 1):
. 2
X=F(A, x) =A+1"(O)~.
of
ax (A, x) = o. (2.17)
2.3. Local Perturbations Near Equilibria 45
CI'(A) <0
CI'(A) = 0
CI'(A) >0
~~ (>., ,¢(>.)) = 0
and, moreover, every solution (>., x) of Eq. (2.17) with 11>'11 < 8 and Ixl < 'TJ
is given by x = '¢(>.).
For each fixed >., the function F(>', x) has a minimum at x = '¢(>.) if
1"(0) > 0, or a maximum if 1"(0) < O. The number of equilibrium points
of Eq. (2.15) depends upon the extreme value a(>.) == F(>', '¢(>.)) of the
function F. In Figure 2.18 we have drawn the flows for these two cases
with several values of a(>.) (compare with Figure 2.2). These results can
be summarized analytically by saying that when a(>.)f"(O) < 0 there are
two hyperbolic equilibria near the origin, a(>.) = 0 implies that there is a
nonhyperbolic equilibrium at the origin, and when a(>.)f"(O) > 0 there are
no equilibrium points about the origin.
It is important to observe in the discussion above that the qualitative
structure of the flow of the perturbed equation (2.15) is determined from a
single function of the parameter >., namely, the function a(>.) correspond-
ing to the extreme value of F(>', x). Thus, even though there may be k
components of the (vector) parameter>' = (>'ll >'2,' .. ,>'k), the bifurcation
behavior of the perturbed equation (2.15) depends on a single number,
a(>.). When this situation occurs in a bifurcation problem, we say that the
original vector field f is a codimension-one bifurcation.
46 Chapter 2: Elementary Bifurcations
where A = (At, A2) are two small parameters. The function a(A) for this
example corresponds to the minimum value ofthe function F(A, x) = Al +
A2X + x 2, and it is given by
Thus the bifurcations occur as we cross the curve Al = A~/4 in the (At, A2)-
plane: there are two equilibrium points if Al < A§j4, and none if Al >
A~/4. I)
Example 2.10. In applications, a single parameter may affect several
terms in the Taylor expansion of a perturbation. For instance, consider the
following perturbation of the vector field I(x) = x 2 given by
of
F(O, x) = I(x), ax (0, 0) = 0,
(2.19)
fPF 0 3 F (0, 0) =
ox 2 (0, 0) = 0, ox
3 1III( 0) -=I- O.
(2.20)
with c(O) = 0, d(O) = 1111(0) -=I- 0, and, for any E > 0, there are 8 > 0 and
> 0 such that the function G satisfies IG(A, x)1 < Elxl 3 for IIAII < 8 and
'T}
Ixl < 'T}.
As usual, we begin by finding the bifurcation values of the parameters,
that is, the values of A = (A1' A2) for which the function F(A, x) given
by Eq. (2.20) has multiple zeros. To accomplish this we must solve the
following two equations:
(2.21 )
of
0= -(A, x)
x2
= A2 + C(A)X + d(A)-2 + -;:;-(A, x).
oG (2.22)
ax uX
We view Eqs. (2.21) and (2.22) as a system of two equations defining A1
and A2 parametrically in terms of x. To obtain the solutions locally, we
can use essentially the method of Gaussian elimination in conjunction with
repeated applications of the Implicit Function Theorem: using Eq. (2.22)
solve locally for A2 as a function of A1 and x, then substitute the result
into Eq. (2.21) to determine A1 in terms of x.
48 Chapter 2: Elementary Bifurcations
~ ~ 00
-(A, x) == H(All x, A2) = A2
& + C(A)X + d(A)-2 + ~(A,
~
x).
x2
0= J(x, ).,1) == ).,1 + 'l/J2().,t, x)x + C().,l, 'l/J2().,1, x))"2
x3
+ d().,l, 'l/J2().,1, x))6 + G().,l, 'l/J2().,1, x), x)
and observe that J(O, 0) = 0 with oJ(O, 0)/0).,1 = 1 =1= o. Thus, by the
Implicit Function Theorem, there is a C3 function 'l/J1(X) defined for x small
so that
J(x, 'l/J1(X)) = O.
Moreover, every small (x, ).,1) satisfying J(x, ).,t} = 0 is given by ).,1 =
'l/J1(X). From the special form of J, it is not difficult to see, again with
some computing, that
'l/J1 (0) = 'I/J~ (0) = 'I/J~ (0) = 0, 'l/Jt (0) = 21'" (0).
Thus, the Taylor expansion of the function 'l/J1 (x) has the form
(2.24)
2.3. Local Perturbations Near Equilibria 49
(2.26)
If 1111(0) < 0, then the equation above is essentially that of the cusp in
Eq. (2.8). Therefore, the qualitative structure of the flows of the perturbed
equation (2.20), for small values of A, are the same as the ones given in
Figure 2.12. For practice, you should draw the flows for the case when
1"'(0) > O.
Equation (2.20) is an example of a "good" two-parameter perturbation
of a vector field with cubic degeneracy in the sense that the bifurcations
are determined only by the constant and the linear terms of the Taylor
expansion of the vector field. The term C(A)X2/2 did not enter into the
first approximation to the cusp in the (Ai, A2)-plane; see Eqs. (2.25) and
(2.26). This is because the function C(A) is differentiable in A and c(O) = O.
Thus, the Taylor expansion for C(A) must be given by
where Cl and C2 are constants; hence, the term C(A)x2 has the form
with a(O) = b(O) = 2(0) = 0, d(O) = 1"'(0) =I- 0, and, for any f > 0, there
3
are 8 > 0 and 'fJ > 0 such that the function G satisfies IG(f.J" x)1 < flxl for
~ ~
To make certain that (J-Lb J-L2) cover all possible small values of the
constant and the linear terms, that is, the range of the vector-valued func-
tion (a(J-L), b(J-L)) cover a neighborhood of zero for small J-L, we suppose that
the Jacobian of (a(J-L), b(J-L)) with respect to J-L at J-L = 0 is not zero:
(2.28)
in a neighborhood of A = (AI, A2) and J-L = (J-Lb J-L2) equal to zero. For any
J-L, the constant A is uniquely defined by Eq. (2.29). To know that relation
(2.29) is a good transformation of parameters we need to make certain
that, given A, a constant J-L is uniquely defined by Eq. (2.29). Condition
(2.29) and the Implicit Function Theorem imply that this is the case in a
neighborhood of zero. In fact, if we define the functions
(2.30)
where J-LI and J-L2 are two real parameters. For the perturbed vector field
(2.30), the functions in Eq. (2.27) are given by a(J-L) = J-LI + J-L2, b(J-L) =
3
J-L2 + J-LI' c(p) = J-Lb d(J-L) = -1, and G(J-L, x) = O. The Jacobian (2.28) of
~ ~
so that Eq. (2.30) attains the special form of Eq. (2.20) as follows:
(2.31)
where the function C(A) is the unique solution /11 C(A) of the cubic
equation
-/1r + /11 - Al + A2 = 0
which vanishes for (AI, A2) = (0, 0). The Implicit Function Theorem guar-
antees that such a solution exists and is unique.
The bifurcation curve ofEq. (2.31) in the (AI, A2)-plane near the origin
is approximately the cusp 8A~ = 9A~, which follows from Eq. (2.26). Thus,
in the original (/11, /12)-plane this curve becomes
which, near the origin, is again a cusp with approximately the same shape
as the one in the (AI, A2)-plane. <>
Example 2.12. Consider the two-parameter perturbation of the vector
field f(x) = -x3 /6 given by
(2.32)
where /11 and /12 are two real parameters. As a special case of Eq. (2.27),
we have a(/1) = /1I, b(/1) = /1~, 2(/1) = /12, and ;1(/1) = -1. Notice that the
condition (2.28) is not satisfied. However, we can analyze Eq. (2.32) in the
following way.
From our previous discussion of the cusp, it is natural to eliminate the
x 2 term. If we introduce the new variable y == /12 - x, then
\ _ /11 + 3/12'
Al =
4 3 (2.33)
then the differential equation above is put into the form of the special
perturbation (2.20):
Y = Al + A2Y - r,Y .
. \ \ 1 3
To find the bifurcation curve of Eq. (2.32) in the original (Ml, M2)-
plane, we substitute the transformation (2.33) into the cusp and obtain
8 ( 23M22) 3 = 9 (Ml 4 3) 2
+ 3M2 .
Thus, the bifurcation values of Eq. (2.32) in the (Ml, M2)-plane form cubic
curves, not a cusp. Of course, the reason for this deviant behavior is that
the parameters (Ml, M2) do not enter into the equation in a nice way, that
is, the determinant in Eq. (2.28) is zero-"bad." <>
As we mentioned earlier, a complete analysis of the cubic degeneracy
is beyond the intended scope of our book. Therefore, we end this rather
long section with an example of a three-parameter perturbation.
Example 2.13. Three parameters: Consider the three-parameter pertur-
bation of f(x) = -x 3 /6 given by
where Ml, M2, and M3 are three parameters. As in the previous example,
the x 2 term can be eliminated by introducing the new variable y == M3 - x.
Then, the differential equation above in the new variable becomes
8 (M2 + 2M3
1 2) 3
= 9 ( Ml + M2M3 + 3M3
1 3) 2
2.3. Local Perturbations Near Equilibria 53
in the three-dimensional (JLl, JL2, JL3 )-space. As one crosses this surface,
the number of equilibrium points changes from one to three. 0
Exercises - - - - - - - - - - - - - - - -____ .. \? 0
2.9. On quadratic degeneracy: For each of the vector fields below, draw the
bifurcation diagrams with the corresponding phase portraits:
(a) F(c, e, x) = c + ex + x 2. This is Example 2.9, but this time use the
transformation x = y - e/2.
(b) F()', x) = a + (), - a)x + x 2. Draw x versus), for various values of a.
Notice the difference between a < 0 and a > O.
(c) F()', x) = ),2 + 2a),x + x 2. Draw x versus), for various values of a.
Notice the difference between lal < 1 and lal > 1.
(d) F()', x) = ), + 2a),x + x 2, for any fixed value of a.
(e) F()', x) = ),4 + 2a),x + x 2, for any fixed value of a.
2.10. On cubic degeneracy: Obtain the bifurcation curves of the following one-
and two-parameter perturbations of the cubic vector field f(x) = _x 3 and
sketch some representative phase portraits:
(a) x = 1 + JL1 + 2/1-1X - x 3 ; (b) x = 1 + /1-t + JL1X - x 3 ;
(c) X = /1-1 + /1-2 + (/1-1 - /1-2)X - x 3 ; (d) x = /1-1 + /1-2 + (/1-1 - /1-~)x - x 3 ;
(e) X = /1-1 + /1-~ + /1-2X - x 3 ; (f) x = /1-1 + /1-2 + /1-~X + /1-2X2 - x 3 •
2.11. On quartic degeneracy: This exercise is on various one- and two-parameter
perturbations of the quartic vector field f(x) = X4. A complete discussion
of the most general perturbation of the quartic is very difficult. For each of
the following perturbations, draw the bifurcation diagrams along with the
corresponding vector fields:
(a) F(c, x) = c + X4; (b) F(d, x) = dx + X4;
(c) F(e, x) = ex 2 + X4; (d) F(c, d, x) = c + dx + X4;
(e) F(c, e, x) = c + ex 2 + X4; (f) F(d, e, x) = dx + ex 2 + x4.
2.12. Obtain the bifurcation curves of the following one- and two-parameter dif-
ferential equations and sketch some representative phase portraits:
(a) x = /1-1 - x 2/(1 + x 2); (b) x = /1-1 - x 2/(1 + X2)2;
(c) X = /1-1 - x 3/(1 + x 3) for x > -1;
(d) x = /1-1 + /1-2X - x 3/(1 + x 3) for x > -1.
2.13. Unfolding a pitchfork: As we have seen in Example 2.5, the pitchfork bifur-
cation occurs for the vector field F(/1-, x) = _x 3 + /1-X at /1- = 0 where the
number of equilibria changes from one to three or vice versa. An important
practical question is what happens to the bifurcation diagram if this one-
parameter vector field is subjected to small perturbations. Surprising as it
may seem, one can in fact write down a "most general" perturbation of this
vector field near the origin using only two additional parameters:
°
~1 - ~V27. Notice that this is a special case of Example 2.6.
(b) For ~1 = and fixed ~2 < 0, obtain the bifurcation diagram x versus
p. and compare your result with Figure 2.16.
(c) For 01 = 0, that is, ~1 < ~V27 and ~2 < 0, obtain the bifurcation
diagram and, again, refer to Figure 2.16. Do the same analysis for
~1 < ~V27 and ~2 > 0.
(d) Discuss some of the remaining cases.
Reference: For more details on this example, and the general theory of
unfoldings of bifurcations, see Golubitsky and Schaeffer [1985].
Idl < - e
0
OIl • I
I
I
• Idl = - e
0
0
I
I
OIl • I. • • • Idl> lei
• • Idl = e
0
Figure 2.19. Phase portraits of x = c + dsinx on IR and 8 1 .
Idl <e
0
periodic on 8 1 . Since cp( t, xo) is implicitly defined by
t= lXo
'P(t, xo)
----
dx
e + dsinx
and e+dsinx f- 0 for all x, there is a unique T such that cp(T, xo) = xo+271",
l
that is,
T-
xo +271" dx
-
1271" - -dx- -
- Xo e + d sin x - 0 e + d sin x .
If cp(t, xo) is a solution of Eq. (2.34), then so are cp(t, xo) + 271" and cp(t +
T, xo). At t = 0 these two solutions are equal; thus, by uniqueness (Theo-
rem 1.4), they are identical for all t ~ O. This proves the assertion above
and, moreover, shows that the period T is independent of the initial con-
dition.
The time periodicity of the solutions of Eq. (2.34), when Idl < lei,
can easily be observed if we plot trajectories in the (t, x)-plane; see Fig-
ure 2.20.0
Exercises _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4<:;.0
2.14. Draw on the circle the phase portraits of the periodic differential equation
x= -A + 2 + cos(2x) - 3cosx
and verify the qualitative features in the indicated ranges of the scalar pa-
rameter A:
56 Chapter 2: Elementary Bifurcations
xl -12.000000
(a) All solutions are periodic if>. < -1/8 and >. > 6.
(b) There are four equilibria if -1/8 < >. < o.
(c) There are two equilibria if 0 < >. < 6.
2.15. No equilibria: Suppose that f is a 27T-periodic C 1 function with the property
that f(x) "I- 0 for all x E ffi.. Show that every solution of the differential
equation :i; = f(x) is periodic on the circle obtained by identifying the end
points of an interval of length 27T. Derive a formula for the period(s) of the
solutions.
F(A, x) = o.
Using the observation described earlier, if necessary, let us first determine
a pair of values (AO, xo) with F(AO, xo) = O. Let us assume that in a
neighborhood of (AO, xo) on the (A, x)-plane the zeros of F(A, x) lie on
a smooth curve. We can parametrize this curve by t so that there will
be two C 1 functions A(t) and x(t) satisfying A(O) = AO, x(O) = Xo, and
F(A(t), x(t)) = 0 for t in a neighborhood of zero.
To represent this parametrized curve as an orbit of a differential equa-
tion, we differentiate the equation F(A(t), x(t)) = 0 with respect to t:
aF . aF
aA (A, X)A+ ax (A, x)x = o.
To satisfy this identity, ()., x) must be a constant multiple of the vec-
tor (- ~; (A, x), ~f (A, x)) . If we choose the parametrization of the curve
(A(t), x(t)) in such a way that this constant is one, then we arrive at the
pair of differential equations
. aF
A= - ax (A, x)
(2.35)
aF
x= aA(A, x)
The zero set of this function is, of course, the cubic curve >. = -x + x 3 .
However, for the sake of practice, let us try to implement the three steps
above.
In Step 1, let us fix, for example, >'0 = -5. Then, using the scalar
differential equation ;i; = -5 + x - x 3 , we find that the set of its equilibria
consists of a single point: E-5 = { -1.904 ... }.
In Step 2, differential equations (2.35) for the function (2.36) become
.x = -(1 - 3x2 )
(2.37)
;i; = 1
with the initial data A(O) = -5 and x(O) = -1.904 .... Numerical integra-
tion of this initial-value problem gives the cubic curve shown in Figure 2.21.
Forward integration (using positive time and positive step size) of the pos-
itive orbit gives the piece of the cubic above the line >'0 = -5 and the
backward integration (using negative time and negative step size) for the
negative orbit yields the remaining lower part.
Although there is no need for Step 3 in this simple example, let us
see what happens for another value of >'0. If we fix >'0 = 0, for example,
then we have Eo = { -1,0, 1}. Numerical computation of the orbits of
Eq. (2.37) through anyone of the points (0, -1), (0,0), or (0, 1) gives the
same cubic curve. <>
Example 2.17. Consider the equation
The pair of differential equations (2.35) for the function (2.38) are
..i. = -A + 3x 2
(2.39)
±= x.
If we choose Ao = 0, then Eo = {1}. The orbit of Eq. (2.39) through
the point (0, 1) gives the upper curve shown in Figure 2.22. Unlike the
example above, the zero set of Eq. (2.38) is not obvious. So, let us ex-
periment with another choice of Ao. For Ao = 2, computations give E2 =
{ -1, -0.618 ... , 1.618 ... }. The orbits of Eq. (2.39) through the points
(2, -1) and (2, -0.618 ... ) coincide and they both give the lower curve.
The orbit through (2, 1.618 ... ) yields the same upper curve we obtained
with the choice Ao = O. Further experimentation with other choices of AO
does not alter Figure 2.22. <>
Example 2.18. Computing pitchfork: As the final example of this section,
let us try to recover numerically the bifurcation diagram of the pitchfork
bifurcation, that is, compute the set of zeros of the equation
F(A, x) = AX - x 3 • (2.40)
The pair of differential equations (2.35) for this example are
..i. = -A+3x2 (2.41)
±= x.
Notice that Eq. (2.41) is the same as Eq. (2.39), but the bifurcation curves
are determined by special choices of initial values. To compute the bifur-
cation diagram of Eq. (2.40), if we choose AO = -1, then E-l = {O}. The
orbit of Eq. (2.41) through (-1, 0) is the negative part of the A-axis; see
Figure 2.23.
60 Chapter 2: Elementary Bifurcations
equations:
(a) x = X2 - A; (b) x = A2 + X2 -1;
(c) x = (A+2)2 +X2 -1; (d) x = [AX -X3)[(A+2? +x2 -1];
(e) x = 2A 3 + 3A2X - X3; (f) x = 2A 3 + 2.9A2X - X3;
(g) X = 0.1 + 2A2 + 3A2X - X3; (h) x = (A + X - X3)(A2 + X2 - 1).
which, for each t, at least takes one flow to the other, that is,
h (cp(t, xo)) = 'Ij; (t, h(xo)) (2.42)
for all t as long as the flows are defined. Furthermore, since we are looking
for an equivalence relation, it is clear that h should be an invertible map
so that h- 1 takes'lj; to cpo What additional properties should we require of
h so as to capture the qualitative features of the flows? This is a rather
delicate question because if we do not restrict h sufficiently, we may not be
able to distinguish two qualitatively different flows; at the other extreme,
if we restrict h severely, then two flows with the same orbit structure may
not be equivalent. One "natural" choice would be to require both h and
h- 1 to be C 1 functions.
Definition 2.19. A C 1 function h : IR -. IR with a C 1 inverse is called a
C 1 diffeomorphism of IR.
To appreciate some of the possible consequences of requiring h to be a
C 1 diffeomorphism, let us study a specific example. Consider the following
linear vector fields
x= -x, x = -2x. (2.43)
62 Chapter 2: Elementary Bifurcations
These two vector fields have the same orbit structure because they each
have one asymptotically stable equilibrium point. For these two vector
fields, Eq. (2.42) becomes
h (e-txo) = e- 2t h(xo).
If we differentiate this equation with respect to Xo and evaluate the result
at Xo = 0, then we obtain
e-th'(O) = e- 2t h'(0).
Since we require h to be invertible, h'(O) =j:. 0 and thus we arrive at the
disturbing implication that -1 = -2. Consequently, we cannot have h to
be a 0 1 function with a 0 1 inverse if we are to consider the flows of ± = -x
and ± = -2x to be qualitatively equivalent. We settle for the next best
thing.
Definition 2.20. A continuous map h : IR -+ IR with a continuous inverse
is called a homeomorphism of IR.
Definition 2.21. Two scalar differential equations ± = f(x) and ± = g(x)
are said to be topologically equivalent if there is a homeomorphism h of IR
such that h takes the orbits of one differential equation to the orbits of the
other and preserves the sense of direction in time.
For the purposes of comparing the qualitative features of flows of scalar
differential equations, it is not a loss to require h to be merely a homeo-
morphism.
Theorem 2.22. Two scalar differential equations ± = f(x) and ± = g(x)
each with a finite number of equilibrium points are topologically equivalent
if and only if they have the same orbit structure.
Proof. Let us first point out that if two vector fields are topologically
equivalent, then the corresponding homeomorphism takes an equilibrium
point of one vector field to an equilibrium point of the other. With this
observation, it is clear that topological equivalence implies that the two
vector fields have the same orbit structure. We will indicate how to prove
the converse implication. Let Xl, ... , xn be the equilibrium points of the
vector field f with their ordering on the line, and similarly, let Xl, ... , xn
be the equilibrium points of g. Let us choose points aI, ... , an+l so that
al < Xl, Xn < a n +1, and ai+l lies in between the consecutive equilibria
(Xi, Xi+1)' Similarly, choose the points f3l, ... , f3n+ 1 so that f3l < Xl,
Xn < f3n+1, and f3i+1 E (Xi, Xi+l)i see Figure 2.24.
We will first construct a homeomorphism h : (-00, Xl) -+ (-00, xd
of the two open intervals. For any point Xo in (-00, Xl), there is a unique
value txo of time depending on Xo such that ({)(txo' xo) = al. If we let
h( xo) = 'lj;( -txo' f3l), then h is a homeomorphism. To extend h, let h( xd =
Xl. Since h(xo) I---> Xl as Xo I---> Xl, now the map h : (-00, Xl] -+ (-00, Xl]
so defined is a homeomorphism.
2.6. Equivalence of Flows 63
•
)(,
~, ~2 ~3 ~4
I • I • I • I
x,
h(x)
x x
Figure 2.25. Graphs of homeomorphism h and its inverse for the topolog-
x = -x and x = -2x.
ical equivalence of the flows of
Exercises _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . 0 . <>
2.17. Identify the functions below as homeomorphisms, diffeomorphisms, or nei-
ther in their domains of definition:
(a) h(x) = 2x + 1; (b) h(x) = 2x 2 ; (c) h(x) = x 3 ;
(d) h(x) = ~X5/3; (e) h(x) = e"'; (f) h(x) = arctanx.
2.18. Construct a homeomorphism of the real line to establish the topological
equivalence of the differential equations x = 2x and x = x + 2.
2.19. Consider the equation x = x 2 - 1 + A depending on a real parameter A.
Show that the flow of this differential equation is topologically equivalent to
that of x = x 2 - 1 if -00 < A < 1, to x = x 2 if A = 1, and to x = x 2 + 1 if
A>l.
2.20. Show that if a vector field x = F(A, x) has a finite number of equilibria and
they are hyperbolic, then the vector field is structurally stable.
(3.1)
Given xo, all other approximate values Xl, X2, ... , x n , ... can be computed
in succession using this formula. This sequence of numbers is considered
to be the solution of the difference equation with initial value Xo.
Example 3.1. Logistic and Euler: Let us now illustrate the procedure
above on the logistic equation
x=ax(l-x), (3.2)
c;
to obtain the following difference equation:
f(x) = ax(1 - x)
Given an initial value xo, it is rather easy to calculate the solution xo, Xl.
X2, ••• of Eq. (3.3) on the computer. So, let us perform several simple
numerical experiments. If b is very small, that is, h is very small relative to
a, then, for any initial value in the interval (0, 1), the solution of Eq. (3.3)
converges monotonically to 1 as n -+ +00. For example, the solution of
Eq. (3.3) with b = 0.3 and Xo = 0.567 is tabulated in Figure 3.2a. This is
exactly the same qualitative behavior that is exhibited by the orbit of the
differential equation (3.2).
The difficulties begin to occur when b > 1. Since b = ha, this happens,
for example, for a = 1000.0 and h = 0.002. Numerically, h = 0.002 appears
on the surface to be a small step size, but it is overcompensated by the value
of a in the differential equation. For instance, the solution of the difference
equation (3.3) with b = 1.3 and Xo = 0.567 tabulated in Figure 3.2b has
little resemblance to the corresponding orbit of the differential equation
(3.2). Although the solution of the difference equation converges to 1 as
n -+ +00, it is not monotone and leaves the interval [0, 1]. As the final
numerical experiment, let us consider the solution of Eq. (3.3) with b = 2.8
and Xo = 0.567; see Figures 3.2c and 3.2d. In this case, it is not even clear
that the approximate solution converges to 1 as n -+ +00. We will reveal
its asymptotic fate at the end of this chapter. <:;
The dynamics of the difference equation (3.3) is surprisingly compli-
cated. In fact, a variant of this difference equation, which comes up natu-
rally in biology when modeling a seasonally breeding population, has largely
been responsible for the recent surge of activities in this area. We will try
to convey some of this exciting development in Section 3.5. Let us proceed
with our discussion on the role of difference equations in numerical math-
ematics with a more familiar example-Newton's Method for computing
zeros of functions.
Example 3.2. Roots with Newton: In Section 2.5, we indicated that the
zeros of a function f could be found as the (l- or w-limit sets of the differ-
ential equation :i; = f (x), and that this process could be implemented on
70 Chapter 3: Scalar Maps
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. f(x)
x = - f'(x)"
f(x n )
xn+1 = Xn - f'(x n ) (3.4)
In fact, any solution with initial value Xo > 0 converges to J2, and with
Xo < 0 converges to -J2. For this particular function, an Euler approxi-
mation with a rather large step size gives the expected result. You should
experiment on the computer with various functions and step sizes. <>
In order to explore the exciting dynamics of difference equations a bit
further, we now turn to the task of developing some basic concepts and
theorems.
3.2. Cube root: Determine a difference equation for computing the cube root of
a real number. Put your equation for ~ into PHASER and iterate until
the first five digits settle down. How many iterations do you need?
(3.6)
For the sake of brevity, we will sometimes refer to such a difference equation
as a map f.
Definition 3.3. A positive orbit ofxo is the set of points xo, f(xo), P(xo),
... , and is denoted by,+(xo).
It is important to realize that a positive orbit of Eq. (3.6) is a set of
discrete points, not an interval. In fact, this is the main reason for the rich
dynamics of difference equations even in one dimension.
Analogous to equilibrium points of differential equations, difference
equations have simple distinguished orbits as well.
Definition 3.4. The point x is called a fixed point for f if f(x) = x.
Notice that the fixed points of f remain fixed under iterations of the
map, as in the case of the equilibrium points of a differential equation being
solutions that are independent of t. However, in computing fixed points, one
must determine the zeros of the function f(x) - x, not f(x). This remark
will be important when we study bifurcations of maps in the next section.
We now turn to a geometric method, called stair-step diagrams, for
following solutions of one-dimensional difference equations. We first plot
the graph of the function f as well as the diagonal, the 45° line. Since
Xn+l = f(x n ), we think of the horizontal axis as Xn and the vertical axis
as Xn+l. The vertical line from Xo meets the graph of f at (xo, f(xo)) =
(xo, Xl)' The horizontal line from this point intersects the diagonal at
3.2. Geometry of Scalar Maps 73
(Xl, xd. The vertical line from this point intersects the horizontal axis at
Xl' By repeating the same steps we can obtain X2, X3, etc. Notice that
this procedure is equivalent to visualizing the phase portrait of Eq. (3.6)
on the diagonal. Also, it is important to observe that the fixed points of
Eq. (3.6) correspond to the points of intersection of the graph of f with the
diagonal. Let us now practice drawing stair-step diagrams on linear maps.
Example 3.5. Linear maps: Consider the linear difference equation
(3.7)
where a is a real parameter. It is easy to see that the positive orbit of
an initial value Xo is the set of points Xn = anxo, for n = 0, 1, 2, ....
Typical stair-step diagrams for the parameter values a = 2.0, a = 0.5,
a = -0.5, and a = -2.0 are shown in Figure 3.3 (you should consider the
cases a = ±1). Notice that when a > 0, the positive orbit is monotonically
increasing or decreasing on one side of the fixed point, much like an orbit
of a differential equation. When a < 0, however, a positive orbit jumps al-
ternately to either side of the origin and is no longer monotonic, a behavior
with no counterpart in scalar differential equations. <>
After this graphical diversion, let us return to fixed points and in-
vestigate their stability properties. Analogous to the notions of stability
and asymptotic stability of equilibria of differential equations, we make the
following definitions:
Definition 3.6. A fixed point x of f is said to be stable if, for any f > 0,
there is a 8> 0 such that, for every Xo for which Ixo - xl < 8, the iterates
of Xo satisfy the inequality Ir(xo) - xl < f for all n 2:: O. The fixed point
x is said to be unstable if it is not stable.
Definition 3.7. A fixed point x of f is said to be asymptotically stable if
it is stable and, in addition, there is an r > 0 such that r(xo) ----) x as
n ----) +00 for all Xo satisfying Ixo - xl < r.
In analogy with the linearization about an equilibrium point of a differ-
ential equation given in Theorem 1.14, we expect, under certain conditions,
that the stability type of the fixed point x of a map f(x) to be the same as
the stability type of the fixed point at the origin of the linear map f'(x)x.
It is evident from the stair-step diagrams of the linear map in Figure 3.3
that the fixed point at the origin is asymptotically stable if lal < 1, and
unstable if lal > 1. This suggests the following linearization theorem about
a fixed point.
Theorem 3.8. Let f be a 0 1 map. A fixed point x of f is asymptotically
stable if 1f'(x)1 < 1, and it is unstable if 1f'(x)1 > 1.
Proof For convenience, we first translate the point (x, x) = (x, f(x)) to
the origin (0, 0). Let u be the new variable defined by u == x-x. Then
74 Chapter 3: Scalar Maps
~----------------Af~~--~~-------!'"
0=2.0
'0
0=0.5
" '0
0=-0.5
0=-2.0
(3.8)
Example 3.13. More nonhyperbolic fixed points: Consider the cubic maps
(3.10)
(3.11)
(3.12)
(3.13)
(a)
(b)
X n +l = -VX;;.
Are the solutions defined for all n? What is an appropriate existence and
uniqueness theorem for initial value problems of difference equations?
3.4. Do the following for each map below: locate fixed points, find the values
of the parameter at which fixed points are not hyperbolic, determine the
3.2. Geometry of Scalar Maps 79
(a)
(b)
(c)
Figure 3.7. N onhyperbolic fixed point at the origin: (a) stable in f (x) =
x - x 3 , (b) unstable in f(x) = x + x 3 , (c) unstable in f(x) = -x - x 3 , and
(d) stable in f(x) = -x + x 3 .
80 Chapter 3: Scalar Maps
stability types of fixed points, draw typical stair-step diagrams near each
fixed point. To create good pictures, you should use PHASER.
(a) f(A, x) = Ax(l - x) for A > 1;
(b) f(A, x) = Ax(l - x) + 0.1 for 0 :s; x :s; 1;
(c) f(A, x) = A - x 2; (d) f(A, x) = A2 - x 2 ;
(e) f(A, x) = eX - A; (f) f(A, x) = -(A/2) arctan x;
(g) f(A, x) = x - AX(X - ~)(x - ~)(x -1). In this example, do not forget
to consider the parameter values 1, 4.6, 9.1, 13.6, and 27.1.
3.5. Show that the fixed point of the map obtained from Newton's method for
computing ~ is asymptotically stable. Accomplish this by estimating an
interval in which ~ is assured to be.
3.6. Compute the fixed points of the map (3.3),
Determine the values of the parameter b for which a given fixed point is
unstable or asymptotically stable.
3.7. A unique asymptotically stable fixed point: Here is a useful setting for estab-
lishing the existence of such a fixed point. Let f : [a, b] -> [a, b] be a map
of an interval into itself. Show the following:
(a) If f is continuous, then it has at least one fixed point in the interval
[a, b].
(b) If, in addition, f is differentiable with If' (x) I < 1 for all x in [a, b],
then f has a unique fixed point in [a, b]. This fixed point is, of course,
asymptotically stable.
Hint: To show existence, apply the Intermediate Value Theorem to the
function f(x) - x. For uniqueness, suppose that there are two fixed points
and use the Mean Value Theorem to arrive at a contradiction.
3.8. Show that if f : lR -> lR is continuous and there is an Xo such that r(xo) ->
if as n -> +00, then if is a fixed point of f.
Xo = 0,
For a given value of >.., the sequence Xn converges to ,¢(>..) if>.. is sufficiently
small.
To define T(>", x), we use the Taylor expansion of F(>.., x):
The hypotheses of the theorem imply that b( >..) =f 0 for >.. sufficiently small.
Therefore, F(>", x) = 0 is equivalent to the equation x = T(>", x) with
where >.. is a scalar parameter. Then the function '¢(>..) of the Implicit
Function Theorem is ,¢(>..) = ->... Recover this function using the method of
successive approximations described above. For this purpose, compute that
T(>-', x) = ->"(1 + >..)-1 - (1 + >..)-lX 2 • Then, take>.. = 0.1, >.. = 0.3, >.. = 0.5,
etc., and iterate with initial value Xo = O. Do you observe any difference in
the rate of convergence for different values of >..?
Lemma 3.14. If 1 is a C 1 function with 1'(x) > 0 for all x in the domain
of definition of 1, then 1 is a monotone map, that is, the positive orbit
I'+(xo) of any initial condition Xo is a monotone sequence.
Proof. From the Mean Value Theorem, we have
Xn+1 - Xn = l(x n ) - l(xn-1) = 1'(xn)(xn - xn-d
for some x n . Therefore, Xn+1 - x n , for every positive integer n, has the
same sign as that of Xl - Xo· <t
For the purposes of dynamics, we will require 1 to be at least C 1 with
positive derivative, and refer to such an 1 simply as a monotone map.
If 1 is monotone, then 1-\ the inverse of 1, exists. We will use the
notation I- n to denote the n-fold composition of 1-1 with itself.
Definition 3.15. If 1 is monotone, then the negative orbit of Xo is the set
of points xo, 1- 1 (xo), 1- 2 (xo), ... , and is denoted by I'-(xo). The orbit I'
of Xo is defined to be I'(xo) == 1'+ (xo) U 1'- (xo).
The geometry of orbits of a monotone map is very similar to that of a
scalar differential equation: the fixed points act like equilibria, and we can
use arrows to indicate the direction of other orbits under forward iteration.
Consequently, to study bifurcations of fixed points of monotone maps we
need only to reinterpret the results in Section 2.3, as we shall do now.
For simplicity of notation, let us assume that the map 1 has a fixed
point at x = 0; if not, we can change coordinates to make it so. Fur-
thermore, suppose that 1'(0) > 0 so that 1 is monotone in a sufficiently
small neighborhood of the origin. Consider the perturbed map F(A, x)
depending on k parameters A == (AI, A2, ... , Ak):
F: lRk X lR ---.lR; (>.., x) f--t F(A, x) with F(O, x) = l(x).
If F(A, x) is a C 1 function, then it follows that F(A, x) is also monotone
in x for each small value of A. Now, for each fixed A, the key observation
is that the analysis of fixed points of F(A, x) is equivalent to the analysis
of the zeros of the function
F(A, x) - x.
In Section 2.3 we have analyzed the bifurcations of zeros of a function,
or, equivalently, the bifurcations of equilibria, under various types of hy-
potheses on the linear, quadratic, and cubic terms. We now translate those
results for bifurcations of fixed points of monotone maps.
Then, for 11>"11 sufficiently small, the perturbed map F has a unique fixed
point near zero whose stability type is the same as the stability of the fixed
point zero of the unperturbed map f.
For each value of the parameter >.., there is a unique hyperbolic fixed point
whose stability type is the same as the fixed point for>.. = O. See Figure 3.8
for the stair-step diagrams of the perturbed map F for several values of
the parameter >... <;
84 Chapter 3: Scalar Maps
Exercises - - - - - - - - - - - - - - - - - - - - . 0 . 0
3.12. A trans critical bifurcation: Show that the map F(>", x) = (1 + >..)x + x 2
undergoes a transcritical bifurcation at the parameter value>.. = o. Compare
this map with the differential equation in Example 2.3.
3.13. A saddle-node bifurcation: Show that the map F(>", x) = eX - >.. undergoes
a saddle-node bifurcation at the parameter value>.. = l.
3.14. Find a value of the parameter>" at which the map F(>", x) = >.._x2 undergoes
a local bifurcation. Identify the bifurcation and draw three representative
stair-step diagrams to illustrate your bifurcation.
3.4. Period-doubling Bifurcation 87
x*
f(x*)
[F2(>., x)]" = F"(>', F(>', x)) [F'(>., x)]2 + F'(>., F(>', x)) F"(>', x).
In particular, at the origin, we have
3·4· Period-doubling Bifurcation 89
It follows from these formulae that the Taylor expansion of p 2 (,X, x) about
the origin is given by
a(O) = 0,
1 a('x) b('x)
- [p 2(,X, x) - x] = 'x(2 +,X) + - x + _ x 2 + .... (3.14)
x 2 6
Since b(O) =I- 0, the analysis of the zeros of this function is identical to
the one we have already given in Case II of Section 2.3 for bifurcations
of nonhyperbolic equilibrium points with quadratic degeneracy. In fact,
if ,X [j2(0)] III > 0, then there is no zero of Eq. (3.14). If,X [j2(0)] III < 0,
then there are two zeros of Eq. (3.14) which correspond to a single period-2
orbit, {x~, P('x, x~) }.
In the case ,X [j2(0)] III < 0, to determine the stability type of the
period-2 orbit, we consider the cubic function p 2 (,X, x) - x and its three
simple zeros, in a neighborhood of the origin, given by 0, x~, and P('x, x~).
If the slope (1 + ,X)2 of p2(,X, x) at x = 0 is less than one (,X < 0 and 0 is
stable), then the slope of p2(,X, x) - x at x~ and P('x, x~) must be greater
than one and thus the period-2 orbit is unstable. Similarly, if (1 + ,X)2 > 1,
that is, ,X > 0 and 0 is unstable, then the period-2 orbit is stable. <>
Example 3.22. Continuation of Example 3.12: Let us now return to the
map in Example 3.12 and consider its one-parameter perturbation given
by
f(x) = -x - 3x 2, P('x, x) = -(1 + 'x)x - (3 + ,X)x2.
Since j2(x) = x - 18x3 - 27x4, we have [j2(0)]111 = -108 =I- O. It is
easy to verify that all of the remaining conditions of Theorem 3.21 are
satisfied. Thus, for each small positive value of ,X, there is a unique periodic
orbit of minimal period 2 which is asymptotically stablej see Figures 3.11
and 3.12. <>
We conclude this section with several remarks regarding Theorem 3.21.
1. For notational simplicity, we assumed x = OJ this can always be
achieved by translation.
2. The nonvanishing assumption on the third derivative cannot be re-
placed by a condition on the second derivative because [j2(0)]" = 0
is always satisfied.
90 Chapter 3: Scalar Maps
(a)
(b)
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L~
(c)
/ \
Figure 3.12. Graphs of the map F(>', x) = -(1 + >.)x - (3 + >.)x 2 near the
origin: (a) for>. < 0, asymptotically stable fixed point at the origin, (b) for
>. = 0, the origin is not hyperbolic but still attracting, (c) for>. > 0, there
is a unique asymptotically stable periodic orbit of period 2 (continued).
3.4. Period-doubling Bifurcation 91
(a)
(b)
(c)
This map is known as the logistic map and has been the subject of recent
intensive studies. Despite its innocuous looks, the logistic map exhibits
3.5. An Example: The Logistic Map 93
Figure 3.13. Asymptotically stable fixed point of the logistic map for the
parameter value>. = 1.8.
(a)
(b)
(c)
Figure 3.14. The logistic map near the first period-doubling bifurcation:
(a) A = 2.8, (b) A = 3, and (c) A = 3.2 (continued).
96 Chapter 3: Scalar Maps
(a)
(b)
(c)
Figure 3.14 Continued. The second iterates of the logistic map for the
parameter values as in the previous page.
3.5. An Example: The Logistic Map 97
Now, this map is precisely the one we already studied in Example 3.22.
=} 3.449 < A < 3.570 : The appearance of the period-two orbit as A is increased
through 3 is only the beginning of a fascinating sequence of bifurcations
that lead to very complicated dynamics. For example, at A = 3.449 (ap-
proximately) the period-two orbit loses its stability and gives rise to an
asymptotically stable period-four orbit, as seen in Figure 3.15. In fact,
there is an increasing sequence of parameter values Al < A2 < A3 < ... at
which the logistic map repeatedly undergoes a period-doubling bifurcation:
as A is increased through Ak the asymptotically stable periodic orbit of
period 2k becomes unstable and a stable periodic orbit of twice the period
bifurcates from the orbit of the lower period; see Figure 3.15. The first
several of the bifurcation values approximately are
Figure 3.16. Apparent chaos, or a periodic orbit with a very long period,
in the logistic map for>. = 3.891. Iterates between 1000-3000 are plotted.
3 !> 5 !> 7 !> ... !> 2 . 3 !> 2 . 5 !> 2 . 7 !> ... !> 22 . 3 !> 22 . 5 !> 22 . 7!> ...
To describe in words, write all the odd numbers except 1, then 2 times the
odd numbers, 22 times the odd numbers, 23 times, etc. Finally, write the
powers of 2 in decreasing order, with 1 at the end. Despite its strangeness,
this list includes all of the positive integers.
Theorem 3.24. (Sharkovskii) Let f : IR - t IR be a continuous map.
Suppose that f has a periodic point of minimal period m. If m !> n in
the Sharkovskii ordering, then f also has a periodic point of minimal pe-
riod n. <>
A noteworthy consequence of this theorem is that if f has a periodic
point of minimal period 3, then it has periodic points of every minimal
period. In particular, the logistic map at A = 3.839, in addition to the
period-3 orbit, has periodic orbits of all minimal periods. However, because
of their instability, they are not readily detectable on the computer.
=* A > 3.839 : As the parameter A is increased, the period-3 orbit undergoes
a period-doubling bifurcation and gives up its stability to an asymptoti-
cally stable period-6 orbit; see Figure 3.18. If A is increased further, there
100 Chapter 3: Scalar Maps
Exercises - - - - - - - - - - - - - - - - - - - .. \), 0
3.18. Another form for logistic: In the mathematical literature, the logistic map
Xn+1 = '>"Xn(1- Xn) is often transformed to the map Xn +l = J-t - x;'. Find
the transformation. Identify the values of the parameter J-t at which the
latter map undergoes a saddle-node or a period-doubling bifurcation.
3.5. An Example: The Logistic Map 101
is used in genetics involving one locus with two alleles. This map undergoes
a sequence of period-doubling bifurcations which accumulate at the value
of the parameter A = 3.5980... . Experiment on the machine to deter-
mine several successive values of the parameter at which the map undergoes
a period-doubling bifurcation and attempt to find a universal constant of
Feigenbaum type. Be warned that this could be tedious experiment because
of the geometric nature of the sequence of bifurcation values.
Help: This map is stored in the library ofPHASER under the name discubicj
just set the parameters. Also, see Rogers and Whitley [1983].
3.20. A map with a period 5 but no period 3 orbit: However unusual it may appear,
the ordering of periods in Sharkovskii's theorem is sharp. Consider, for
example, the map f : [1, 5] -> [1, 5] defined by
ax if X < 0.5
g(x} ={ a(1.0 - x} if x 2:: 0.5,
h(x} = ~ arcsiny'X,
7l"
verify that the tent map for the parameter value a = 2.0 is topologically
conjugate to the logistic map f(x} = AX(1.0-X} for the parameter value
3.5. An Example: The Logistic Map 103
>. = 4. Notice that the tent map is piecewise linear while the logistic
map is quadratic. It is usually the case that the mathematical analysis
of a piecewise linear map turns out to be somewhat easier than that
of a nonlinear map even if they are topologically conjugate. Indeed,
Ulam and von Neumann [1947] showed the existence of an "invariant
measure" for these maps, at the indicated parameter values, using the
topological conjugacy above. Even if you do not know what an invariant
measure is, you may still want to look up this famous paper.
(c) The tent map is stored in the library of PHASER under the name tent.
Compute some orbits numerically for the two maps at the indicated
parameter values where the two maps are topologically conjugate. Do
you notice a rather strange thing happening in the numerical compu-
tations with the tent map? Try the parameter value a = 1.9999; what
is going on this time? The answer may elude you, but ponder about it
anyway.
3.22. A one-hump map with two attractors: Consider the quartic map
where a is a scalar parameter. This map has a unique maximum on the unit
interval; visually it looks very much like the logistic map or the tent map,
the so-called "one-hump" maps. Initially, it was hoped that a one-hump
map depending on a parameter could be shown to have at most one stable
periodic orbit for a given value of the parameter. However, the Singer map
has both an asymptotically stable fixed point and an asymptotically stable
periodic orbit of minimal period 2 for the parameter value a = 1.0. Find
them on the computer by trying various initial conditions. Also, explore the
dynamics of this map by changing the parameter a.
Help: This map is stored in the library of PHASER under the name singer.
Also, see Singer [1978] where this map first appeared.
Bibliographical Notes _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ~
The dynamics community has been dealing with monotone maps since the
days of Poincare, as we shall see in the next chapter. The study of nonin-
vertible real scalar maps, however, became popular in the early seventies,
and eventually turned into a large industry. The review article by Whitley
[1983], and the books by Collet and Eckmann [1980] and Devaney [1986]
cover some of the basics and more.
It is interesting to note that one of the most remarkable theorems on
scalar maps was already proved in Sharkovskii [1964] before the subject
became vogue; see also Stefan [1977]. A special case was rediscovered in
an article with a provocative title by Li and Yorke [1975].
The surprising geometric nature of the successive period-doubling bi-
furcations in logistic-like maps was observed numerically by Feigenbaum
[1978 and 1980]. This "universal" property was later proved by Lanford
104 Chapter 3: Scalar Maps
[1982 and 1987], using the machine in a novel way. Universality can be
used to make quantitative predictions about bifurcations in a physical sys-
tem where an explicit formula for a map governing the system may not
be available. Such a prediction has been demonstrated by Libchaber and
Mauer [1982] in experiments on Rayleigh-Benard convection.
An elementary exposition of the algorithms of Euler and Newton are
contained in, for example, Conte and deBoor [1972]. Recently, ideas from
dynamical systems have begun to open new avenues in numerical analysis,
as exemplified in Smale [1981 and 1987]; see also Saari and Urenko [1984]
and Ushiki [1986].
Iterating a number on a computer 64 times, let alone 1,000,000 times,
can lead to spurious results because of finite floating point arithmetic. It
has long been known that certain well-behaved maps exhibit shadowing
property: near a numerically computed orbit, there exists a true orbit
which, however, may be the orbit of a different initial value than the one
intended; see Anosov [1967] and Bowen [1975]. This important, and com-
forting, property has been established at some parameter values of the
logistic map in Hammel et al. [1987].
~D
4 -_ _ __
Scalar
Nonautonomous
Equations
Ii 1001
In this chapter, we begin our study of nonautonomous
"al", diff",ential equation, and develop a geometric the-
ory analogous to the one given in Chapter 1 for auton-
omous equations. After a brief general introduction, we
focus our attention on equations with coefficients that are
periodic in time. For such equations, we show that Ct- and
w-limit sets, if they exist, are periodic solutions. With the help of this fun-
damental result, we then illustrate in several specific examples how to estab-
lish the existence of periodic solutions. Finally, we investigate the stability
of periodic solutions using the theory of scalar maps developed in Chapter 3.
108 Chapter 4: Scalar Nonautonomous Equations
x=f(t,x) (4.1)
will be the subject of this and the following chapter. Such an equation
is called nonautonomous because the function f depends explicitly on t as
well as x. In this section, we point out some of the basic similarities and the
important differences between solutions of Eq. (4.1) and the autonomous
case discussed in Sections 1.1, and 1.2.
The notion of a solution of the nonautonomous differential equation
(4.1) and the initial-value problem for Eq. (4.1) are defined in the same way
as in the autonomous case. However, for given initial data (to, xo) E JRxJR,
it will be necessary to denote a solution of Eq. (4.1) through Xo at to by
cp(t, to, xo) with cp(to, to, xo) = Xo. The necessary generalization of the
existence and uniqueness theorem for an initial-value problem holds, as
stated in the Appendix, that is, cp(t, to, xo) is uniquely defined and is
continuous together with its first derivatives with respect to t, to, and Xo
in all variables (t, to, xo).
The direction field of Eq. (4.1) is defined as before, and the trajectory
through (to, xo) is defined to be the set
where Ito,xo is the maximal interval of definition of the solution cp(t, to, x).
It is, of course, also possible to define the orbit of Eq. (4.1) through
Xo at to as the set {(cp(t, to, xo)) : t E Ito,xo} c JR. However, orbits
of nonautonomous equations do not enjoy the same properties as those of
autonomous ones. More precisely, in the autonomous case, there is a unique
orbit through Xo due to the fact that the solution cp(t, to, xo) through Xo
at to satisfies cp(t, to, xo) = cp(t - to, 0, xo). In the nonautonomous case,
as illustrated by Example 4.2 below, orbits are not uniquely defined by Xo.
The main implication of the discussion above is that, in the qualita-
tive study of nonautonomous differential equations, we cannot effectively
use the orbits in JR, but must consider the trajectories in JR x JR. The
trajectories are, of course, uniquely defined by (to, xo). Although, they are
not monotone in t, solutions of nonautonomous differential equations still
have the useful property of monotonicity with respect to the initial data Xo.
This fact is the content of the lemma below which is a direct consequence
of the uniqueness of solutions (compare with Lemma 1. 7).
4.1. Geneml Properties of Solutions 109
Lemma 4.1. Let cp(t, to, xo) and cp(t, to, Yo) be two solutions ofEq. (4.1)
through Xo at to and Yo at to, respectively. Then
cp(t, to, xo) < cp(t, to, Yo) for t ~ to if Xo < Yo· 0
Let us now give several examples of nonautonomous differential equa-
tions to illustrate some of their differences from autonomous ones.
Example 4.2. Orbits do not suffice: Consider the simple differential equa-
tion
x
= sint.
Two trajectories of this equation are shown in Figure 4.1 to illustrate the
monotonicity with respect to the initial data Xo. Notice that the orbit of
the solution cp(t, 0, xo) = Xo + 1 - cos t is the interval [xo, Xo + 2] for any
Xo. Thus, if Xo is near Yo, then the orbits of the solutions cp(t, 0, xo) and
cp(t, 0, Yo) overlap but are not equal. Also, observe that every solution is
periodic with period 27r, the same period as the vector field. 0
Example 4.3. Finite time: Consider the differential equation
x= (cost)x 2 .
If Isin to + l/xol > 1, then the solution is again defined for all t. Otherwise,
the solution is defined only on a finite interval; see Figure 4.2. For to = 0,
notice that every solution with Ixol < 1 is periodic with a period of 27r. 0
110 Chapter 4: Scalar Nonautonomous Equations
x = -x + cost.
The flow of this equation is given by
where a(t) and b(t) are scalar continuous functions. The solution <p(t, to,xo)
of Eq. (4.2) is given by
If x(t o) = xo, then y(to) = Yo and the differential equation (4.2) in the new
variable becomes
y=e -1''0 a(u) du bet).
To obtain the solution of this differential equation, we simply integrate
with respect to t :
yet) = Yo + i to
t
e
- 1'0 a(u) du b(s) ds.
8
Now, returning back to the variable x, we recover the solution (4.3). If you
enjoy performing integrations, you may wish to use this formula to obtain
some of the explicit solutions we have used in the examples above. <>
4.3. Formula for linear equations: Show that the solution of ± = a(t)x + f(x)
with x(O) = Xa satisfies
x(t) = e1'0,
a(u) du
Xa + it I'
a e s
a u du
() f(x(s)) ds.
: to + 1
I
I
If, moreover, r.p(t + T, to, xo) =I- r.p(t, to, xo) for any 0 < T < T, then T is
called the minimal period.
In our subsequent discussion of I-periodic differential equations, we
will be interested chiefly in I-periodic solutions. Let us proceed with a
simple lemma.
E and not on to) such that, for every Yo for which Iyo - xol < h, the
solution <p(t, to, Yo) of Eq. (4.4) through Yo at to satisfies the inequal-
ity l<p(t, to, Yo) - <p(t, to, xo)1 < E for all t ~ to. The periodic solution
<p(t, to, xo) is said to be unstable if it is not stable.
Definition 4.10. A periodic solution <p(t, to, xo) of Eq. (4.4) is said to
be asymptotically stable if it is stable and, in addition, there is an T > 0,
independent oEto, such that l<p(t, to, Yo) - <p(t, to, xo)1 -+ 0 as t -+ +00 for
all Yo satisfying Iyo - Xo I < T.
Of course, the point IIk(xo) is the same as the point ¢k(O); hence, the
Poincare map is monotone. Also, it is important to notice that, from
differentiable dependence on initial values as stated in the Appendix, the
Poincare map is differentiable with nonnegative derivative.
The chief importance of the Poincare map stems from the following
restatement of Lemma 4.8: a point Xo is the initial value of a l-periodic so-
lution of Eq. (4.4) if and only if Xo is a fixed point of the Poincare map, that
is, II(xo) = Xo. Naturally, the stability properties of a l-periodic solution
of Eq. (4.4) are the same as the stability properties of the corresponding
fixed point of the Poincare map. Thus, we can apply the previous results
on monotone maps (Section 3.3) to the Poincare map of Eq. (4.4) in our
investigation of periodic orbits, as we shall do subsequently.
4.2. Geometry of Periodic Equations 117
Exercises - - - - - - - - - - - - - - - -_ _ _ eft\;). 0
4.8. Scaling time: Rescale the time variable so that the 27r-periodic differential
equation x = -x + cos t becomes 1 periodic.
4.9. Suppose that b(t) is a I-periodic continuous function and let bo = fol b(s) ds.
Show that all solutions of x = b(t) are 1 periodic if bo = OJ otherwise, they
are all unbounded.
Hint: The Poincare map is II(xo) = xo + boo
4.10. Consider a I-periodic scalar differential equation x = f(t, x) satisfying
f(t, 0) = O. If there is an r > 0 such that for Ixol < r the solutions
r.p(t, to, xo) -+ 0 as t -+ +00, then show that the zero solution x = 0 is
stable.
4.11. Fredholm's Alternative: Suppose that a(t) and b(t) are I-periodic continu-
ous functions and let ao = fol a(s) ds. Show the following properties of the
differential equation x = a(t)x + b(t):
1. If ao i- 0, then there is a unique I-periodic orbit which is asymptotically
stable when ao < 0 and unstable when ao > O.
2. Let Co = fol exp{ - f a( u) du }b( s) ds. If ao = 0, then every solution is
I-periodic if and only if Co = O.
3. If ao = 0, then every solution is unbounded if Co i- O.
4. Why do you think this problem is called the Fredholm's Alternative?
You may want to look it up in a mathematical encyclopedia or in a
good book on linear algebra.
Hint: From the variations of the constants formula, the Poincare map is
given by
4.13. Can you relate the Fredholm Alternative for I-periodic solutions to the
results on the boundary-value problem in the exercises at the end of Sec-
tion 4.I?
Hint: Consider a(t) and b(t) for the boundary-value problem as defined only
on [0, 1], and extend them as I-periodic functions to JR. Even though the
new coefficients are discontinuous in t, the theory remains valid if we allow
discontinuities in the derivatives at the integers.
0=1 (4.8)
x= f((}, x).
It is clear from the form of the first equation that the orbits of Eq. (4.8)
x
on the (x, (})-plane correspond to the trajectories of = f(t, x). We will
undertake a detailed study of a pair autonomous differential equations on
the plane in Part III of our book. However, Eq. (4.8) is rather special, so
let us examine it more closely.
Observe that the first equation is periodic with any period. Let us take
this period to be 1. Thus, from the considerations in Section 1.4, it can
be viewed as a differential equation on the circle 8 1 . Since f is I-periodic,
f((} + 1, x) = f((}, x), if we identify the points () with points () + k, for any
integer k, the second equation remains unchanged. Therefore, the orbits of
Eq. (4.8), hence the trajectories of Eq. (4.4), can be conveniently viewed
on 8 1 x JR, a cylinder.
Since (}(t) = to + t is an increasing function of t, an orbit of Eq. (4.8)
continues to wind around the cylinder as time goes on. The function <Pk(t),
o s:; t s:; 1, defined in Eq. (4.7), represents one revolution around the
cylinder. Because <Pk(l) = <Pk+l (0), the sequence of functions {<pd fit
together to make a "spiral" on the cylinder. Iterations of an initial point
under the Poincare map is the successive intersection of this spiral with a
vertical line on the cylinder.
Also, Theorem 4.11 can be reinterpreted as follows: if a positive orbit
(t 2:: 0) of Eq. (4.8) is bounded, then it must approach a periodic orbit as
t ---+ +00. Of course, a periodic orbit wraps around the cylinder only once.
4.3. Periodic Equations on a Cylinder 119
t = 0
Figure 4.6. Poincare map and two solutions of the I-periodic differential
equation :i; = sin 27rt on the cylinder.
0=1
:i; = sin 27r0.
We have drawn in Figure 4.6 the flow of this system on the cylinder. Notice
that all orbits are periodic with period 1 which wrap around the cylinder.
Consequently, the Poincare map is the identity map, II(xo) = xo, and thus
all points are fixed points. <:;
Example 4.14. Consider the I-periodic differential equation
Exercises - - - - - - - - - - - - - - - - - - - , . \ : ? 0
4.14. To obtain numerical solutions of a I-periodic scalar differential equation,
especially on PHASER, it is often necessary to convert such an equation
to a pair of autonomous equations using Eq. (4.8). Convert the I-periodic
equation in Example 4.3 and reproduce Figure 4.2 using PHASER. To re-
cover the entire figure, you will have to run orbits of the planar differential
equations backward in time.
4.15. Experiment numerically, on PHASER, of course, with the following periodic
differential equations and try to locate as many periodic solutions as you
can; also to sharpen your imagination, when you get an exciting picture,
visualize it as a flow on the cylinder:
(i) :i; = x 2 + a + bsin t; (ii) :i; = a sin t + (b + eeos t)x - x 3 ;
(iii) :i; = asint - sin x; (iv) :i; = asint _x 5 ;
(v) :i; = asint + bsin(v'2t) - sinx.
Suggestions: You should first convert these equations to pairs of autono-
mous differential equations. When entering your equations into PHASER,
do so with parameters. Then systematically fix the values of the parame-
ters and experiment with different initial values. You will easily locate the
asymptotically stable periodic solutions with forward integration. To locate
the unstable periodic solutions, however, integrate backwards in time.
4.4. Examples of Periodic Equations 121
Xo--r----------------------------------,
n(xol
n2(xol
n'lxol
(b)
(c)
~_ _--1--periodic
fixed-
point orbit
(d)
Figure 4.7. Of the 1-periodic equation:i; = -x + cos 271"t - 271" sin 271"t
(a) several solutions, (b) one solution and its translations to the left by in-
teger amounts, (c) one solution on the cylinder, and (d) the graph of the
Poincare map.
122 Chapter 4: Scalar Nonautonomous Equations
+00
L enei211"nt,
n=-CX)
then
+00
<I>(t) = L en
1 + i21l"n e
i211"nt
n=-oo
is a I-periodic solution (of course, one can obtain a real solution from the
complex one above, if so desired). Since the solution of Eq. (4.9) through
Xo at t = 0 is
cp(t, 0, xo) = e- t [xo - <1>(0)] + <I>(t),
it follows that <I>(t) is the unique I-periodic solution. In addition, this
formula for the solution through Xo shows that the Poincare map is given
by
II(xo) = e- 1 [xo - <1>(0)] - <1>(0),
and its graph is similar to the one given in Figure 4.7d. Thus, the I-periodic
solution <I>(t) is asymptotically stable.
124 Chapter 4: Scalar Nonautonomous Equations
(4.11)
et(cp + cp + M) ~ O.
Finally, multiply through with e- t and rearrange the terms to arrive at the
inequality (4.11).
The uniqueness of <P(t), and thus its asymptotic stability, is easy to
establish. If x(t) and y(t) are two solutions, then z(t) = x(t) - y(t) satisfies
the linear autonomous differential equation z = -z, whose flow is given by
z(t) = e-tz(O). If x(t) and y(t) are I-periodic, then so is z(t). Consequently,
z(O) = 0 and thus z(t) = 0 for all t. This establishes the uniqueness of <P(t).
Its asymptotic stability also follows from the fonn of the flow of z = -z. 0
Since Eq. (4.9) is linear, it may appear that the latter argument above
is unnecessarily convoluted. After all, one could use the explicit formula
given in Example 4.6. Although no such formula exists for nonlinear equa-
tions, qualitative reasoning similar to the one above, however, can be used
successfully to exhibit I-periodic solutions of certain nonlinear I-periodic
differential equations. We now give an instance of such a situation.
Example 4.18. Qualitative analysis of a nonlinear equation: Consider the
nonlinear differential equation
One can regard this differential equation as a model for the growth of a
population where the intrinsic growth rate r(t) and the carrying capacity
k(t) exhibit periodic (for example, seasonal) fluctuations.
Let us first show that this equation has at least two I-periodic solu-
tions. The trivial solution x(t) == 0 is obviously a I-periodic solution. To
look for a second solution, observe that if x(t) is a solution with x(O) < 0,
then x(t) < 0 for all t ~ 0 and thus there is no I-periodic solution with
negative initial data. Consequently, we must show that there is a I-periodic
solution with positive initial data. To this end, let Km S k(t) S KM for
some constants K m , KM. Then for any solution x(t) satisfying x(t) > K M ,
we have x(t) < O. Also, if x(t) is a solution satisfying 0 < x(t) < K m , then
x(t) > O. Therefore, any solution x(t) with initial data 0 < x(O) < Km
must be bounded for all t E JR, approach zero as t ---+ -00, and approach a
I-periodic solution <I>(t) as t ---+ +00. Furthermore, this I-periodic solution
has the property Km S <I>(t) S K M .
It is possible to show that <I>(t) is the only I-periodic solution with
positive initial data; hence, it is asymptotically stable. It is not entirely
trivial to prove this fact; see the exercises. <>
We end this section with some remarks and problems related to possi-
ble generalizations of the examples above. For instance, the Riccati equa-
tion with periodic coefficients,
dx
dr = b(r) + a(r)x - c(r)x2,
A
where a, b, and c are continuous, I-periodic functions with c(r) > 0 for all
r, generalizes Example 4.19. The form of this equation can be simplified
somewhat if we change the independent variable r to t through the formula
r = J~ c- 1 (s)ds. Then the Riccati equation becomes
where a = ale and b = hie. If the initial data for Eq. (4.13) is very large
in absolute value, then the solution is decreasing. In the case where the 1-
periodic solutions are hyperbolic [that is, II'(xo) -j. 1], there must be a finite
number of them because they are isolated; see the exercises. Furthermore,
the number is even. It is possible to prove that there are no more than two
such solutions; see the exercises.
Consider the following generalization of Example 4.18:
where c(t) and d(t) are arbitrary continuous functions of period 1. Using
arguments similar to the ones given in Example 4.17, one can show that
every solution of Eq. (4.14) is bounded for t :::::: O. However, the argument
for uniqueness is no longer valid for this generalization. In fact, we saw in
Section 2.1 that for c and d constants the uniqueness did not hold and there
were sometimes three solutions. If the 1-periodic solutions of Eq. (4.14)
are hyperbolic, then there must be an odd number of solutions, but are
there no more than three 1-periodic solutions? The answer is yes, but a
complete discussion of Eq. (4.14) is difficult; special cases are contained in
the exercises.
Exercises _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ .. \/.0
4.16. Show that the equation ± = _x 5 +c(t), where c(t) is a continuous I-periodic
function, has a unique I-periodic solution and is asymptotically stable.
4.17. If c(t), d(t), and e(t) are continuous I-periodic functions, show that the
equation
± = _x 3 + c(t)x 2 + d(t)x + e(t)
has at least one I-periodic solution. Also, if this I-periodic solution is un-
stable, show that there must be another I-periodic solution.
Hint: Show that ± < 0 if x is large enough, and ± > 0 if -x is large enough.
4.18. Suppose that f is a C 1 function with !,(x) > 0 for all x and satisfying
f(x) --+ +00 as x --+ +00, and f(x) --+ -00 as x --+ -00. Show that the
differential equation ± = - f(x) + c(t), where c(t) is a continuous I-periodic
function, has a unique I-periodic solution which is asymptotically stable.
Hint: Use the Mean Value Theorem.
4.19. Hyperbolic is isolated: A I-periodic solution <p(t, 0, xo) is called hyperbolic
if II'(xo) oil. Show that hyperbolic I-periodic solutions are isolated from
other I-periodic solutions, that is, the corresponding hyperbolic fixed points
of the Poincare map are isolated.
4.20. Riccati Equation: If a(t) and b(t) are I-periodic continuous functions, prove
that the Riccati equation
± = b(t) + a(t)x - x2
128 Chapter 4: Scalar Nonautonomous Equations
{1
and
1
v(l) < v(O)exp r(s)(l- x(s)/k(s)) dS}.
where the subscripts m and M denote the minima and maxima of the ap-
propriate functions, respectively.
Hint: Note that the equation can be written as
that is,
a
a: (t, 0, xo) = exp[}o
t of ]
ax (s, cp(s, 0, xo)) ds . (4.16)
ocp
-(t, 0, xo) = 1 +
lot -a
of ocp
(s, cp(s, 0, xo)) -a (s, 0, xo) ds.
oXo 0 x Xo
130 Chapter 4: Scalar Nonautonomous Equations
Now, if we let z(t) == 8cp(t, 0, xo)/8xo, then the equation above becomes
t8f
z(t) = 1 + io 8x (s, cp(s, 0, xo)) z(s) ds.
Retaining only the first term of the Taylor expansion of this vector field
yields the differential equation (4.15).
Lemma 4.21. Let cp(t, 0, xo) be the solution of a i-periodic equation
± = f(t, x) with cp(O, 0, xo) = Xo and II be the Poincare map. Then the
derivative of the Poincare map is given by
r
18f
II'(xo) =exp[io 8x(t, cp(t, 0, xo))dt]. (4.18)
Proof. From the definition of the Poincare map, II(xo) cp(l, 0, xo).
Differentiating both sides with respect to Xo yields
, _ dII 8cp
II (xo) = -d (xo) = -8 (1, 0, xo).
Xo Xo
Now, the conclusion of the lemma follows from Eq. (4.16). 0
Theorem 4.22. Let cp(t, 0, xo) be a i-periodic solution of the i-periodic
equation (4.4) and define
r18 f
ao == io 8x (t, cp(t, 0, xo)) dt.
Then,
(i) cp(t, 0, xo) is asymptotically stable if ao < 0, and
(ii) cp(t, 0, xo) is unstable if ao > 0.
Proof. Since cp(t, 0, xo) is a I-periodic solution, the point Xo is a fixed
point of the Poincare map II and, from Lemma 4.21, II' (xo) = eao . Thus,
the conclusion follows from Theorem 3.6 because eao < 1 if ao < 0, and
eao > 1 if ao > 0. 0
Analogous to Definition 3.10 for the case of autonomous equations, we
introduce the following terminology for I-periodic solutions satisfying the
hypotheses of the theorem above.
4.5. Stability of Periodic Solutions 131
and its 1-periodic solution cp(t, 0, 0) = sin 27ft. Notice that this equation
is a particular instance of Example 4.13. A simple computation yields
ao = r -3sin2
10
1
27ftdt = -~ < 0.
2
and its 1-periodic solution cp(t, 0, 1) = cos 27ft. This periodic solution
corresponds to the fixed point of the Poincare map at Xo = 1. A simple
computation shows that
Therefore, the periodic solution is not hyperbolic and thus its stability type
cannot be determined from the linearization of the Poincare map. We will
return to this example in the next chapter. (;
x. = x 2 - -1
2 . 47f t
cos42 7f t - sm
47r
has the I-periodic solution cp(t, 0, I/(27r)) = [I/(27r)] cos 2 27rt. Show that
this solution is hyperbolic and II'(I/(27r)) = exp(I/(27r)). There is another
I-periodic solution which is asymptotically stable. Can you find it explicitly?
Are there any more I-periodic solutions?
Hint: See the hint for the Riccati equation.
132 Chapter 4: Scalar Nonautonomous Equations
has the 1-periodic solution cp(t, 0, 0) = sin 27rt. Show that this solution is
unstable with II'(O) = exp(1/2). How many more 1-periodic solutions can
you guarantee?
°
4.26. Suppose that a(t) is 1-periodic with < a(t) < 1. Verify that the equation
x = x(1 - x)[x - a(t)] has at least three 1-periodic solutions.
Hint: Show that x(t) = °
and x(t) = 1 are asymptotically stable and
therefore there is a 1-periodic solution in the interval (0, 1).
Bibliographical Notes - - - - - - - - - - - - - - - - - - I f ®
As we have seen in this chapter, the qualitative behavior of the solutions of
scalar I-periodic differential equations is nearly as simple as the behavior
of scalar autonomous equations once the dynamics is reduced to that of
the Poincare map. If equilibria are replaced by the fixed points of the
Poincare map, the orbit structure is easy to classify because the Poincare
map is monotone. On the other hand, determining the precise number
of I-periodic solutions for a given I-periodic equation is not an easy task
because the differential equation must be integrated to obtain the Poincare
map. For a given polynomial vector field with I-periodic coefficients, the
maximal number of its I-periodic solutions is not known. Pliss [1966] gives
an example of a fourth degree polynomial with at least five I-periodic
solutions. See also Lloyd [1972] for an extensive discussion of these matters.
An important extension of the theory of I-periodic scalar equations
is to take f(t, x) to be quasiperiodic in t; for example, f(t, x) = sint +
sin v'2t. Although there is an extensive theory for such equations (see, for
example, Fink [1974], Sell [1971], and Yoshizawa [1974]), very little is known
about the number of quasiperiodic solutions that can exist. If f(t, x) =
g(x) + c(t) and g'(x) -10 for all x, there is a unique quasiperiodic solution,
nicely generalizing the result in Section 4.3. On the Riccati equation with
quasiperiodic coefficients, see Johnson and Sell [1981].
5 - -_ __
Bifurcations of
Periodic
Equations
where
F: lRk x lR x lR ---+ lR; (>', t, x) f-t F(>', t, x)
satisfying
Exercises _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ "'c:;;. 0
5.1. Consider the autonomous differential equation ± = f(x) with an equilibrium
point x, that is, f(x) = O. When such an autonomous differential equation
is viewed, trivially, as a 1-periodic differential equation in t, the equilibrium
point of the autonomous equation becomes a 1-periodic solution of the 1-
periodic equation. Show that, as such, the Poincare map II(xo) = <p(1, 0, xo)
of the 1-periodic differential equation satisfies II(x) = x and II'(x) = ef'(x).
5.2. Show that the differential equation ± = x 2 - 1 + Acos 21l"t has at least
two 1-periodic solutions for each Awhen IAI is sufficiently small. Using the
linear variational equation, discuss the stability properties of these 1-periodic
solutions. Finally, show that there are exactly two 1-periodic solutions for
each Awhen IAI is sufficiently small.
5.3. Consider the differential equation ± = ao + cox 2 + G(A, t, x), where A
is a scalar parameter, ao and Co are constants satisfying aoCo < 0, and
5.2. Stability of Nonhyperbolic Periodic Solutions 135
G is a continuous 1-periodic function in t with IG(A, t, x)1 ::; IAI for all
(t, x) E lR x lR. Show that there is a AO > 0 such that, for IAI < AO, there
are exactly two 1-periodic solutions '1/JI(A, t) and W2(A, x) which have the
properties WI(O, t) = J-ao/co and W2(0, t) = -J-ao/co. Show also that
these 1-periodic solutions are hyperbolic and that WI is unstable and W2 is
asymptotically stable if Co > O.
Hint: Use the first exercise above and the properties of the flow when A = O.
5.4. Consider the differential equation
Then the first several terms of the variational differential equation (4.17)
for i will have the form
(5.3)
where the coefficients are I-periodic functions, and the notation O(z4) de-
notes the terms of order z4 and higher; see the Appendix. Notice that
the study of the neighborhood of the zero solution of Eq. (5.3) is equiva-
lent to the study of the neighborhood of the I-periodic solution rp. Now,
let us assume that rp is nonhyperbolic, that is, ll'(O) = 1, or equiva-
lently f; b(s) ds = O. Then the function y(t) == f;
b(s) ds is I-periodic
in t. To wit, consider y(t + 1) - y(t) = ftt+! b(s) ds. Since the inte-
gral over a period of a periodic function is invariant under translation,
Itt+! b(s) ds = f; b(s) ds = O. Thus, y(t) is I-periodic.
In order to eliminate the linear term in the variational equation (5.3),
introduce the I-periodic change of variable u(t) defined by
(5.5)
where c(t) and d(t) are I-periodic functions given by
c(t) = efot b(s)ds c(t) and d(t) = e 2 fat b(s)ds d(t). (5.6)
where d(t) is as in Eq. (5.6). Thus, the fixed point at xo, hence, the
corresponding periodic solution rp(t, 0, xo) of Eq. (4.4), is unstable if
do > 0 and asymptotically stable if do < O.
5.2. Stability of Nonhyperbolic Periodic Solutions 131
Thus, it follows from the theorem above that the nonhyperbolic I-periodic
solution cp(t, 0,1) = cos27rt of Example 4.25 is unstable; see Figure 5.1. <>
Proof of Theorem 5.1. The key idea of the proof is to reduce the variational
equation (5.3) to a differential equation with constant coefficients by using
successive transformation of variables. In the choice of the transformations,
138 Chapter 5: Bifurcations of Periodic Equations
where wet) is the new variable and f3(t) and ,et) are I-periodic functions to
be determined. For simplicity of notation, in the formulae below, we will
omit the variable t if there is no danger of confusion. A few calculations
show that the differential equation for w is given by
Using the series expansion (1 - X)-1 = 1 + x + x 2 + ... for Ixl < 1, the
differential equation above for tV can be simplified to the following form:
Let us now choose f3 and I in a such a way that the differential equa-
tion (5.8) becomes as simple as possible. It is tempting to put /3 = c
so that the coefficient of the w 2 term will vanish. Unfortunately, this
choice makes f3(t) a I-periodic function if and only if Co = J01 c(s)ds = 0.
In fact, the solutions of /3 = c are given by f3(t) = f30 + J~ c(s) ds and
°
f3(t + 1) = f3(t) if and only if = ftH1 c(s) ds = J~ c(s) ds. However, if
we chose f3(t) = (constant) + J~ [c(s) - col ds, that is, f3(t) satisfies the
differential equation
/3 = c(t) - co, (5.9)
then f3(t) is I-periodic. So, we choose f3 satisfying the differential equa-
tion (5.9). We may also require that J01 f3(s) ds = 0.
Suppose now that f3 has been chosen as above. To simplify the co-
efficient of the w 3 term in Eq. (5.8), we can determine the function ,et),
satisfying J~ ,( s) ds = 0, from the differential equation
where
m = d+ 2{3/3, f..L = 10 1m(s) ds.
We note that
f..L = do == 10 1 d(s) ds (5.11)
when applied to Eq. (5.5) yields the following differential equation we have
been seeking:
(5.13)
For initial data Wo small, the Poincare map of Eq. (5.13) can be ap-
proximated by computing the Taylor expansion of w(t) in Wo near Wo = O.
We now indicate how to do this for the first two terms. The solution of
Eq. (5.13) satisfying w(O) = Wo is given by
Thus, using the chain rule and changing the order of integration and dif-
ferentiation we have
aw(t) = 1 +
awo
t
[2c W(s) aw(s)
10 o awo
+ O(w2)] ds. (5.14)
aw(t) = l. (5.15)
awo
Differentiating Eq. (5.14) once more gives the second derivative
a2w~t)
awo
= t
10
[2Co(aw(S))2
awo
+ 2COW(S/:w~s) + O(w 2)] ds.
uWo
Evaluating this at Wo = 0 and using the fact (5.15) results in
140 Chapter 5: Bifurcations of Periodic Equations
8 3 w(t) _ 6 t
J:l 3 - jL.
uWo
Uo = Wo + ,8(0)w6 + ,(O)w~.
If we let n be the Poincare map of Eq. (5.5), then the transformation
(5.12) and the fact that ,8 and, are I-periodic imply that, for initial data
Uo small,
n(uo) = II(wo) + ,8(0) [II(wo)]2 + ,(0) [II(wo)]3
(5.17)
= Uo + cou6 + jLU~ + O(u~).
Now, the conclusions of the theorem follow immediately if we differentiate
Eq. (5.17) and use the fact in Eq. (5.11).0
Exercises - - - - - - - - - - - - - - - - - - - - .. Q, 0
5.6. Consider the I-periodic equation
Verify that 'P(t, 0, 0) = sin 27rt is a I-periodic solution. Show that II' (0) =
1, III/(O) = 0, and IIII/(O) < o. Establish that this I-periodic solution is
asymptotically stable. You may wish to experiment numerically to see if
there are any additional I-periodic solutions.
5.7. Show that the zero solution of:i; = [(cos 2 t)(sin 2 t)] x 3 is asymptotically
stable.
5.8. Show that the zero solution of:i; = [sint] x 2 is unstable. Notice that Theo-
rem 5.1 is not applicable.
5.3. Perturbations of Vector Fields 141
Theorem 5.3. Suppose that Aoco ~ 0. Then, for A near zero, the differ-
ential equation (5.18) has
(i) no I-periodic solution if AAoco > 0;
(ii) one I-periodic solution if AAoco = 0;
(iii) two I-periodic solutions if AAoco < 0. <)
We now turn to the somewhat difficult task of computing the partial
derivatives of Eq. (5.21). In order to give the idea of the technique for
calculating such things let us start with a general lemma.
Lemma 5.4. If cp(A, t, 0, xo) is the solution of Eq. (5.18) with initial data
cp(A, 0, 0, xo) = xo, then 8cp(A, t, 0, XO)/8A is the solution of the following
initial-value problem for a linear differential equation:
8F 8F
i = 8x (A, t, cp(A, t, 0, xo))z + 8A (A, t, cp(A, t, 0, xo)),
(5.23)
z(o) = 0.
Proof. The proof is similar to that of Lemma 4.20. Observe that
z(o) = 0.
°
This is a nonhomogeneous linear equation and thus can be solved ex-
actly; see Example 4.6. Since cp(O, t, 0, 0) = and B(O, t) = 0, we obtain
from the solution of Eq. (5.24) that
8cp
8A (0, t, 0, 0) = io
t 8A
8A (0, s)ds. (5.25)
Therefore,
8Il (18A
8A (0,0) = io 8A (0, s)ds == Ao· (5.26)
Eq. (5.19). Differentiating Eq. (5.24) with respect to Wo and knowing that
z = a'P (>.. , t, 0, wo)/a>.., we see that a2'P(>", t, 0, wo)/a>..awo is the solu-
tion of the initial-value problem (for simplicity of notation, variables are
omitted)
u
• a'P -a'P + -a B -a'P ,
= [ B + 2C'P + O( 'P 2 )] u + 2C-
awo a>.. a>.. awo (5.27)
u(o) = 0.
v = [B + 2C'P + O('P2)]V,
(5.28)
v(O) = 1,
and B(O, t) = 'P(O, t, 0, 0) = 0, we have
a'P
-a
Wo
(0, t, 0, 0) = l. (5.29)
~'P r {8M M
a>..awo (0, t, 0, 0) = io [2co io a>.. (0, r)dr + a>.. (0, s)]ds. (5.30)
Consequently,
a2rr (l aA r aB
a>..awo (0,0) = io [2co io a>.. (0, r)dr + a>.. (0, s)]ds == Bo· (5.31 )
Since J;
2(cos27l"s)ds (sin 27l"t)/7l", we introduce the new variable u(t)
defined by
z(t) = e(sin 2?rt)/7r u(t) (5.33)
To put this equation into the form of Eq. (5.19), that is, to make the
coefficient of the u 2 term constant, we introduce the transformation
Imitating the calculations in the proof of Theorem 5.1, we obtain the fol-
lowing differential equation for w:
and, from Example 5.2, Co > O. Thus, it follows from Theorem 5.3 above
that near the periodic solution cos 27l"t the perturbed equation (5.32) under-
goes a saddle-node bifurcation: it has two I-periodic solutions if>. < 0, and
no I-periodic solution if>. > OJ see Figure 5.2. You may wish to compute Bo
and determine the Poincare map (5.22) of the perturbed equation (5.35). <>
We end this section with a remark on the general theory. In Theorem
5.3 we assumed that both AD and eo were not zero. If AD = 0 but eo =1= 0,
for example, then we need further computations to determine the nature of
the bifurcation point>. = O. In particular, it will be necessary to compute
the partial derivative 8 2 ll(0, 0)/8>.2 in order to determine the quadratic
terms in >. of the function ll(>', 0). This is a relatively routine but tedious
task, thus we refrain from giving the details.
where Co =I 0, and a(t) and b(t) are I-periodic continuous functions satisfying
Jo1 a(s) ds = Jo1 b(s) ds = 1. Show that there is a cusp in the (A1, A2)-plane
near (A1, A2) = (0, 0) for which there are three I-periodic solutions inside
the cusp and one I-periodic solution outside the cusp. Find approximate
formulas for the cusp and for the solutions.
Suggestions: Integrate the differential equation formally to obtain the fol-
lowing expression for the Poincare map:
146 Chapter 5: Bifurcations of Periodic Equations
Observe that
8II
8)..28x o (0, 0, 0) = )..2.
Then, obtain the expansion for the Poincare map near zero:
and describe how the solutions of Eq. (6.1) can be viewed as orbits of a
pair of autonomous differential equations on a torus, 8 1 x 8 1 . It is not a
restriction to assume the periods to be 1; if not, we can always rescale the
variables t and x to make it so.
We begin our discussion by pointing out two key properties of solu-
tions of Eq. (6.1) which facilitate the viewing of solutions on a torus. Let
cp(t, to, xo) be a solution of Eq. (6.1) with cp(to, to, xo) = Xo· Then, from
the uniqueness of the solution of the initial-value problem and the period-
icity properties of f, we have
0=1 (6.4)
:i; = f(B, x).
It is clear from the form of the first equation that the orbits of Eq. (6.4)
correspond to the trajectories of Eq. (6.1). The first equation is periodic
with any period; let us take its period to be 1. As we saw in Section 4.3,
the relation (6.2) implies that if we identify B with B + k, for any integer
k, then the orbits of Eq. (6.4), hence the trajectories of Eq. (6.1), can be
viewed as smooth curves on the cylinder 8 1 x IR. Using the relation (6.3),
if we further identify x with x + k, for any integer k, then the orbits of
Eq. (6.4) become smooth curves on the two-torus 8 1 x 8 1 which we denote
by T2. For simplicity, let us take Bo = a and summarize this construction:
Translate all unit squares whose corners lie at the integers to the unit
square [0, 1] x [0, 1], then glue the top of this square to its bottom, and its
left side to its right, while preserving their orientations; see Figure 6.1.
6.1. Differential Equations on a Torus 149
,:.:1.~~:1:t~.
~... ~.... '
.:.::::::::::: ..
8
Figure 6.1. Gluing the opposite sides of a square, while preserving their
orientations, gives rise to a torus.
Next, we define the notion of the Poincare map of the flow of Eq. (6.4)
on the torus. If we modify Definition 4.12 to take into account the peri-
odicity in x, then we are led to define the Poincare map II of the flow of
Eq. (6.1) to be
II(xo) = cp(l, 0, xo) (mod 1),
where (mod 1) means that only the fractional part of the value of the
solution is retained. In terms of the orbits of Eq. (6.4) on the (B, x)-plane
this has the following geometric meaning: After all the unit squares with
integer corners are translated to the first unit square, the image of a point
(0, xo) on the left side of the first unit square under the Poincare map
is the x-coordinate of the first point at which the orbit through (0, xo)
leaves the right side of the first unit square. As such, the Poincare map
is a map of the unit interval into itself, II : [0, 1] ~ [0, 1]; it usually has
a jump discontinuity, but is convenient to iterate on the computer. If
we now perform the identification described earlier by gluing the opposite
sides of the first unit square, then the unit interval becomes a circle of
circumference 1 (the dark circle on the torus in Figure 6.1). Hence, the
Poincare map becomes a diffeomorphism of the circle into itself, II : 8 1 ~
8 1 . The properties of the orbits of Eq. (6.4) on the torus are, of course,
reflected in the properties of the iterates of the Poincare map on the circle.
In fact, the periodic points, not just the fixed points, of the Poincare map
correspond to the orbits of Eq. (6.4) that are closed curves.
For the purposes of visualization, it may sometimes be more convenient
to denote a point on the circle by its usual angle ¢ measured in radians.
In this case, a point is determined up to (mod 27r), that is, by any angle of
the form ¢ + 27rk for any integer k.
Let us now illustrate these ideas on the simplest possible example.
Example 6.1. Parallel flow: Consider the differential equation :i; = c,
where c is a given real number, which is trivially I-periodic both in x in t.
150 Chapter 6: On Tori and Circles
(6.5)
x=c.
For reasons which will become self-evident momentarily, the flow of this
system on the torus is called a parallel flow.
The orbit of Eq. (6.5) through the point (0, xo) is the line x = cO+xo·
By taking 0 = 1, we see that its Poincare map is given by II(xo) = c+xo
(mod 1), which is a rotation by arclength of c around the circle. We will
subsequently analyze the geometry of the orbits of Eq. (6.5) on the torus
and its Poincare map for all possible values of c. For the moment, let us
consider a particular case.
For c = 1/2, the orbit of Eq. (6.5) through the point (0, xo) is the line
x = ~O+xo.
As seen in Figure 6.2a, this orbit repeats itself in every other unit square
it passes through. So, when these squares are translated to the first unit
square the entire orbit consists of two line segments. If we now identify
the opposite sides of this square, we obtain a closed curve on the torus
which goes through the hole twice before returning to its initial positionj
see Figure 6.2b. All orbits are closed curves of this type.
For c = 1/2, the Poincare map of Eq. (6.5) is given by
where). is a real parameter, and try to determine the flow of the equivalent
system
0=1
(6.7)
x= sin(27l'x) + ). sin(27l'O)
on the torus.
6.1. Differential Equations on a Torus 151
(a) (b)
I1(x) t----..".----~/'/
/'
/'
~----....,/,.(/
/'
/
/'
/'
/'
/'
/'
/'
/'
/'
Xo X1 x
(e) (d)
Figure 6.2. For iJ = 1, :i; = 1/2: (a) a solution on the (9, x)-plane and
its translation to the first unit square, (b) a solution on the torus, (c) its
Poincare map on the unit interval, and (d) its Poincare map on the circle.
equilibria correspond to the fixed points of the Poincare map of Eq. (6.8).
From the linear variational equation in Section 4.4, it is not difficult to
determine that these fixed points are hyperbolic with the same stability
type as the corresponding equilibria. Consequently, Eq. (6.8) has two hy-
perbolic periodic orbits on the torus, one asymptotically stable and the
other unstable, and all the remaining orbits approach one of these periodic
orbits either in forward or reverse time.
For >. sufficiently small, the fixed points of the Poincare map of the
unperturbed equation retain their stability types because they are hyper-
bolic. It is possible to show that under small perturbations the entire flow
of the perturbed system (6.7) remains qualitatively the same as the flow of
the unperturbed equation (6.8) on the torus; see Figure 6.3. <)
Let us now return to Example 6.1 and determine its flow and the
Poincare map for all values of the parameter c.
Theorem 6.3. Let rr be the circle map defined by
o (a)
(b)
(c)
Figure 6.3. For equation x = sin(27rx) + Asin(27rt): (a) Poincare map and
the How on the torus for A = 0, (b) How on the unit square for A = 0, and
(c) How on the unit square for A = 0.15.
154 Chapter 6: On Tori and Circles
/' /"
(a)
/" ,./
(b)
Figure 6.4. For x = 0.51234 ... (irrational!): (a) its Poincare map on the
unit interval, and (b) a dense orbit on the torus.
xo, rrk(xo), rr2k (xo), ... divide 8 1 into arcs of length less than c. Since c
was arbitrary, it follows that the orbit {rrn(xo) } is dense in 8 1 . <>
The implications of this theorem for the geometry of the parallel flow
of Eq. (6.5) are easy to deduce. When c is rational of the form c = p/q,
the orbits are closed curves on the torus which go around the torus p times
while going through the hole of the torus q times. When c is irrational,
the orbits never close up and any single orbit comes arbitrarily close to any
given point on the torus; see Figure 6.4.
(mod 1).
Iterate the initial point Xo = 0.33 under this map several times by handj
yes, by hand. Then iterate the same point in the computer. Do you see a
difference in the asymptotic behavior of the orbit?
Aid: Use the one-dimensional map named mod stored in the library of
PHASER.
Exercises - - - - - - - - - - - - - - - - - - - - " 0 . 0
6.5. Find the rotation numbers of the following vector fields on the torus:
(a) f(t, x) = cos 27rt;
(b) f(t, x) = 1 + cos 27rx;
(c) f(t, x) = AX (mod 1), where A is a real parameter. Is the rotation
number rational or irrational for A = 1? You may have to do this
problem numerically. Set A = 1 and use an explicit solution to compute
the iterates of the Poincare map to estimate the rotation number. Also,
try several different initial values; do you get the same rotation number?
6.6. Suppose that II>. satisfies the conditions of Theorem 6.6 and that all fixed
points of 110 are hyperbolic. Show that p(II>.) is constant for IAI small.
where a and b are real parameters, and attempt to investigate its dynamics
numerically. For this purpose, if we use Euler's algorithm with step size h
158 Chapter 6: On Tori and Circles
°
with the end points and 1 of the unit interval identified. As such, this
map is referred to in the mathematical literature as the standard or the
canonical circle map. To see the effect of the choice of the step size, you
may wish to compare the dynamics of this circle map described below with
For °:s
that of the differential equation (6.10) on the circle.
10 < 1, the map (6.13) is a diffeomorphism of the circle because
F(w, 10, x + 1) = F(w, 10, x) + 1 and (d/dx)F(w, 10, x) > 0. At 10 = 1, it is
a homeomorphism, and for 10 > 1, it is no longer one-to-one.
The notion of rotation number can be generalized for homeomorphisms
:s :s
of the circle. Indeed, when 0 10 1, the rotation number p(w, c) of the
standard map is defined to be the following limit:
where n is integer and the values of the iterates of Xo are used without
(mod 1). It can be proved that all the conclusions in Theorem 6.5, which
we stated for Poincare maps only, remain valid with this definition of the
rotation number. When 10 > 1, rotation number is not defined as the limit
above no longer exists; we will say more about this later.
Let us first focus our attention on the parameter range 10 1
where the notion of rotation number is well defined and try to describe the
°:s :s
behavior of the rotation number for various values of the parameters w and
c. We begin with two very special cases where only one of the parameters
is present. When 10 = 0, this map is simply the rotation around the circle
by arclength wand we have already studied its dynamics in Theorem 6.3.
At the other extreme, when w = 0, there are two fixed points at x =
and x = 1/2. They are both hyperbolic and the first one is unstable while
°
6.3. An Example: The Standard Circle Map 159
0.8
-....~
.'
~ 0.6
3- .... _ ... 0 . 2 4 [ 2....
1
Q. 0.4
... -" 0.22 .. -.,
0.2 a= • .-
....... 0.20 .. '
0.250.26 0.27
00 0.2 0.6 0.8
w
the second is asymptotically stable as long as 0 < c < Ij compare with the
Poincare map of Example 6.2.
For each fixed c = t, with 0 < t ~ 1, one can prove the following
properties of the rotation number p(W, t) :
• p(W, t) is a nondecreasing and continuous function
ofwj
• for each rational number p/q, there is an interval
I p / q with nonempty interior such that for all W E
I p / q we have p(w, t) = p/qj
• for each irrational number a, there is a unique W
such that p(w, t) = a.
A typical graph of p(w, t) for fixed 0 < t ~ 1 is shown in Figure 6.5,
where the properties listed above are readily visible. This striking graph,
which is an example of a Cantor junction, has also been dubbed as the
"devil's staircase."
In Figure 6.6, we have plotted some of the important features of the
bifurcation diagram of the standard map on the (w, c)-plane. From each
rational number on the w-axis, there originates a sharp widening wedge
with nonempty interior in which the rotation number is this constant ra-
tional. Furthermore, none of these wedges with rational rotation numbers
overlap when 0 ~ c ~ 1. From each irrational number on the w-axis, how-
ever, there originates a continuous curve with no interior and extends to
c = 1.
The dynamics of the standard map within each wedge is rather simple.
Let us examine, for instance, the wedge emanating from the origin. Since
the rotation number is zero in this wedge, the map must have a fixed point.
Indeed, for c t=- 0, the fixed points of the map are given by
160 Chapter 6: On Tori and Circles
1.25
.,o .,
1
1.0
€ 0.75
0.50
0.25
0.0 ~-~.LJe-l-L-...L...Ll.---1-----L-l...:1-~L.......I---1...-=-0'-::8.L.LL...----:-'
0.0 0.2 0.4 0.6 .
Figure 6.6. Bifurcation diagram of the standard map. Inside each wedge,
the so-called "Arnold tongues," the rotation number is rational.
-27fW
sin(27fx) = --.
c
If we fix c and increase W from 0, we see that two fixed points, the in-
tersections of the graphs of sin(27fx) and a horizontal line, one stable and
the other unstable, eventually coalesce and disappear. At this moment, we
move out of the wedge with rotation number 0 and the orbits on 8 1 become
dense. We have illustrated this saddle-node bifurcation of fixed points in
Figure 6.7, which should be compared with Figure 2.19. A similar dynam-
ical phenomenon occurs in the other wedges, but the role of fixed points
is replaced by that of periodic points of appropriate periods. Figure 6.8
shows the saddle-node bifurcation of period-2 points in the large wedge
with rotation number 1/2.
For c > 1, the dynamics of the standard map becomes rather compli-
cated and some of the interesting observations still remain to be numerical.
As we observed before, in this parameter range the standard map is no
longer a homeomorphism and the rotation number is not defined. The
qualitative dynamics of the map cannot be captured by the rationality or
the irrationality of a single number. The limit in Eq. (6.14) is not unique
but takes values on an interval of real numbers. Consequently, if we try
to extend the bifurcation diagram above the line c = 1, we notice that the
6.3. An Example: The Standard Circle Map 161
Figure 6.8. Stair-step diagrams of F and the graphs of F2 (see the fol-
lowing page) depicting the saddle-node bifurcation of period-2 orbits of the
standard map in and out of the wedge with rotation number 1/2.
6.3. An Example: The Standard Circle Map 163
Figure 6.9. Constructing a universal constant for the standard circle map.
rational wedges overlap and new wedges grow from irrational rotation num-
bers; see Figures 6.6 and 6.9. This signals the presence of very complicated
orbits, a situation often called chaos.
In the mist of this bewildering complexity, however, there are several
"universal constants" which mark the transition to chaos for a large class of
maps, including the standard map. Since these constants are independent
of the precise form of a map, they could be of practical use in applications,
much like the universal constant of Feigenbaum for interval maps. We
now describe one such universal constant. Consider the irrational number
(v'5 - 1) /2, the golden mean, and its continued fraction expansion
v'5 -1 1
-----=---------
2 1
1+--
1
1+--
see Figure 6.9. Then, the numerical experiments point to the fact
.
11m Wn - W n -1
= -2.834 ....
n-++oo W n +1 - Wn
It has been proved that this number turns out to be the same for a large
class maps of the circle at their points of transition to chaos. However, the
standard map is not yet one of them.
Exercises - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ eT-Y'. 0
6.8. On the shape of a tongue: For the standard circle map, in a region of the
(w, c)-plane with a rational rotation number there is an asymptotically sta-
ble periodic orbit. To determine an approximate shape of such a region,
one can follow the fixed point while changing the parameters. Using this
observation, determine numerically the shapes of the two large regions with
rotation numbers 0 and 1/2.
Help: The standard circle map is stored in the library of PRASER under
the name arnold.
Since its inception by Poincare, the study of toral flows, circle diffeomor-
phism and rotation number in particular, has evolved into a deep theory
due to the efforts of Denjoy, Arnold, Herman, and others. Arnold [1983], a
good source on this topic, contains proofs of the basic properties rotation
numbers, as well the structural stability result in Theorem 6.7; see also
Coddington and Levinson [1955], Devaney [1986], Hale [1980], Hartman
[1964], and Nitecki [1971].
For C 2 vector fields, there are essentially only two types of qualitative
behavior as described in Theorem 6.5. It is a remarkable fact that there
may be other types of behavior if the vector field is only a C1 function; see
Denjoy [1932].
It is difficult to determine rotation numbers in specific equations and
one must often resort to numerical computations. Such computations re-
quire special algorithms and a great deal of care; see, for example, Van
Veldhuizen [1988]. After all, what is irrational in floating point arithmetic?
Since the seminal paper of Arnold [1965], the two-parameter standard
map has captured the attention of many mathematicians and scientists.
Prompted by the discoveries of Feigenbaum on interval maps, numerical
experiments (Shenker [1982]) eventually led to the universal scaling proper-
ties of the standard map (Rand [1988]). Circle maps similar to the standard
map are encountered in applications; indeed, the phenomenon of "Arnold
tongues" is used as a paradigm to explain frequency locking phenomena in
certain oscillatory systems (Bak [1986]).
166 Chapter 6: On Tori and Circles
If a circle map is not a diffeomorphism, one can still consider the limit
to define a rotation number, but the limit depends on the initial point. For
piecewise monotone circle maps, one can generalize the concept of rotation
number to one of rotation interval; see Newhouse et al. [1976] and Levi
[1981).
Our investigation of scalar equations that are I-periodic in t and x
led us naturally to flows on the torus which do not have any equilibrium
points. Toral flows with equilibria, however, do occur in other contexts and
they may exhibit quite different dynamics; see Cherry [1938) and Palis and
de Melo [1982).
7 - -_ __
Planar
Autonomous
Systems
/
/
/
Xl = X2
. (7.2)
X2 = -y9 sIn
.
Xl.
7.1. "Natural" Examples of Planar Systems 171
Xl = X2
(7.3)
X2 = -Xl·
This approximate system is known as the linear harmonic oscillator. Be-
cause of its simplicity we will use Eq. (7.3) to illustrate certain basic geo-
metric concepts in the next section. 0
The equations of the planar pendulum (7.2) and the linear harmonic
oscillator (7.3) are examples of "conservative" systems. We will explain
the meaning of this label later in this chapter and also, because of their
importance, devote Chapter 14 to this important class of systems.
Example 7.3. Competing species: In certain applications the use of planar
differential equations may in fact be necessary to describe models. For
instance, let us consider two interacting populations: a prey species Xl and
its predator X2.
For the sake of simplicity, it is plausible to assume that in the absence
of any interactions between the two species, the prey could grow without
bounds and the predator would become extinct. In the presence of interac-
tions, however, the growth rate of Xl should be impaired while the growth
rate of X2 improves. The simplest such mathematical model arising from
these assumptions is called the predator-prey system of Volterra and Lotka:
Xl = alxl - a2 X l x 2 (7.4)
X2 = -a3 X 2 + a4 X I X 2,
where aI, a2, a3, and a4 are positive constants.
To obtain a somewhat more realistic model of two interacting species
we can modify Eq. (7.4) to include the effects of competition of the prey
among themselves for their limited amount of resources, and the compe-
tition among the predators for the limited amount of prey. The resulting
system
(7.5)
X2 = -a3X2 + a4XIX2 - b2X~,
where bl and b2 are again positive constants, is called the competing species
model.
Naturally, both of these systems for modeling populations are mean-
ingful only in the positive quadrant of the (Xl, x2)-plane. We will shortly
172 Chapter 7: Planar Autonomous Systems
~c
-- L
The function f is called the characteristic of the resistor and its exact form
depends on the type of the material the resistor is made from. The voltage
and the current across an inductor satisfy Faraday's law
VR + VL - Vc = o.
Using the relations above, we can eliminate all but two of the variables ,
iL and VL, which satisfy the system of differential equations
Xl = X2 - (0/L)1/2 f(xl)
X2 = -Xl
Exercises - - - - - - - - - - - - - - - - - - - - .. <::'.0
7.1. Where is the mass? Look up in a physics book to learn how the differential
equation for the pendulum is derived. Does it bother you that the differential
equation (7.1) is independent of the mass of the bob at the end of the
pendulum? Consult Galileo.
7.2. Lienardform: Convert the second-order Lienard's equation ii+ f(x)iI+Y =0
to the first-order system
Xl = X2 - F(xI)
X2 = -Xl
for i = 1,2
for i = 1, 2
be two given real-valued functions in two variables. In the following several
chapters we will undertake a geometrical study of a pair of simultaneous
differential equations of the form
Xl = I1(X1, X2)
(7.8)
X2 = h(xl, X2).
Let us begin our study of the general planar system (7.8) by developing
some basic notations and geometric concepts. In this discussion, it will
be convenient to use boldface letters to denote vector quantities. For
instance, if we let x = (Xl, X2), x = (Xl, X2), and f = (11, h), then
Eq. (7.8) can be written as
x= f(x). (7.9)
7.2. Geneml Properties and Geometry 175
This equation now looks the same as the scalar equation x = I(x) consid-
ered in Part I, but we must keep in mind that x is a two-vector and f is
a vector-valued function. We will follow the convention of using subscripts
to denote the components of a vector and superscripts to label different
vectors, e.g., Xl = (xl, x~). In particular, an initial-value problem for
Eq. (7.9) will be indicated by
satisfies all the requirements of a norm listed above. The sup-norm and the
Euclidean norm, however, are considered to be equivalent to one another
because for any x E IR2 we have
Figure 7.3. Equivalence of the Euclidean norm 0 and the sup norm D.
x,
(a)
x,
(b)
x, (t)
-14.000000
Figure 7.4. For the linear harmonic oscillator (7.3): (a) a trajectory in
the three-dimensional (t, Xl, X2)-space, (b) circular orbit resulting from
projecting the helical trajectory onto the (Xl, x2)-plane, and (c) graphs of
XI(t) and X2(t) vs. t.
178 Chapter 7: Planar Autonomous Systems
of Eq. (7.9) onto the (Xl, x2)-plane. More precisely, to each point x on the
(Xl. x2)-plane, where f(x) is defined, we can associate the vector f(x) =
(h(x), hex)) which should be thought of as being based at x. In other
words, we can assign to the point x the directed line segment from x to
x + f(x). In the case of the linear harmonic oscillator (7.3), for example,
at the point (5, 5) we picture an arrow pointing from (5, 5) to (5, 5) +
(5, -5) = (10, 0). The collection of all such vectors is called the vector
field generated by Eq. (7.3), or simply the vector field f. Projections of
trajectories onto the (Xl. x2)-plane are called orbits. More specifically, we
make the following definition:
Definition 7.6. The positive orbit ,+(xO), negative orbit ,-(xO), and
orbit ,(XO) ofxo are defined, respectively, as the following subsets of1R.2
[the (Xl. x2)-plane]:
,+(xo) = U
tE[O, .axo)
<pet, xO),
,-(xo) = U
tE(oxo, OJ
<pet, xO),
,(XO) = U
tE(oxo, .axo)
<pet, xO).
Figure 7.5. (a) The vector field, and (b) the phase portrait of the linear
harmonic oscillator (7.3) in the two-dimensional phase plane (Xl, X2). A
family of concentric periodic orbits encircling an equilibrium point is called
a center.
Definition 7.8. A solution cp(t, xO) ofx = f(x) is called a periodic solu-
tion of period p, with p > 0, if cp(t + p, xO) = cp(t, xO) for all t E IR. The
°
minimal period p is that period with the property that cp(t, xO) #- xO for
< t < p. The orbit ')'(xO) = {cp(t, xO), t E IR} of a periodic solution
cp( t, x O) with period p is said to be a periodic orbit (also closed orbit) of
period p.
It is evident from this definition that a periodic orbit is a closed curve
on the (Xl, x2)-plane. Also, any orbit of x = f(x) that is a closed curve
must correspond to a periodic solution.
Example 7.9. Linear harmonic oscillator continued: We now return to
180 Chapter 7: Planar Autonomous Systems
Figure 7.6. Phase portrait of the planar pendulum (7.3). Notice the pres-
ence of centers.
the linear harmonic oscillator and rigorously justify the picture of its phase
portrait as depicted in Figure 7.4b.
Notice first that the only equilibrium point is the origin. Let x(t) =
(Xl(t), X2(t)) be the solution through x(O) = xO =f. 0 and consider the
square of the distance of the solution from the origin, Ilx(t)112 = [Xl(t)]2 +
[X2 (t) j2, as a function of t. Then,
7.2. General Properties and Geometry 181
Figure 7.S. Phase portrait of the competing species equations (7.5) in the
positive quadrant for al = a2 = a3 = a4 = 1 and b1 = 0.4, b2 = 0.2.
Figure 7.9. Phase portrait of the oscillator of Van der Pol [Eq. (7.6)].
The periodic orbit attracting the nearby orbits is called a limit cycle.
and so
Ilx(t)112 = IIxol1 2 for all t.
Consequently, the solution x(t) must remain on the circle of radius Ilxoll
with its center at the origin. Since the circle contains no equilibrium points,
the solution must be periodic of some period.
The direction of the arrows on the circular orbits can easily be deter-
mined from the differential equation. For example, on the positive xl-axis
the vector field points down, hence the solutions move in the clockwise
direction. The resulting phase portrait consisting of a family of periodic
orbits encircling an equilibrium point is called a center.
182 Chapter 7: Planar Autonomous Systems
Xi YS. Ti",.:
-2.999909
Figure 7.10. Graphs of Xl (t) vs. t of two different solutions ofthe linear
harmonic oscillator. Notice that both solutions have the same period.
4. D9CiJ'UiJliJ
x. -4.1109999
In narrative form, to find w(xO) keep discarding the "tail end" of the closure
of the positive orbit of xo. The equivalence of these two definitions of the
w-limit set is not immediately obvious; this you may want to ponder.
The concept of the a-limit set a(xO) of an orbit ')'(xO) can be defined
similarly by reversing the direction of time. More precisely, a point y E
a(xO) if there is a sequence tj with tj ---- axo as j ---- +00 such that
rp(tj, xO) ---- y as j ---- +00. The geometric definition is
a(xo) = n ')'-(rp(T, xO)).
To illustrate the concepts of limit sets, let us consider several simple,
but common, situations.
Let x be a point in IR? such that rp(t, xO) ---- x as t ---- +00. In this
case, we can choose the sequence {tj } to be any increasing sequence with
tj ____ +00 as j ____ +00 to show that x E w(xO). Also, x is an equilibrium
point (why?) and w(xO) = x.
As the next example of a planar limit set, let us consider the case of
a point on a periodic orbit. More specifically, suppose that rp(t, xO) is a
periodic solution of minimal period p. Then ')'(xO) is a closed curve and
w(xO) = ')'(xO) = a(xO). In fact, if y E ,),(xO), then there is a ty E [0, p)
such that y = rp(ty, xO). If we now take the sequence tj = jp + ty, where
j = 1, 2, ... , then y = rp(tj, XO) for all j and y E w(xO). It is evident that
no other points can be in w(xO). A similar argument shows that a(xO) =
')'(XO).
We will meet equilibria and periodic orbits as limit sets in many spe-
cific planar systems. Examples of more complicated limit sets and their
Q.
complete classification will be given in Chapter 12.
Exercises _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ • <>
7.3. Numerical dangers: As we saw in Chapter 3, numerical solutions of scalar
differential equations can sometimes be tricky. Similar difficulties may be-
come more pronounced in planar systems. To appreciate some of the possible
184 Chapter 7: Planar Autonomous Systems
dangers, try to compute with PHASER one of the nontrivial periodic or-
bits of the linear harmonic oscillator using Euler's algorithm with step size
h = 0.1. Do you believe what you see? What is the source of the difficulty?
Recompute the same orbit with Euler, but use smaller step sizes, for exam-
ple, 0.05, 0.01, and 0.001. Will you ever get the correct result in a finite
time? Try these experiments with negative step sizes also. Finally, compute
the same orbit using Runge-Kutta's algorithm with step size h = 0.1.
7.4. Equivalent norms: Even though it may not be an exciting exercise, try to
establish the inequalities (7.12) regarding the equivalence of the Euclidean
and sup norms.
7.5. Recreating pictures: The differential equations of the linear harmonic os-
cillator, predator-prey model, and planar pendulum are stored in the 2D
library of PHASER under the names linear2d, predprey, and pendulum, re-
spectively. Recreate the illustrations in this section using PHASER. Also,
perform experiments to estimate the periods of the periodic orbits of these
systems. Do the periods increase monotonically as you move away from an
equilibrium point?
7.6. Limit sets: Suppose that the vector-valued function x(t) given in each case
below is the solution of some differential equation on the plane. Find the 0;-
and w-limit sets if they exist:
p(t) = 1- ( 1
~
t)2
1 + et
(7.13)
Notice first that there is no equilibrium point. Since XI(t) moves with
uniform speed and X2(t) is constant, the orbits are the lines parallel to the
Xl-axis. See Figure 7.12 for the phase portrait of Eq. (7.13). 0
Despite its simplicity, the product system (7.13) captures the local
dynamics of any planar system away from an equilibrium point. A point
x in the plane is called an ordinary point of x = f(x) if f(x) "# o. By
the continuity of f, there is a neighborhood of x containing only ordinary
points. In such a neighborhood, we have the following theorem:
Theorem 7.12. (Flow Box Theorem) In a sufficiently small neighborhood
of an ordinary point of the planar system x = f(x) there is a differentiable
186 Chapter 7: Planar Autonomous Systems
- h
i v,
I--_-+---~
: I
change of coordinates y = y(x) such that in the new coordinates the orig-
inal system becomes the product system YI = 1, Y2 = o.
Proof. We may first change coordinates (translation, rotation, scaling in
time) so that the ordinary point of interest is moved to the origin, and the
vector field at the origin is f(O) = (1, 0), that is, the vector field at the
origin is pointing along the Xl-axis. In a sufficiently small neighborhood of
the origin, consider the mapping h: lR2 ____ lR2 given by
h(YI, Y2) == <P(YI, (0, Y2)),
where <P is the flow of x = f(x). From the fundamental theorem given
in the Appendix, the map h is differentiable. Also, observe that h is the
identity map on the Y2-axis, and on the Yl-axis its derivative at the origin
is f(O) = (1, 0). Hence, from the Inverse Function Theorem, h has a
differentiable inverse, that is, h is a diffeomorphism. Consequently, h maps
an open neighborhood of the origin in the (YI, Y2)-plane, which we take to
be the box shown in Figure 7.13, diffeomorphically to an open neighborhood
of the origin on the (Xl. x2)-plane.
Now, using the differentiable mapping h-l, let us pull back the flow
<p(t, (x~, xg)) to the (Yl. Y2)-coordinates. Let (y~, yg) be the unique point
satisfying h(y~, yg) = (x~, xg). Then,
h-l<p(t, (x~, xg)) = h-l<p(t, h(y~, yg))
= h-l<p(t, <p(y~, (0, yg))
= h-l<p(t + y~, (0, yg))
= h-Ih(t+y~, yg)
= (t + y~, yg).
7.3. Product Systems 187
Figure 7.14. Flow box for the product system Xl = Xl, X2 = -X2 about
an ordinary point x.
Since the vector-valued function (t + y~, yg) is the flow of the product
system ill = 1, Y2 = 0, the desired coordinate system is (YI, Y2). <>
It is evident in the proof above that the Flow Box Theorem guarantees
the existence of a box-like neighborhood of any ordinary point of any planar
system such that the orbits of the system enter at one end of the box and
flow out through the other-flow box. Moreover, no orbit leaves through
the sides of the box. Let us illustrate these ideas on a simple example.
Example 7.13. A flow box: Consider the product system
Xl = Xl (7.14)
X2 = -X2·
Xl = aXI
(7.15)
X2 = bX2·
188 Chapter 7: Planar Autonomous Systems
The phase portraits of (7.15) on the (Xl, x2)-plane can be determined quite
easily; see Figure 7.15. Let us fix a < 0 and consider all possible values
of b. You might like to draw the phase portraits for the cases when a > O.
If b < a < 0, then both Xl (t), X2(t) --+ 0 exponentially as t --+ +00,
that is, the w-limit set of any orbit is the origin (0, 0) which is the unique
equilibrium point. Furthermore, all the orbits except the x2-axis are asymp-
totically tangent, as t --+ +00, to the xl-axis because
If b = a < 0, then all orbits tend to the origin along straight lines as
t --+ +00. The phase portrait of the case a < b < 0 is similar to that of
b < a < 0 with the roles of the axes interchanged. In all of these cases the
equilibrium point (0, 0) is called a stable node because the w-limit sets of
all the orbits is the origin. Typical phase portraits of these three cases are
shown in Figures 7.15a--c. If a < 0 = b, then all the points on the x2-axis
are equilibrium points; see Figure 7.15d.
If a < 0 < b, then the equilibrium point at the origin is called a saddle
and the flow is shown in Figure 7.15e (see also Example 7.13). It has a
completely different structure than a node since there are two orbits whose
w-limit sets are (0, 0) and two orbits whose a-limit sets are again the origin.
All other orbits leave a neighborhood of the origin in the directions of both
increasing and decreasing t.
The linear product system (7.15) has a nice geometric interpretation
as a gradient system. Let us consider the function
(7.16)
7.3. Product Systems 189
(a)
b<a<O
(b)
b= a<O
(c)
a<b<O
Figure 7.15. Typical phase portraits ofthe product linear system Xl
aXI, X2 = bX2 (continued).
190 Chapter 7: Planar Autonomous Systems
(d)
a<O= b
(e)
a<O<b
where the vector field is now written as the negative of the gradient of F :
X= -VF(x).
If (Xl(t), X2(t)) is a solution of Eq. (7.16), then using the chain rule we
have
Thus, F is always decreasing along the solutions of Eq. (7.16) and can be
thought of as a "potential" function of Eq. (7.16). The graphs of the surface
z = F(Xl' X2) for several choices of a and b are shown in Figure 7.16. A
7.3. Product Systems 191
Figure 7.16. Graphs of the potential function F(Xl' X2) = -~(ax~ + bx~)
for several choices of a and b.
192 Chapter 7: Planar Autonomous Systems
particle starting on these surfaces moves "downhill" under the flow defined
by Eq. (7.16). For example, when a < 0, b < 0, the particle goes to the
minimum of F. However, when a < 0 < b, the surface has the shape of
a saddle and the particle misses the critical point unless it is started at a
special position. <>
It is also easy to construct nonlinear examples on the plane using
products of nonlinear scalar equations. We present one such example below.
You might like to select some other nonlinear scalar equations from Part I
and draw the phase portraits of their products.
Example 7.15. Product logistic: Let us consider the following system
consisting of a copy of the logistic equation (3.2) on each axis:
Xl = XI(l- Xl)
(7.17)
X2 = x2(1 - X2)'
We have plotted the phase portrait of Eq. (7.17) in Figure 7.17 which you
are invited to decipher. <>
where ro = r(O) and (}o = (}(O). Notice that solutions are defined for all
t E(-00, +00).
Let (ro, (}o) be an initial value withro =f O. Then it is easy to see
that any point (1, (}) on the unit circle is an w-limit point of (r2' (}o) by
simply taking the sequence tj in Definition 7.10 to be tj = ((}o - ()) + 27r'j.
Consequently, the w-limit set of any initial value with ro =f 0 is the unit
194 Chapter 7: Planar Autonomous Systems
circle. Hence, all the orbits of Eq. (7.19), except the equilibrium point at
the origin, spiral onto the unit circle with increasing time. The unit circle
itself is a periodic orbit with period 271". Finally, the origin is the a-limit set
of orbits with initial value ro < 1 (what is an appropriate sequence tj?). 0
Exercises _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ .IV, 0
7.8. Sketch the phase portraits and discuss the a- and w-limit sets of orbits of
the following product systems:
(a) Xl = -Xl, X2 = x~; (b) Xl = x~, X2 = -X2;
(c) Xl = -Xl, X2 = X2 - x~; (d) Xl = Xl - x~, X2 = X2 - x~.
7.9. Find a function F so that the vector field in Example 7.15 can be written
as a gradient system. Plot the graph of this function.
that is not constant on any open subset of JR.2 is called a first integml of
a planar differential equation x = f(x) if the function H is constant along
every solution, that is, for any solution x(t) with initial value x(O) = xo,
the composite function satisfies H(x(t)) = H(xO) for all t for which the
solution is defined.
We should remark that in the definition of a first integral above we
required the domain of H to be JR.2 (or, the domain of the definition of
the vector field); for this reason, H as defined above is often said to be a
global first integral. In our presentation in this chapter we will simply use
the term a first integral. The notion of a local first integral is discussed in
the exercises.
7.4. First Integrals and Conservative Systems 195
. 8H 8H
H(x) == VH(x)· f(x) = ~(x) h(x)
UXI
+ -(x)
8X2
h(x) = O. (7.20)
Figure 7.19. Graphs of the first integrals of (a) the linear harmonic oscil-
lator, (b) planar pendulum, and (c) predator-prey.
open subset of the entire plane, then a level set of H is, in general, a one-
dimensional subset of the plane and the orbit through xO is a connected
piece of this one-dimensional set. Now, by just sketching the level sets of a
first integral, one can infer a considerable amount of information about the
shapes of the orbits and hence the phase portrait. The exercise of sketching
curves-algebraic geometry-can at times be difficult even for polynomial
equations. For example, while it is clear that the level sets of the first
7.4. First Integrals and Conservative Systems 197
integral of the linear harmonic oscillator are concentric circles about the
origin, we will have to work a little to determine the level sets of the first
integral of the pendulum.
Unlike the impression we may have given in the example above, "most"
differential equations do not possess nontrivial first integrals. Here is one
such example.
Example 7.19. Nonexistence of a first integral: Consider the planar sys-
tern
Xl = -Xl
(7.21)
X2 = -X2·
As we saw in Example 7.14, the orbits of this system are rays terminating
at the origin. Any first integral must take on a constant value on anyone of
these rays. However, the requirement that a first integral be continuous, in
particular, at the origin, implies that the first integral must have the same
constant value on all the orbits. Thus, any first integral of this system
must be the constant function. <;
We now turn to the important question of the existence of first inte-
grals. Since a first integral gives information about the shapes of orbits
and not their time parametrizations, it is convenient to eliminate the time
variable from the system x = f(x). This is easy to do locally as we now
explain. If xO is not an equilibrium point, then at least one of h (xO)
or h(xO) is not zero. Let us suppose that h(xO) i- O. Then, there is
an open neighborhood of XO such that h (x) i- 0 in this neighborhood.
Therefore, the first component of the orbit <pet, xO) is strictly monotone
in t and it makes sense to replace the t-parametrization of the orbit by a
parametrization in the first coordinate, that is, in the neighborhood of xO,
the orbit through xO can be defined as a solution of the nonautonomous
scalar equation
dX2 h(xI, X2)
(7.22)
dXI = h(XI, X2)"
Similar remarks apply when h(xO) i- O. If xO is an equilibrium point,
then the orbit through xO is the point xO itself and no parametrization is
necessary.
Now, suppose that
(7.23)
dH 8H 8H dX2
-=0=-+--,
dXI - 8XI 8X2 dXI
198 Chapter 7: Planar Autonomous Systems
which, unfortunately, is not 0 1 on the plane and thus not a first integral
of the differential equation (7.21). <>
The most notable examples of differential equations possessing first
integrals arise in mechanical systems without friction and the existence of a
first integral is most apparent in their so-called Hamiltonian formulations.
For a given 0 1 function H IR2 ---t IR, a planar system of differential
equations of the form
. aH
Xl = - -
aX2
. aH
X2 = ---
aXl
is called a Hamiltonian system with the Hamiltonian H. The total energy of
a mechanical system, up to a multiplicative or additive constant, can often
be taken as the Hamiltonian of the system. It is clear from the special
form of the equations that the Hamiltonian function is a first integral-
conservation of energy. A special class of Hamiltonian systems known as
conservative systems comes from a second-order differential equation of the
form ii + g(y) = O. Indeed, it is easy to verify that
7.4. First Integmls and Conservative Systems 199
Xl = X2
X2 = -g(Xl).
For any xg, the curve defined by Eq. (7.25) is symmetric with respect to
the Xl-axis. Therefore, we need to plot only the curves
(7.26)
and then reflect through the xl-axis. Notice that the range of the values
of Xl needed to define the curve are given by the inequality
With these observations, we can effectively construct the orbits of the pen-
dulum by considering the values of xg. The directions of the orbits can
easily be inferred from the vector field.
200 Chapter 7: Planar Autonomous Systems
29
"£(1 -cos x,)
x,
Figure 7.20. Constructing the orbits of the planar pendulum from a first
integral.
For xg = 0, Eq. (7.26) gives the equilibrium points (-71", 0), (0, 0), and
(71", 0).
For 0 < (xg)2 < 4g/l, the range of Xl is an interval of length less than
271" and symmetric about the origin; see Figure 7.20. The curve defined by
Eq. (7.26) when reflected about the xl-axis yields a closed curve on the
plane. Since there are no equilibrium points on it, this closed curve is a
periodic orbit corresponding to the oscillation of the pendulum about the
equilibrium position (0, 0); see Figure 7.21.
For (xg)2 = 4g/l, the curve defined by Eq. (7.26) and its reflection
about the Xl-axis is again a closed curve. However, on this closed curve
there are several orbits. In particular, the equilibrium points (-71", 0) and
(71", 0) are on this curve. These equilibria correspond to the vertical position
of the pendulum while the pendulum is "sitting on its head." There are
two other special orbits: one whose a-limit set is (-71", 0) and w-limit set
is (rr, 0), the other, which is the reflection of this orbit, whose a-limit set
is (71", 0) and w-limit set is (-71", 0). These special orbits are called hetero-
clinic orbits and they correspond to the motions of the pendulum from one
equilibrium point to the other, in infinite time. Because of its importance
in dynamical systems, we record here the definition of a heteroclinic orbit
for future reference.
Definition 7.22. An orbit whose a-limit set is an equilibrium point and
w-limit set is another equilibrium point is called a heteroclinic orbit.
If (xg)2 > 4g/l, then the range of Xl is unrestricted and the curve
defined by Eq. (7.26) is a 271"-periodic graph over the Xl-axis. The level set
7·4· First Integrals and Conservative Systems 201
-7r
-B
H(x) = h = O - ...._.,~
on the
Figure 7.21. Flow of the pendulu m as viewed on the plane and
cylinder.
There
consists, of course, of this graph and its reflection about the Xl-axis.
and they corresp ond to the orbits of the
are no equilib ria on these curves
velocit y so large that the pendul um
motion s of the pendul um with initial
revolves around and around withou t end.
the
Let us reexam ine the qualita tive feature s of the phase portrai t of
202 Chapter 7: Planar Autonomous Systems
pendulum and present a view of the flow on a cylinder. This is quite nat-
ural, and also convenient. Indeed, the physical state of the pendulum is
determined by its angle of deviation from the vertical position and by its
velocity. Two physical states of the pendulum that differ in () by 27l' should
be considered the same. This, of course, is reflected in the differential equa-
tion describing the evolution of the states of the pendulum: Equation (7.2)
remains the same under the change of variables (Xl, X2) f---+ (Xl + 27l', X2)'
If we now take the first variable Xl mod 27l', the flow of the pendulum takes
place on the cylinder 8 1 x lR; see Figure 7.21. Notice that the orbits that
go around and around now become periodic orbits encircling the cylinder.
Besides its physical appeal, this observation has an important theoretical
consequence, as we shall see in the next chapter: all orbits are bounded.
Also, a heteroclinic orbit on the plane turns into another type of special
orbit on the cylinder called a homo clinic orbit whose formal definition will
appear in the following section.
The qualitative description of the flow of the equation of pendulum is
now complete. However, there is one important quantitative question that
still remains: what are the periods of the periodic orbits of the pendulum?
Unlike in the case of the linear harmonic oscillator, the concentric periodic
solutions, say, near the origin, of the pendulum have different periods; the
period is a monotone function of the amplitude. Further details on the
periods are contained in the exercises. <:)
Exercises - - - - - - - - - -_ _ _ _ _ _ _ _ _ _ .. 0 . <:)
7.10. On the line: Define the notion of a first integral for a scalar autonomous
x
differential equation. Does the scalar equation = -x have a first integral
that is not identically constant? Can you characterize all conservative scalar
autonomous differential equations?
7.11. A first integml for predator-prey equations: Consider the predator-prey
equations (7.4) in the positive quadrant. Its orbits away from the two equi-
libria (0, 0) and (a3/a4, aI/a2) are the solution curves of the scalar equation
al - a2X2 dX2
X2 dXl
and integrate it to obtain the first integral
Although it is not immediately obvious, show that the level sets of H are con-
centric closed curves encircling the equilibrium (a3/a4, al/a2), as depicted
in the numerically computed phase portrait in Figure 7.7.
7.4. First Integrals and Conservative Systems 203
Hint: Write the level set H(Xl, X2) = k as the product f(Xl) g(X2) = K,
where the functions f and 9 are given by f(Xl) = x~3/ea4"'1 and g(X2) =
x~1/ea2"'2, and K is some constant. Now, determine the graphs of these
functions by finding their critical points, etc.
7.12. Local first integrals: Show, using the Flow Box Theorem, that in a suffi-
ciently small open neighborhood of a regular (nonequilibrium) point there
always exists a local first integral. As seen in Example 7.19, however, a
local first integral need not exist in an open neighborhood of an equilibrium
point.
7.13. Draw the orbits and the direction of the flows of the following differential
equations:
(a) Xl = X2(X~ - x~), X2 = -Xl(X~ - x~);
(b) Xl = x2(1- x~ - x~), X2 = -xl(l- x~ - x~).
Warning: Watch your division when computing dX2/dxl.
7.14. Sketch the phase portraits of the following equations and discuss the a- and
w-limit sets of orbits:
(a)8+9+93 =0; (b)~+9-93=0;
(c) 8 + 9 - 92 = 0; (d) 9 + 9(1- 9)(A - 9) = 0, 0 < A < 1/2;
(e) Xl = sinx2, X2 = -sinxl.
7.15. Period in conservative systems: Consider the second-order equation jj +
g(y) = 0, or the equivalent first-order system
Xl = X2
X2 = -g(Xl).
(a) Verify that this is a conservative system with the Hamiltonian function
H(Xl, X2) = xV2 + G(Xl), where G(Xl) = fO"'l g(u) duo
(b) Show that any periodic orbit of this system must intersect the xl-axis
at two points, say, (a, 0) and (b, 0) with a < b.
(c) Using the symmetry of the periodic orbits with respect to the xl-axis,
show that the minimal period T of a periodic orbit passing through
two such points is given by
T = 2jb --;=:=:::::;:::::du=:::::::=:~
a J2[G(b) - G(u)]
4 1"'/2 d1;
T = Viii 0 J1 - sin2(b/2) sin2 e
Hint: Use the symmetry with respect to the x2-axis, the half-angle
formula cosu = 1 - 2sin2(u/2), and the change of variable sin(u/2) =
sine sin(b/2).
204 Chapter 7: Planar Autonomous Systems
:h =X2
•
X2 = -Xl - 3
Xl'
(a) If you have not done so already, sketch the phase portrait.
(b) Obtain the integral formula for the period of the periodic orbits.
(c) Compute the first two terms of the Taylor expansion of the period
when b is small and observe that the frequency increases with b. This
behavior is referred to as a hard spring.
(d) Show that, for b large
T -
- b
~ 11 VI -
0
dv
v4
[1 + 0(-b1 ) ] .
(e) What happens to the frequency in the limit as b ~ +oo?
Xl = A + xi
(7.27)
X2 = -X2·
Observe that the second equation is linear with X2(t) --+ 0 as t --+ +00.
Thus, all the orbits of Eq. (7.27) eventually approach the xl-axis where
the dynamics of the system are governed by the first equation. We should
7.5. Examples of Elementary Bifurcations 205
point out, however, that the first equation is simply Example 2.2. The
phase portraits of the flow of Eq. (7.27) for various parameter values are
now easy to construct. For A < 0, there are two equilibria. One of these
equilibria is a saddle point because, as in the linear system in Example 7.13,
there are two orbits near the origin such that the w-limit sets of these orbits
are the origin, and there are two orbits whose a-limit sets are again the
origin. We will undertake a detailed study of the geometry of flows near a
saddle point in Chapter 9. The other equilibrium point is a node because
the w-limit sets of all the orbits starting near this equilibrium point is the
origin. At A = 0 the two equilibria coalesce into one, and for A > 0, the
equilibrium point disappears; see Figure 7.22.
In the study of bifurcations of scalar autonomous equations, bifurca-
tion diagrams were a very convenient way to portray much information
about the qualitative dynamics of the flow. We want to continue to em-
ploy similar bifurcation diagrams in higher dimensions. For the present
example, since the dynamics take place in the first equation we can follow
the equilibrium points by drawing the xl-coordinates of the equilibrium
points as functions of the parameter A. The resulting bifurcation diagram
is then the same as Figure 2.3. <>
Saddle-node bifurcations of equilibria occur quite commonly in non-
product systems as well. The example above, despite its simplicity, cap-
tures the essential features of the general case. Unraveling the meaning of
this remark will be the subject of Chapter 10.
Example 7.24. Pitchfork bifurcation: Consider the product system
Xl = -AXl - x~
(7.28)
X2 = -X2·
A<O
A= 0
A>O
Figure 7.22. Phase portraits of the saddle-node bifurcation.
7.5. Examples of Elementary Bifurcations 207
"""'"" J \( I/"""
~
.----/ "--
,,<0
~ /
~ ~
~/
~~
,,>0
Figure 7.23. Phase portraits of the pitchfork bifurcation.
208 Chapter 7: Planar Autonomous Systems
When ,\ < 0, all solutions spiral clockwise into the origin with increasing t.
For ,\ = 0, this is the harmonic oscillator and as we have already seen, all
solutions are periodic so that the origin is a center. Since at this value of
°
the parameter the number of periodic orbits changes from none to many,
we consider ,\ = a bifurcation value. For,\ > 0, all solutions spiral out
clockwise without bounds,i. see Figure 7.24. Let us now plot a bifurcation
diagram for the periodic orbits of Eq. (7.29). Since every periodic orbit
encircles the origin, it is convenient to represent a periodic orbit by its
"amplitude," that is, by the a at which point the periodic orbit intersects
the xl-axis. With this convention, the equilibrium point at the origin is
view~d as a degenerate periodic orbit of zero amplitude. The resulting bi-
furcation diagram, which is sometimes aptly dubbed as vertical bifurcation,
is shown in Figure 7.25. <:;
r = r('\ - r2)
(7.32)
0=-1.
1\=0
1\>0
------r
Xl = X2
(7.33)
X2 = Xl + AX2 - xi·
At this point in our book it is somewhat difficult to construct the phase
portraits of this system except for A = O. Luckily, as you may suspect,
A = 0 will turn out to be a bifurcation value and thus is a good place to
start.
For A = 0, the system (7.33) is conservative with the first integral
(7.34)
1;\<0
;\=0
;\>0
----------f-------------A
When .A f:. 0, the center is destroyed but the saddle remains. The loop
consisting of the homoclinic orbit and the equilibrium point at the origin,
however, is broken. The manner in which the loop breaks depends on the
sign of the parameter .Aj see Figure 7.29. <>
Unlike in the case of scalar equations, the precise notion of qualitative
change, or bifurcation, in planar systems is not easy to formulate. Some-
times bifurcations are clearly marked with a change either in the number
of equilibrium points or the number of periodic orbits. At other times,
there can be subtler bifurcations that are "global" in character and thus
7.5. Examples of Elementary Bifurcations 213
A<O
A>O
Figure 7.29. Breaking a homoc1inic loop in Eq. (7.33).
214 Chapter 7: Planar Autonomous Systems
Q.
class of vector fields-linear.
Exercises _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ .. <)
7.18. Recreating pictures: The differential equations of saddle-node, pitchfork,
vertical, and Poincare-Andronov-Hopf bifurcations are stored in the 2D li-
brary of PHASER under the names saddlenod, pitchfork, linear2d, and hopf.
Recreate the illustrations in this section using PHASER.
7.19. A product system: Draw some representative phase portraits of the product
system
Xl = >. + xi
X2 = >. + x~
for negative, zero, and positive values of the scalar parameter >..
7.20. A product in polar: Consider the following planar system depending on two
real parameters >. and W
Xl = X2
. 2
X2 = Xl - Xl.
7.5. Examples of Elementary Bifurcations 215
Obtain the period function and discuss its behavior in the limit near the
homoclinic loop; and also near the origin.
We will reconsider many of the topics of this chapter in more detail later,
where we will also provide a wide choice of references. For the moment,
here are some sources pertaining mostly to applications which you might
find interesting.
Natural examples of differential equations abound. Mechanics is dis-
cussed from a contemporary viewpoint in Arnold [1978). The two species
problem is analyzed in Hirsch and Smale [1974). For various ecological
models, consult D'Ancona [1954], May [1973), Maynard Smith [1968 and
1974], and Pielou [1969). For Van der Pol's equation, the original source is
Van der Pol [1927); we will return to this famous equation in Chapter 12.
A general mathematical formulation of electrical circuits is given in Hirsch
and Smale [1974) and Smale [1972); engineering details are available in, for
example, Desoer and Kuh [1969). For general applications, see Andronov,
Vitt, and Xhaikin [1966).
Various norms on IRn and their equivalence can be found in, for in-
stance, Smith [1983]. Appropriate formulations of the existence, unique-
ness, and dependence on initial data of solutions of systems of ordinary
differential equations are in the Appendix.
8 - -_ __
linear
Systems
then the system (8.1) can be written in the equivalent but more convenient
vector notation
x=Ax. (8.2)
Lemma 8.1. The solutions of a linear system x = Ax are defined for all
t E JR. <)
Definition 8.2. Two solutions x 1 (t) and x 2 (t) of Eq. (8.2) are said to be
linearly independent if, for each t E JR, the relation CIX1 (t) + C2X2(t) = 0
implies that Cl = 0 and C2 = o.
Linear independence of x 1 (t) and x 2 (t) is equivalent to the fact that
the determinant of the 2 x 2 matrix whose columns consist of these two
vectors is nonzero:
(8.4)
Proof. (i) First observe that the solution of the initial-value problem x(O) =
o for Eq. (8.2) is identically zero: r.p(t, 0) = o. Now, suppose that there
are constants Cll C2, and 7" such that CIX1 (7") + C2x2(7") = 0, where x 1 (t)
and x 2(t) are the columns of X(t). Then, CIX1 (t + 7") + C2x2(t + 7") is also
a solution for Eq. (8.2), which for t = 0 is zero. Therefore, by uniqueness
of solutions, we have
(ii) The right-hand side of Eq. (8.4) is a solution of Eq. (8.2) be-
cause it is a linear combination of the solutions xl(O) and x2(O). Since
X(O) X(O)-1 = I, it also satisfies the initial condition. Thus, from the
uniqueness theorem, it is the solution. <>
The superposition principle implies that the set of all solutions of
Eq. (8.2) is a vector space. The lemma above shows that the dimension
of this vector space is two. After discovering some general facts about the
flows of linear systems, we will determine explicit bases for the vector space
of solutions of Eq. (8.2) for any given coefficient matrix A.
As we saw in the first example of our book, the flow of the scalar
linear differential equation x = ax is given by the exponential function
cp(t, xo) = eatxo. To obtain an analogous formula for the flow of linear
planar systems we introduce the notation
where X(t) is any fundamental matrix solution ofEq. (8.2). Then, Eq. (8.4)
for the flow of Eq. (8.2) can be written as the matrix exponential
(8.6)
hence establishing the desired analogy. Of course,
e AO = I,
and thus eAt is a principal matrix solution of Eq. (8.2).
We now collect several important properties of the principal matrix
solution eAt and provide the reason for this choice of notation.
Lemma 8.5. The principal matrix solution eAt satisfies the following
properties:
(i) eA(t+s) = eAt e As ;
(ii) (eAt) -1 = e-At;
Since P(t) appears on both sides of this equation, we attempt to find P(t)
iteratively by using successive approximations. If we take as our initial
guess
p(O)(t)XO = Ixo,
and compute the successive iterates with
then as the kth iterate, for k = 0, 1, 2, ... , we obtain the polynomial ex-
pression
1 1
p(k)(t)xO = Ixo + AxOt + ,A2xOe + ... + k,AkxOt k.
2. .
We now show that the sequence of vectors p(k)(t)xO given by this
formula converges as k ---+ +00. The first observation is that it suffices to
show the convergence of p(k) (t)xO for all xO on the unit circle 8 1 == {x :
Ilxll = I} only, because any vector in JR2 can be written as a scalar multiple
of some vector on 8 1 . The second observation is the fact that, since IIAxo II
as a function of xO is continuous and 8 1 is a closed bounded set, there
exists a constant a > 0 such that
and
Because of the uniform convergence of the series this process can be justified
rigorously to obtain the integral equation
Therefore, cp(t, XO) is the solution of Eq. (8.2) satisfying the initial condi-
tion cp(O, xO) = xO. Since, from Eq. (8.6), cp(t, xO) = eAtxO, the sequence
pk(t) converges to eAt as k -) +00. <>
It is evident from the lemma above that the principal matrix solution
eAt enjoys many of the properties of the analogous scalar exponential func-
tion. Despite these similarities, however, there are also major differences
between the scalar and matrix exponential functions, and care must be
exercised in computations. For instance, in general, eAt. eBt ::f. e(A+B)t for
two 2 x 2 matrices A and B. Here are two such matrices.
Example 8.6. Noncommuting matrices: Consider the matrices
eAt = I + At = (1 t)
o 1 '
eBt = I + Bt = (1 0)
-t 1 '
and
eAt . eBt = (1- -t
t2
B.l. Properties of Solutions of Linear Systems 223
The power series of e(A+B)t is also easy to sum, if we recall that the
power series expansions of the scalar functions sin t and cos t are given by
. +00 n t2nH t3 t5 t7
smt= ~)-1) =t--+---+ ...
n=O (2n + I)! 3! 5! 7!
and
+00 t2n t2 t4 t6
cost = ~)_l)n_ = 1- - + - - _ + ...
n=O (2n)! 2! 4! 6! .
Again, using part (iv) of Lemma 8.5 we obtain rather easily that
To compute eAt we will utilize Lemma 8.7. We can write the matrix A as
the sum of two commuting matrices:
The diagonal part of the matrix can be exponentiated as above and the
exponential of the second matrix was already computed in Example 8.6.
So, the desired formula for eAt follows from Lemma 8.7.
Let us now draw the phase portrait of the corresponding linear sys-
tem. We will suppose that >. < 0 and leave the remaining cases >. = 0 and
>. > 0 to you to practice with. We first observe that the origin is the only
equilibrium point. Next, using the explicit formula cp(t,xO) = eAtxo for
solutions we see that all orbits approach the origin as t -+ +00. further-
more, since dX2/ dXl -+ 0 as t -+ +00, all orbits eventually become tangent
to the xl-axis at the origin. The resulting phase portrait as depicted in
Figure 8.1 is often called a (stable) improper node.
B.l. Properties of Solutions of Linear Systems 225
A= -1
A=Q
A= 1
Figure 8.1. Phase portraits of the linear system Xl = AX1 + X2, X2 = AX2
for A = -1, A = 0, and A = 1.
226 Chapter 8: Linear Systems
A= ( 0:
o
0)+( -130 (3)
0: 0
Exercises _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ .. Q. <>
8.1. An animation sequence: Draw the phase portraits and indicate the direction
of the flows of the following linear systems:
(a) x= ( 0
-1 ~) x; (b) x= (~ -1)
o x;
a<O, (3)0
o
a= 0, (3 >0
a>O, (3)0
Hint: Verify that BeAt and eAtB are matrix solutions of x = A x and use
uniqueness.
8.4. Prove that if AB = BA, then e(A+B)t = eAt e Bt .
Hint: Using the previous exercise, verify that e(A+B)t and eAte Bt are matrix
solutions of the linear system x = (A + B) x.
8.5. A square under the flow: Draw the image of the unit square in the first
quadrant under the flow of the linear system Xl = -Xl, X2 = X2 at t = 1
and also at t = -1. What are the areas of the images?
y = P-1APy. (8.9)
It is also easy to determine the flow in the new coordinates using Eq. (8.6):
and thus
eAt = PeP-1 APtp- 1 . (8.10)
°
e-o3t ) y.
Now, using formula (8.10), we can readily determine the flow in the original
coordinates:
Av = AV.
The vector v is called an eigenvector of A corresponding to the eigen-
value A.
For future reference, we should point out that, since A is a real matrix,
a real eigenvalue has a corresponding real eigenvector. A complex eigen-
value, however, has a necessarily complex eigenvector of the form vI + iv 2 ,
where both vI and v 2 are real and nonzero vectors.
230 Chapter 8: Linear Systems
Figure 8.3. Phase portraits of Eq. (8.12) and its canonical form.
(A - AI) v = 0 (8.13)
Observe now that these two vectors are linearly independent, and the trans-
formation matrix P we chose in Example 8.9 has its columns as these
eigenvectors. <:;
Some embellishments of the computations in this example yield a con-
structive proof of the following general result for transforming an arbitrary
2 x 2 matrix to a canonical matrix.
Theorem 8.12. Let A be a 2 x 2 matrix with real entries. Then, there
exists a real invertible 2 x 2 matrix P such that
P- 1 AP=J,
Since we assumed that A1 - A2 =f. 0, and v l is not the zero vector, this
implies that C1 = 0; and now Eq. (8.16) yields C2 = o. Thus, the eigen-
vectors v l and v 2 corresponding to two distinct eigenvalues are linearly
independent.
Now, routine matrix multiplications suffice to exhibit the desired trans-
formation property of P:
where
J = (~1 ~2).
Since P is invertible, if we multiply both sides of this matrix equation by
p- 1 , we obtain the desired similarity
(ii) Equal eigenvalues: There are two cases to consider. First, sup-
pose that A is a double eigenvalue but there are two corresponding linearly
B.2. Reduction to Canonical Forms 233
p- l AP =J = (Ao 0)
A .
(8.18)
where
J=(~ ~).
Since P is invertible, if we multiply both sides of this matrix equation by
P-l, we obtain the desired similarity
p- l AP =J = (Ao 1)A .
Let us now show the existence of v 2 satisfying Eq. (8.18). Since
(A - AI) is not the zero matrix (why?), if there is no such vector v 2 , then
the range of (A - AI) must contain a vector u 1 so that u 1 and v 1 are lin-
early independent. Let u 2 be such that (A - AI) u 2 = u 1 . Consequently,
any vector, and in particular u 2 , can be written as a linear combination of
u 1 and v 1 :
u2 = C1V 1 + C2U 1 .
If we substitute this into the equation (A - AI) u 2 = u 1 and use the fact
that v 1 is an eigenvector, we obtain
exist a vector v 2 satisfying Eq. (8.18) from which the linear independence
of vI and v 2 is almost immediate.
(iii) Complex eigenvalues: Suppose that A = a + i{3, with {3 ::f 0,
is a complex eigenvalue of A with a corresponding complex eigenvector
vI + iv 2 , that is,
(8.19)
where VI and v 2 are two nonzero real vectors. Furthermore, these two real
vectors are linearly independent. Indeed, suppose that they were linearly
dependent. Then there would be two nonzero real constants Cl and C2
such that CIV I + C2V2 = 0, or equivalently VI = (C2/Cl)V 2 . Using this
in Eq. (8.19) we obtain Av 2 = (a + i(3)v 2 . Since the left-hand side of
this equation is real and the right-hand side is complex, we arrive at a
contradiction. So, the vectors VI and v 2 must be linearly independent.
As a transformation matrix P, consider the matrix whose columns
consist of the real and imaginary parts of the complex eigenvector:
where
J=(_~ ~).
Since P is invertible, if we multiply both sides of this matrix equation by
p- l , we obtain the desired similarity
This concludes the proof of the theorem and our somewhat extended ex-
cursion into linear algebra. <>
We remarked in Section 8.1 that the space of solutions of a planar
linear system is a two-dimensional vector space. The efforts in the proof
of the theorem above easily yield a basis for this vector space. If one is
B.2. Reduction to Canonical Forms 235
Exercises _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ .,.0.0
8.6. Changing coordinates: Consider the linear system Xl = Xl, X2 = -X2.
Transform this system to new coordinates y = p- 1 x for various choices of
P below and sketch the phase portraits in these new coordinate systems:
8.7. Find the Jordan Normal Form and also compute eAt for each of the following
matrices:
where p and w are constants, and you will have to consider their relative
magnitudes.
8.8. Determining fundamental matrix solutions: In this exercise we outline how
to find two linearly independent solutions of x = A x using eigenvalues and
eigenvectors.
(a) Show that if '\1 =I- '\2 are two real eigenvalues of A and v l and v 2
are eigenvectors corresponding to '\1 and '\2, respectively, then e A1 tv l
and e A2t v 2 are two linearly independent solutions of the linear system
x=Ax.
(b) Show that if '\1 = '\2 are the eigenvalues of A and there is only one
independent eigenvector v l corresponding to '\1 and v 2 is any solution
of (A - '\lI)v 2 = v l , then eA1tv l and e A1t (v2 + tv l ) are two linearly
independent solutions of the linear system x = A x. As a specific
example, find the solution of the initial-value problem
. 12" (2 1)
x= -1 0 x, x(O) = (~) .
(c) If,\ = a+i(3, with (3 =I- 0, is a complex eigenvalue of A and v = v l +iv 2
be a corresponding complex eigenvector, then
* = (1 -1)
5 -3 x, x(o) = (~) .
8.9. All ellipses: Show that every orbit of the linear system
. (0 -2)
x= 8 0 x
is an ellipse. The origin is, of course, a degenerate ellipse. Draw the phase
portrait of the system.
8.10. A second-order equation: Consider the second-order differential equation
ii + PiJ + w2 y = 0, where p and w are real constants. Convert this differen-
tial equation to an equivalent planar linear system and find a fundamental
matrix solution.
8.11. On chamcteristic polynomial: Show that the characteristic polynomial of a
2 x 2 matrix A can be written as
where
and )..1 and )..2 are the roots of the equation det(A - ).. I) = o.
8.12. A useful estimate: As an application of Theorem 8.12 on normal forms,
establish the following estimate:
If the eigenvalues of a 2 x 2 matrix A have negative real parts, then there
are positive constants k and 0: such that, for all x E IR?,
for t 2:: o.
The utility of this result will become apparent in the next section.
8.13. A "stiff" linear system: Since the exact solutions of linear systems can
readily be written down, they often serve as testing grounds for various nu-
merical algorithms for solving initial-value problems for ordinary differential
equations. For example, try to solve the following linear system using Euler,
Improved Euler, Runge-Kutta, and various step sizes:
. _ (998.0 1998.0)
x- -999.0 -1999.0 x.
Any success? Compute the eigenvalues and the eigenvectors of the sys-
tem to obtain explicit solutions; compare your theoretical findings with the
8.3. Qualitative Equivalence in Linear Systems 237
numerical ones. By the way, the term stiff does not really have a precise
mathematical definition; it usually means that the solutions are doing inter-
esting things on a very short, as well as on a long, time scale. To capture
such a behavior numerically, one must use either a "very small" step size,
and wait, or detect the places where the drastic changes are and vary the
step size accordingly. How do the magnitudes of the eigenvalues of the lin-
ear system above compare? For more information on this linear system, see
Forsythe et al. [1977], p. 124.
(8.20)
Figure 8.4. (a) All linear systems whose eigenvalues have negative real
parts-sinks-are topologically equivalent (continued).
B.3. Qualitative Equivalence in Linear Systems 241
x2 x2 x2
x, x,
~ "-
'\ (
x,
• • •• • ••• --
.... ...'.
-----iHHH-t--t--t--- x ,
Case (iii): A = (-0>' _\). For any c > 0, we can assume that A =
(-0>' _\) by the transformation of coordinates x 1--+ (~ ~)x. If we let D =
1/(2A) I, then
-1
ATD+DA= ( /( c/(2A)) == -E.
c 2A) -1
Now, choose c so that 1 - [c/(2A)j2 > 0 and introduce the change of co-
ordinates x 1--+ E- 1/ 2x. Then the matrix C == E- 1/ 2DE-1/2 possesses the
desired properties. <>
We now turn the proof of our first main result of the section.
Proof of Theorem 8.15. Let us first observe that the conditions on the
coefficient matrices A and B are necessary for topological equivalence. For
this purpose, it is not difficult to persuade oneself that if a homeomorphism
takes a bounded positive orbit of x = A x to a bounded positive orbit
of x = B x, then the homeomorphism also takes the w-limit set of one
positive orbit to that of the other. A similar remark holds for a-limit sets
of bounded negative orbits also. Consequently, if A has both eigenvalues
with negative real parts and A is topologically equivalent to B, then every
solution of x = B x approaches zero as t -+ +00. Thus, both eigenvalues
of B must have negative real parts. A similar argument holds, as t -+ -00,
if both eigenvalues of A have positive real parts. If A has one positive and
one negative eigenvalue, then there must be one nonequilibrium solution
x
of = B x that approaches zero as t -+ +00 and another solution that
approaches zero as t -+ -00. Consequently, B must have one negative and
one positive eigenvalue.
To prove sufficiency, let us begin with Case (iii) as it is the simplest.
From the previous section, we may assume that if A and B have one
negative and one positive eigenvalue, then they can be put, by linear change
of coordinates, into the following Jordan Normal Forms:
-A1 -J.L1
A-
- ( 0 B= (
o
where each Ai > 0 and J.Li > O. Now, recall from Section 2.6 that the two
linear scalar differential equations
Therefore, any nonequilibrium solution x(t) ofx = A x crosses the level sets
of XTCAx inward. A similar computation shows that any nonequilibrium
solution x(t) of x = B x crosses the level sets of XTCBx inward also; see
Figure 8.6.
It is now easy to define a homeomorphism h : JR2 - 0 --+ JR2 - 0 that
takes the orbits ofx = Ax to the orbits ofx = Bx. Let h: {x: XTCAx =
1} --+ {x : x T CBx = 1} be a given homeomorphism of the two ellipses.
For any xO =f:. 0, there is a unique time txo such that Xl == e-Atxo XO lies on
the ellipse XTCAX = 1. Now, define h(xO) = eBtxOh(xl); see Figure 8.7.
It is evident, from continuous dependence of solutions on initial data, that
~
XI
it. = Ax it. = Bx
Figure 8.6. Orbits of a planar linear system whose eigenvalues have neg-
ative linear parts cross the level curves of an appropriate positive definite
quadratic function, ellipses, inward.
Therefore, for any c > 0, there is 8 > 0 such that Ilh(xO) II < c so long as
(xOfCAxO ~ 8. Since XTCAx is norm equivalent to the Euclidean norm,
h is continuous at the origin. As we saw in an example above, however, h
cannot always be made a diffeomorphism at the origin. <:;
We now give an outline of a proof of the classification of nonhyperbolic
planar linear systems.
Proof of Theorem 8.16. It is not difficult to establish that a nonhyperbolic
planar linear system is topologically equivalent to one of the five cases
by putting the coefficient matrix in Jordan Normal Form and using the
ideas from the proof of Theorem 8.15. In the last case it is necessary to
reparametrize the periodic orbits. Therefore, it remains to show that no
two cases in the statement of the theorem are topologically equivalent.
Here are the distinguishing qualitative features of each case that makes it
unique:
A = (~1 ~): The w-limit sets of all positive orbits are equilibrium points.
A = (6 ~): The a-limit sets of all negative orbits are equilibrium points.
8.15. On topological and linear equivalence: Show that the following two linear
systems have the same eigenvalues and thus are topologically equivalent;
however, they are not linearly equivalent:
. (-2 0)
x= 0 -2 x,
Draw the phase portraits of these two linear systems and compare.
8·4· Bifurcations in Linear Systems 247
. (-1 0)
X= 0 2 x,
8.17. All the same: Consider the following five coefficient matrices A of the linear
systems :ic = A x:
(-1o 1) -2 .
tr A <0
+
tr A <0 det A > 0
+
det A < 0
A(/.L) = (-1 0)
0 /.L
(8.23)
where J.L is a real parameter near zero. The geometry of this co dimension-
one bifurcation is depicted in Figure 8.9.
Case (iii): Unfolding Ao = (g~). The construction of an unfolding of
this nonhyperbolic matrix is essentially a combination of the two previous
cases above. The matrices that are topologically equivalent to Ao satisfy
the conditions A =I- 0 with det A = 0 and tr A = O. These conditions
8.4. Bifurcations in Linear Systems 251
C) @ 6
A(/l) = ( !, :)
Figure 8.9. An unfolding of (~1 ~) depending on one parameter.
(8.24)
where 11-1 and 11-2 are real parameters near zero. For a pictorial summary
of this bifurcation, see Figure 8.10.
The nonhyperbolic matrix (~ ~) is, in a way, the simplest example
of a codimension-two singularity. Its second unfolding above will be of
paramount importance when we investigate the bifurcations of an equilib-
rium point of a nonlinear system in Chapter 13.
Exercises _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ "Y'. 0
8.18. Open but not dense: Show that the set of planar linear systems for which
the origin is a sink is an open but not dense subset of lR4.
252 Chapter 8: Linear Systems
~1
8.20. Dense open sets are "fat": Prove that if 81, ... , 8 k are open and dense
subsets of lRn , then their intersection 8 1 n ... n 8k is also open and dense
in lRn.
8.21. For the following matrices, discuss the transition from one topological equiv-
alence class to another as the scalar parameter A is varied:
depending on three real parameters /L 1, /L2, and /L3 unfold the codimension-
two singularity
A(O, 0, 0) = (~ ~)?
Moral: It is not just the number of parameters that counts, it is also where
they are inserted.
B.5. Nonhomogeneous Linear Systems 253
(a) Ao= ( -1
0 ~1) , A I = ( -1
0 ~}
(b) Ao= ( -1
0 0)
1 ' Al = (~ ~1}
(c) Ao= ( -1
0 ~2) , A I = ( -1
0 !1) .
x= Ax+g(t), (8.25)
y. -_ - A e -At x + e -At·x.
Now, if we substitute Eq. (8.25) into the equation above, we get the desired
differential equation
(8.27)
Suppose that we specify the initial condition x(to) = X O for the original
equation (8.25). In the new coordinates, this is the same as specifying the
initial condition y(to) = e-AtoxO for Eq. (8.27). To obtain the solution
254 Chapter 8: Linear Systems
y(t) satisfying the initial condition above, we simply integrate both sides
of Eq. (8.27) from to to t and rearrange the terms:
(8.28)
This formula is known as the variation of the constants formula, likely be-
cause, in contrast with the solution of the homogeneous part, the expression
inside the brackets has now become a function of t.
The variation of constants formula (8.28), of course, gives the flow of
the nonautonomous system (8.25) and, when written in the expanded form
reveals an important insight. Observe that the first term is the flow of the
linear part x = Ax of Eq. (8.25). It is easy to verify that the second term
is the "particular solution" of the full equation (8.25) satisfying the initial
condition x(to) = O. We should add in conclusion that our interest in this
explicit solution of nonhomogeneous linear systems stems from its utility
in certain technical estimates related to questions of stability of nonlinear
autonomous systems, as we shall see in the next chapter.
Exercises - - - - - - - - - - - - - - - - - - - - , . < : : 1 . 0
8.26. Find the general solution of the affine system
Xl = +1
X2
X2 = -Xl + 1
(b) Show that 'Ij;(t, yO)rp(t, xO) = yOxO for all t E IR, where rp(t, xO) is the
flow of x = Ax.
8.29. Fredholm's Alternative: Suppose that f(t) is a continuous 27r-periodic vector
function and consider the differential equation x = Ax + f(t).
(a) Show that this equation has a 27r-periodic solution if and only if
1o
271"
y(t) f(t) dt = 0
1 o
271"
f(t)costdt=O, 1 o
271"
f(t)sintdt=O.
256 Chapter 8: Linear Systems
ic = A(t)x, (8.30)
where tr A(t) = all (t) + a22(t), the sum of the diagonal entries of A(t).
Proof· We may assume that X(t) is a fundamental matrix solution of
Eq. (8.30), otherwise the formula is trivially true. Now, let B(t) be defined
as the matrix [X(t)]-l A(t) X(t), or, equivalently, A(t) X(t) = X(t) B(t).
Notice that tr B(t) = tr A(t).
For simplicity of notation, let z(t) == det X(t). To establish the formula
of Liouville, it suffices to show that z(t) satisfies the differential equation
i = [tr A(t)] z.
8.6. Linear Systems with l-periodic Coefficients 257
Now, since X(t) is the principal matrix solution of Eq. (8.32), if x(t) is the
solution of Eq. (8.32) with x(O) = xo, then
x(t) = X(t)xo
and
x(t + 1) = X(t + I)xO = X(t)Cxo.
In words, the action of translating the solution x( t) by 1 unit of time is the
same as translating the initial value XO into the new initial value C xo. In
particular,
(8.34)
c: IR? - IR?;
which should be viewed as the Poincare map of the I-periodic linear system
(8.32), see Figure 8.Il.
We will undertake a detailed study of planar maps, including, of course,
the linear ones, in Chapter 15. For the moment, however, we trust that
you will find the few remarks below acceptable. The notions of a fixed
point and its stability that we have developed in Chapter 3 for scalar maps
are easily generalized to planar maps. This essentially entails replacing
scalar quantities with vectors, and absolute values with norms, in several
definitions in Chapter 3. With similar replacements, Definitions 4.9 and
4.10 are also readily generalized to yield definitions of stability, asymptotic
stability, and instability of a solution of the I-periodic system (8.32).
A periodic solution of Eq. (8.32) corresponds to a fixed point of the
linear planar map (8.34), and the stability type of the periodic solution of
Eq. (8.32) is the same as the stability type of the corresponding fixed point
of the map (8.34). Notice in particular that the zero solution of Eq. (8.32)
corresponds to the fixed point of C at the origin. We now summarize the
main implications of these remarks for the zero solution of Eq. (8.32).
Lemma 8.22. Let f.ll and f.l2 be the eigenvalues of the matrix C as given
in Eq. (8.33). Then
(i) If If.lil < 1, for i = 1, 2, then the zero solution of Eq. (8.32) is asymp-
totically stable.
(ii) If If.lll = 1f.l21 = 1 and f.ll =f f.l2, then the zero solution of Eq. (8.32) is
stable.
(iii) If one of the eigenvalues has modulus greater than 1, then the zero
solution of Eq. (8.32) is unstable. <>
8.6. Linear Systems with 1-periodic Coefficients 259
Definition 8.23. The eigenvalues of the matrix C satisfying Eq. (8.33) are
called the characteristic multipliers of the l-periodic linear system (8.32).
It is possible to prove considerably more about the structure of linear
I-periodic systems, but we refrain from doing so here. Be warned, however,
that unlike the case of linear systems with constant coefficients, the theory
of I-periodic linear systems remains difficult without a complete resolution.
To illustrate the richness of this subject, we conclude this section with an
example of great prominence.
Equation.
2: °
and c are two real parameters, is known as the linear Mathieu
Let X(t) be the principal matrix solution of Eq. (8.35) and C == X(I). It is
easy to discern form Lemma 8.21 that det C = exp J;
tr A(t) dt = 1. Since
the determinant is the product of the eigenvalues, we have the important
relation below between the eigenvalues of C:
J.lIJ.l2 = 1. (8.36)
In light of this fact, Lemma 8.22 yields the following theorem for Mathieu's
Equation.
Theorem 8.25. For Mathieu's linear differential equation (8.35):
(i) If J.ll is not real, then lJ.lll = 1J.l21 = 1 and the zero solution is stable.
(ii) If J.ll is real and lJ.lll < 1, then 1J.l21 > 1 and the zero solution is
unstable. <>
260 Chapter 8: Linear Systems
Imaginary Imaginary
-1 Real Real
• Stable 0 Unstable
Figure 8.13. Shaded regions are the zones of stability in the (0', e)-plane
of the linear Mathieu Equation.
Then, for each such 0'0, there exists an eo > 0 such that, for lei < eo,
the eigenvalues of C remain to be nonreal. This observation follows from
the continuous dependence of distinct eigenvalues on the parameters which
is a consequence of the quadratic formula. Using sophisticated analysis
and numerical computations, it is possible to determine fairly accurately
the shapes of the curves on the parameter plane dividing the regions of
stability and instability; see Figure 8.13. As a numerical experiment, we
have tried in Figure 8.14 to find a point on one of these curves dividing
the regions of stability. We fixed 0' = 0.6540, and plotted a solution and
its Poincare map of Mathieu's Equation with initial data near the origin
for the values of the parameter e = 0.3900, e = 0.3950 and e = 0.4000. It
appears that the point (0.6540, 0.3995) on the (0', e)-plane is almost at the
boundary of stability. <>
We conclude here our study of linear differential equations and next
turn to the role of linear theory in the local analysis of nonlinear vector
fields.
Exercises _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ,.0. <>
8.32. Chamcteristic multipliers of constant coefficients: Consider a constant ma-
trix A as a I-periodic function and find the characteristic multipliers of the
system :X: = Ax.
8.33. Computing chamcteristic multipliers: Find the characteristic multipliers of
the following systems:
(a) Xl = (sin 211't) Xl, X2 = (-1 + cos 211't) X2j
(b) Xl = (sin t) Xl, X2 = (-1 + cost) X2j
(c) Xl = (cos 211't) Xl, X2 = (sin 211't) Xl + (-1 + cos 211't) X2·
8.34. Chamcteristic multipliers are invariants: Show that the characteristic mul-
tipliers are independent of the choice of the fundamental matrix.
262 Chapter 8: Linear Systems
PHASE PORT:
<8>
x Min:
6.990999
-6.99999
y ",ax:
Y Min:
4.909000
-4.00909
:II Min:
~0_ 90900
-HJ.0999
HAP POIHC:
x Min:
-6.09999 6.990009
-4.00099 4.909090
PHASE PORT:
•
x ...,in:
-6.99999 6.999999
y Min: y Max:
-4.99090 4. 09CiUJ99
z "in:
-HLge0g 10.00009
MAP POINC:
)( Min: x Max:
-6.09999 6.990090
y ",in: Y Max:
-4.09900 4.900909
PHASE PORT:
x Min:
-6.99999 6.900000
y .. in: Y Max:
-4.00009 4.000900
Z Min:
-10.0009 19.09099
MAP POINC:
x .dn:
-6.00909 6.009900
Y Min: Y Max;
-4.09000 4.000990
Hint: If X(t) and X(t) are two fundamental matrices, and X(t+1) = X(t) C
and X(t + 1) = X(t) C, then C is similar to C.
8.35. Negative real parts are not enough: As we saw earlier in this chapter, all
solutions of a linear system with constant coefficients eventually go to zero
if the eigenvalues of its coefficient matrix have eigenvalues with negative real
parts. Unfortunately, this is not so in general for linear 1-periodic systems.
Here is an example of this sort from Markus and Yamabe [1960j. Consider
the 1-periodic linear system whose coefficient matrix is
-1+(3/2)cos2t 1-(3/2)Costsint)
A(t) = ( .
-1 - (3/2) sin t cos t -1 + (3/2) sin 2 t
(a) Compute the eigenvalues of A(t) to find >'1,2 = [-1 ± iJ7] /4. Notice,
in particular, that the real parts of the eigenvalues are negative. On the
other hand, verify that the vector (- cos t, sin t)e t / 2 is a solution of the
1-periodic system above; but this solution is unbounded as t -+ +00.
(b) One of the characteristic multipliers is e". What is the other multiplier?
(c) If you are brave, you may wish to tackle the somewhat futile exercise
of finding a second linearly independent solution of this system.
8.36. Floquet representation: If X(t) is a fundamental matrix solution of a linear
system with I-periodic coefficients (8.32), then there exists a 1-periodic
matrix pet) and a constant matrix B, with perhaps complex entries such
that
X(t) = P(t)e Bt .
Show how to reduce Eq. (8.32) to a linear system with constant coefficients.
The catch is that it is very difficult to compute characteristic multipliers.
Hint: Since X(t + 1) = X(t) C and the matrix C is nonsingular, C has a
logarithm B, that is, C = eB. Let pet) = X(t)e- Bt . If x = P(t)y, then
y = B x. When is the matrix B real?
8.37. Mathieu on PHASER: The equation of Mathieu is stored in the library of
PHASER under the name mathieu in three dimensions. Investigate the
zones of stability numerically; in particular, reproduce Figure 8.14.
Bibliographical Notes _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ~
Jordan Normal Form is one of the central results in Linear Algebra, and
you can look up the details for higher dimensions in any good book on Lin-
ear Algebra. However, most algebraists prefer algebraically closed fields-
complex numbers; for the real case and applications to differential equa-
tions, see, for example, Hirsch and Smale [1974). For large systems, one
often has to resort to numerics, in which case care is required; see Golub
and Wilkinson [1976). By the way, we never had to put a matrix larger
than 4 x 4 into normal form.
264 Chapter 8: Linear Systems
Figure 9.1. All solutions approach an equilibrium point as t -> +00 but
the equilibrium point is not stable.
If one of the eigenvalues of the coefficient matrix A has positive real part,
then the equilibrium point x = 0 is unstable. <>
It is evident in the definitions above that the stability type of an equi-
librium point is a local property. Consequently, as in the case of the scalar
equations, it is reasonable to expect that under certain conditions the sta-
bility type of x can be determined from the approximation of the vector
field f with its derivative, which is a linear vector field. For this purpose,
let us suppose that f = (ft, h) is a C 1 function, and let the matrix
y=f(y+x).
(9.5)
Although the function y(t) appears on both sides, we will use this integral
equation to estimate Ily(t)11 in terms of IlyOIl as a function of t.
Let us suppose that the constants K and a are given as in Theorem
9.3, m > 0 is such that mK < a, and c > 0 is chosen so that Eq. (9.4) is
satisfied. Then, from the estimate (9.1) in Theorem 9.3, we have
9.1. Asymptotic Stability from Linearization 269
as long as Ily(s)11 ::; e and 0 ::; s ::; t. Multiplying both sides of this
inequality with eat yields
To finish the proof, select 6 > 0 so that K6 < e. If IlyOIl < 6, then the
inequality (9.6) guarantees that Ily(t)11 < e since a - Km > o. Therefore,
the solution y(t) exists for all t ~ 0 and the equilibrium solution y = 0
of Eq. (9.3) is stable. Also from Eq. (9.6), we have y(t) -+ 0 as t -+ +00
if Ily011 < 6. Consequently, y = 0 is asymptotically stable and the rate of
approach to the equilibrium is exponentially fast. <:;
Example 9.6. The damped pendulum: As an application of the asymp-
totic stability theorem above, let us consider the equation for the damped
pendulum given by
jj + 2aO + w2 sinO = 0, (9.7)
where a > 0 reflects friction and w2 == gil. Using the transformation in
Example 7.1, it is easy to see that this second-order equation is equivalent
to the planar system
Xl = X2
(9.8)
X2 = _w 2 sin Xl - 2aX2.
The equilibrium points of this system are (mr, 0), where n is any integer.
Let us first consider the stability of the equilibrium point at the origin.
The Jacobian of the vector field (9.8) is the matrix
. (0-w - 1) x.
x= 2 2a (9.9)
270 Chapter 9: Near Equilibria
J.Ll,2 = -a ± ..ja 2 - w2 •
The real parts of both eigenvalues are negative for all a > 0 and w > 0
(why?). Therefore, by Theorem 9.5, the equilibrium point (0,0) of the non-
linear equation (9.8) is asymptotically stable. For comparison, the phase
portraits of both the nonlinear equation (9.8) and its linearization (9.9)
near the origin have been plotted in Figure 9.2 for several values of the
parameters a and w.
What about the stability of the other equilibrium points? It is easy to
see that, when n is an even integer, the linearized equations at the equilib-
rium points (mr, 0) are the same as the system (9.9). So the analysis above
applies to these points as well. When n is an odd integer, however, the lin-
earized equations are different. We will consider the stability properties of
such points in the next section. <>
then
f(t) :S K eJ: 9(s)ds, a:S t :S b.
Suggestions: Let h(t) = K + J:
f(s) g(s) ds, and notice that h(a) = K.
Using calculus, obtain h(t) = f(t) get) :S h(t) get). Multiply this inequality
by exp{ - J:
f(s) g(s) ds} to arrive at
Figure 9.2. Local phase portraits of the damped nonlinear pendulum and
its linearization near the equilibrium point at the origin for the values of
the friction coefficients 0.25 and 1.5.
272 Chapter 9: Near Equilibria
9.4. Asymptotic stability under small perturbations: Suppose that the eigenvalues
of the matrix A have negative real parts and B(t) is a I-periodic matrix with
IIB(t) xii:::::; c Ilxll for all t and x. Show that if C is sufficiently small, then all
solutions of x = [A + B(t)] x approach zero as t -+ +00.
Hint: Use the variations of the constants formula and Gronwall's inequality.
Since the eigenvalues of -Df(x) have negative real parts, by Theorem 9.5,
the solution y = 0 of Eq. (9.11) is asymptotically stable. In particular,
there is an r > 0 such that, if Ily011 = r, then the solution cp(t, yO) of
Eq. (9.11) approaches 0 as t -+ +00.
Now, fix e and yO satisfying 0 < e < rand Ily011 = r. Letibe the time,
depending on e and yO, such that Ilcp(i, yO)11 = e. Notice that cp( -t, yO) is
a solution of Eq. (9.10) and cp( -i, cp(i, yO)) = yO. Therefore, the solution
of Eq. (9.10) with initial data cp(i, yO), satisfying Ilcp(i, yO)11 = e, reaches
the circle Ilyll = r in a finite amount of time. Since e can be chosen
arbitrarily small, this implies the instability of the equilibrium solution
y = 0 of Eq. (9.10).
Let us suppose now, as the second case, that the eigenvalues of the Ja-
cobian matrix are real with ILl ::; 0 < IL2. By a linear change of coordinates
(see Section 8.2) the Jacobian matrix can be put into a diagonal canonical
9.2. Instability from Linearization 273
and compute its derivative along the solution curves of Eq. (9.12). If
(Y1 (t), Y2(t)) is a solution of Eq. (9.12), and Ily(t) II < c then using Eq. (9.4)
we make the following estimates:
. dV
V(y(t)) = di(Y1(t), Y2(t))
= Y2Y2 - Y1YI
= J.L2Y~ + Y2 92(YI, Y2) - J.LIY~ - YI 91(Yb Y2)
::::: J.L2Y~ - mllyll (IY11 + IY21) - J.L1Y~
::::: (J.L2 - m)y~ - 2m1Y111Y21- (J.LI + m)y~.
which is the wedge lying above the lines Y2 = ±Yb as shown in Figure 9.3.
Notice that in the region n the level set V- I (C2) lies above the level set
V-I(cd if C2 > C1'
Next, let us choose c > 0 and m so that Eq. (9.4) is satisfied and
J.L2 -4m > O. Let U == { (Yb Y2) : Ilyll < c} be an open neighborhood of the
origin. In the region nnu, we have V(y) > 0 and V(y) ::::: (J.L2-4m)y~ > O.
Suppose that yO E n n U. Then the solution through yO remains in
n as long as yet) E U. To finish the proof, we observe that the solution
through yO has norm equal to c for some value of t, that is, the solution
must hit the boundary of U. This follows because there is {j > 0 such that
V(y) > {j if Y E U and V(y) > V(yO). The proof of the instability of the
equilibrium point of Eq. (9.12) at the origin is now complete. <>
Example 9.S. The damped pendulum continued: Let us consider the sta-
bility type of the equilibrium points (mr, 0) of Eq. (9.8) when n is an odd
274 Chapter 9: Near Equilibria
C2 > C, > 0
v,
Figure 9.3. The level· sets of V and the region n where V (y) > O.
ttl,2 = -a ± -Ja 2 + w2 .
Since one eigenvalue is positive (and the other is negative), it follows from
Theorem 9.7 that these equilibrium points are unstable.
For comparison, we have plotted in Figure 9.4 the phase portraits of the
nonlinear system (9.8) and its linearization (9.13) near one of these unstable
equilibrium points. Of course, the linear system is a saddle. Furthermore,
the nonlinear system looks much like a saddle also, and the linearization is
a good reflection of the local phase portrait of the nonlinear system near
the equilibrium. The preservation of a saddle under small perturbations of
a linear system is true in general and we will explore this fact further in
Section 9.5. <:;
The stability type of an equilibrium point of a nonlinear system cannot
always be determined from linearization. It is evident from Theorems 9.5
and 9.7 that such a situation can occur only if some eigenvalue of the
linearization has zero real part (and the remaining eigenvalue has negative
real part). In this case, we must examine effects of the specific nonlinear
terms of the vector field to determine the local dynamics. Indeed, we
have already encountered an instance of this difficulty in the saddle-node
9.2. Instability from Linearization 275
\
Figure 9.4. Local phase portraits of the damped nonlinear pendulum and
its linearization near one of the unstable equilibria.
bifurcation given in Example 7.23 where one zero eigenvalue was present.
Here is an example with purely imaginary eigenvalues:
Example 9.9. When linearization does not suffice: Consider the system
of nonlinear differential equations
. (0 1)
X= _lOx. (9.15)
The eigenvalues of this linear system are ±i, which have zero real parts.
To analyze the behavior of the nonlinear system (9.14), we compute
the derivative of the square of the distance of a solution from the origin:
Exercises _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ,.0, 0
9.5. Equilibrium of Van der Pol: Consider the Lienard form of Van der Pol's
oscillator
Xl = X2 - >..(xV3 - Xl)
X2 = -Xl,
where>.. is a scalar parameter. Show that the only equilibrium point is at
the origin. Determine its stability type as a function of >...
9.6. Show that the zero solution of the differential equations whose vector fields
are given below is unstable:
9.8. If the origin is a stable but not asymptotically stable equilibrium point of
the planar system x = f(x), can the origin be a saddle point of the linearized
equations?
9.9. Feedback control: Consider the equation for the pendulum of length l, mass
m, in a viscous medium with friction proportional to the velocity of the
pendulum. Suppose now that the objective is to stabilize the pendulum in
the vertical position (above its pivot) by a control mechanism which can
move the pivot of the pendulum horizontally. Let us assume that fJ is the
angle from the vertical position measured in the clockwise direction and the
restoring force v due to the control mechanism is a linear function of fJ and 9
that is, v(fJ, 9) = clfJ + c29. Convince yourself that the differential equatio~
describes the motion of such a pendulum. Show that the constants C1 and
C2 can be chosen in such a way so as to make the equilibrium point (fJ, 9) =
(0, 0) asymptotically stable.
9.10. "Pole" placement: In the problem above, the linearized equations had the
form
x= Ax+ bv,
and the problem was solved by choosing v = eT x = ClXl + C2X2 so that
the eigenvalues of the matrix A + beT have negative real parts. Prove the
following result:
Theorem: If the matrix (b I Ab) is nonsingular, that is, the linear system
above is controllable, then there is a vector e such that the eigenvalues of
A + beT have negative real parts.
Hint: There is a vector e with tr (A + beT) < 0 and det(A + beT) > o.
makes explicit use of nonlinear terms, and usually gives a better estimate
of the basin of attraction of an asymptotically stable equilibrium point.
Variants of this geometric method yield instability results, as well as means
to establish the stability type of an equilibrium point in the presence of
eigenvalues with zero real parts. As we saw in earlier examples, this latter
case cannot be established from linearization. Despite this praise, however,
we should point out that the direct method of Liapunov is not without
certain serious limitations, as we shall see momentarily.
The main idea behind Liapunov's method is to determine how certain
special real-valued functions vary along the solutions of x = f(x). Let us
begin by defining these functions.
Definition 9.10. Let U be an open subset ofIR2 containing the origin. A
real-valued C l function
V- l (k)={xEIR2: V(x)=k},
are closed curves for small k > O. The projection of these level sets onto
the (Xl, x2)-plane results in concentric ovals encircling the origin; see Fig-
ure 9.5. These remarks are, of course, obvious in the examples above, and
they can be made precise locally for certain classes of functions known as
"Morse functions"; namely, those functions V(x) for which the eigenval-
ues of the Hessian matrix (fPV(X)/OXiOXj) evaluated at local minima are
positive. In this case, the Implicit Function Theorem implies that these
9.3. Liapunov Functions 279
V(x"x 2 ) =c 2
V(x, , x 2) = c,
------++--~--++--------.x,
x,
Figure 9.5. Graph and level curves of a positive definite function V near
the origin.
local minima are isolated. Also, one can show that the level sets of V near
the minima are diffeomorphic to circles; see the Appendix. However, a sat-
isfactory characterization of the level sets of an arbitrary positive definite
function is not available.
In simple situations, the standard choice of positive definite functions
come from homogeneous quadratic polynomials (quadratic forms) in two
variables. Here is an elementary test for positive definiteness of such func-
tions:
Lemma 9.11. A homogeneous quadratic function V(XI' X2) = ax~ +
2bxlX2 + cx~, where a, b, and c are real numbers, is positive definite if
and only if a> 0 and ac - b2 > O.
Proof. We will prove the necessity of the conditions on the coefficients;
the sufficiency follows from similar reasoning. Suppose that V is positive
definite. Since V(Xl, 0) > 0 if Xl f:. 0, we must have a > O. If X2 f:. 0 is
fixed, then V(Xl' X2) > 0 for all Xl and there can be no real zeros Xl of
V(Xl, X2). Thus the discriminant 4(b2 -ac) of this quadratic function must
be negative. <>
Now, we would like to determine how the solutions of x = f(x) cross
the level sets of a positive definite function V. If x(t) is a solution of
x = f(x), then
This expression is simply the inner product of the vector f(x) with the
gradient vector '\7V(x) of V at the point x:
where B is the angle between f(x) and '\7V(x). The gradient vector '\7V(x)
is the outward normal vector to the level curve of Vat x. Thus, if V(x) < 0,
then the angle between f(x) and '\7V(x) is obtuse which implies that the
orbit through x is crossing the level curve from the outside to the inside.
Similarly, if V(x) = 0, then the orbit is tangent to the level curve; if
V(x) > 0, the orbit is crossing the level curve from the inside to the outside.
These three possibilities are shown in Figure 9.6. With these observations,
the following basic theorem of Liapunov is quite plausible:
Theorem 9.12. (Liapunov) Let x = 0 be an equilibrium point ofi. = f(x)
and V be a positive definite 0 1 function on a neighborhood U ofO.
(i) IfV(x) :::; 0 for x E U - {O}, then 0 is stable.
(ii) IfV(x) < 0 for x E U - {O}, then 0 is asymptotically stable.
(iii) IfV(x) > 0 for x E U - {O}, then 0 is unstable.
Proof. Because of the geometric remarks above, we will give a formal proof
of (i) only. Let c > 0 be sufficiently small so that the neighborhood of the
origin consisting of the points with Ilxll :::; c is contained in U. Let m be
the minimum value of V on the boundary Ilxll = c of this neighborhood.
Since V is positive definite and the set Ilxll = c is closed and bounded,
we have m > O. Now choose a 8 with 0 < 8 :::; c such that V(x) < m for
Ilxll :::; 8. Such a 8 always exists because V is continuous with V(O) = O.
If Ilxoll :::; 8, then the solution x(t) of i. = f(x) with x(O) = XO satisfies
Ilx(t)11 :::; c for t 2: 0 since V(x(t)) :::; 0 implies that V(x(t)) :::; V(xO) for
t 2: O. This proves the stability of the equilibrium point at the origin. 0
9.3. Liapunov FUnctions 281
Since the mass, length, and the gravitational constants are positive, V is
positive definite in a sufficiently small neighborhood of the origin. More-
over, from Eq. (9.16), we have
V(x) == o.
Thus, from part (i) of the theorem of Liapunov, the origin is a stable
equilibrium point of the planar pendulum (7.2). (;
In our next example, we attempt to determine a basin of attraction of
an asymptotically stable equilibrium point using a Liapunov function.
Example 9.15. Basin of attraction: Consider the second-order differential
equation
z + 2az + z + z3 = 0,
where a is a constant satisfying 0 < a < 1, which is equivalent to the system
Xl = X2 (9.18)
X2 = -Xl - 2ax2 - x~.
The origin is the only equilibrium point and the eigenvalues of the lineariza-
tion at the origin are -a±i(3, where (3 = J1- a 2 • Consequently, it follows
282 Chapter 9: Near Equilibria
y,
from Theorem 9.5 that the origin is asymptotically stable. Let us now try
to estimate the basin of attraction of the origin by using a Liapunov func-
tion. With the intent of determining a quadratic Liapunov function, we
first put the linear part of the vector field into Real Jordan Normal Form.
If we use the new variables y defined by
p-l =~
(3 ((3a 0)
I'
(9.19)
.(
V Yl, Y2
) = - (2
Yl + Y22) - 1 3
a(3YIY2'
Now, the main task is to determine the largest subset of JR2 containing the
origin where - V is positive definite. This is a difficult task, so let us settle
for the largest such disk. Observe that the level sets of the function + y~yr
are circles about the origin and the level sets of the function a~Y~Y2 are
similar to hyperbolas; see Figure 9.7.
9.3. Liapunov Jilunctions 283
that is, TO = va/3. Thus, every solution of Eq. (9.19) with initial value yO
satisfying lIyo II < TO approaches the origin as t ---t +00. The circle of radius
TO becomes an ellipse when transformed back to the original variables.
With a little more work, one could obtain a slightly larger subset of
the basin of attraction of the origin. However, it is clear that - V is not
positive definite on all of lR2 • One could also try other Liapunov functions
to obtain possibly larger subsets of the basin of attraction of the origin.
Since it is not clear what to try, the important question remains: How
large is the basin of attraction of the origin? We will answer this question
after we uncover additional properties of Liapunov functions in the next
section. 0
There is a chapter in the theory of Liapunov functions that is usually
referred to as the conveTse theorems of Liapunov. The basic premise of
these results is that if an equilibrium point is, for example, asymptotically
stable, then there exists an appropriate Liapunov function with the proper-
ties listed in (ii) of Theorem 9.12. Although it may be of limited practical
use, such a result is of considerable theoretical interest, as we shall see
in Chapter 13. In this spirit, let us reprove Theorem 9.5 on asymptotic
stability of an equilibrium point from linearization.
Example 9.16. A Liapunov function fOT an asymptotically stable equilib-
rium point: Let us suppose that the initial transformation of the variables
in the proof of Theorem 9.5 have been made and consider the system
where A, which is the Jacobian matrix, has eigenvalues with negative real
parts, g(O) =·0 and Dg(O) = o. We now construct a Liapunov function
that implies the asymptotic stability of the origin for Eq. (9.20).
Our Liapunov function will be chosen as a quadratic form V (x) =
xTBx, where the symmetric matrix B satisfies A TB + BA = -I. Lemma
8.17 implies that there is such a V that is positive definite. Then, for
Eq. (9.20), we have
V(x) = _XT X + 2xT Bg(x).
For any m > 0, there is a 8 > 0 such that IIg(x)II ~ mllxll if Ilxll ~ 8. Let
{3 be the largest eigenvalue of B, then
I
/
I
I
I
I
\ u
\
\ I
\ /
"'- //'
........ _----/
Figure 9.S. Instability theorem of Cetaev.
If m < 1/(2f3), then -V is positive inside the disk U = {x : Ilxll < 8}.
Therefore, the quadratic function V as constructed satisfies the hypothe-
ses of Theorem 9.12 (ii) and thus the origin is an asymptotically stable
equilibrium point of Eq. (9.20). <>
We conclude this section with an embellishment of the theorem of
Liapunov. The instability part of Theorem 9.12 has the deficiency of con-
sidering a full neighborhood of the origin and thus is not applicable to
equilibria of saddle type. The following theorem of Cetaev, which is remi-
niscent of the latter part of the proof of Theorem 9.7, is a way to remedy
this shortcoming:
Theorem 9.17. (Cetaev) Let U be a sufficiently small open neighborhood
of the origin. If there is an open region n and a C1 function V : n - IR.
with the properties
(i) the origin is a boundary point of n;
(ii) V(x) = 0 for all x on the boundary points ofn inside U;
(iii) V(x) > 0 and V(x) > 0 for all x E n n U,
then the origin is an unstable equilibrium point.
Sketch of Proof. We have illustrated in Figure 9.8 a typical situation de-
scribed by the theorem. From the property (i), there are points in n, hence
in U, that are arbitrarily close to the origin. From (ii) and (iii), no orbit
starting from anyone of these points in n can cross the boundary of n in U.
Thus, also from (iii), such orbits must leave the neighborhood U through
n. Consequently, the origin is an unstable equilibrium point. <>
Example 9.18. Instability with Cetaev: As an application of the theorem
of Cetaev, let us consider the system of differential equations
·32
Xl = Xl +X2 X 1
X2 = -X2 + X~,
9.3. Liapunov Thnctions 285
which has an equilibrium point at the origin. Notice that since the eigen-
values of the linearized vector field at the origin are 0 and -1, none of
the theorems in the previous two sections is applicable. Therefore, let us
consider the function V(Xb X2) = xV2 - xV2 and the open region
Observe that V(x) > 0 for x E fl, and V(x) = 0 on the boundary. Next, we
compute the derivative of V along the solutions of the differential equations
above:
V(Xl' X2) = xt - X2(X~ - x~) + x~.
Xl = xi + XIX2
X2 = -X2 + x~ + XIX2 - xi.
Hint: Consider the function V(XI, X2) = xi!4 - x~/2 and show
X2 = -2X2 + 3xi.
Answer: xi!2 + x~/4 = 1/9.
9.4. An Invariance Principle 287
f:
9.20. Indirect control: Suppose that 'IjJ : JR -+ JR, u I--> 'IjJ(u) , is a C 1 function
satisfying 'IjJ(0) = 0, u'IjJ(u) > 0 if u =f. 0, and 'IjJ(s) ds -+ +00 as lui -+
+00. For k, c, and p positive constants with kp > 0, show that every solution
of the indirect control problem
x = -kx - e, e= 'IjJ(u),
approaches zero as t -+ +00.
The label "indirect control" comes from the fact that in the system above
e
the control variable is not given directly as a function of the state variable
X; instead, it is determined indirectly using another differential equation. In
certain situations, indirect control turns out to be very efficient; on related
matters, see, for example, Lefschetz [1965].
Lemma 9.21. If a positive orbit "1+ (xO) [respectively, "1- (XO)] is bounded,
then the w-limit set w(xO) [respectively, a(xO)] is a nonempty, compact,
and connected invariant set.
Proof. The fact that w(XO) is nonempty and compact are relatively easy
consequences of the definition of omega-limit set. Let us establish the key
property of invariance of w(xO). If y is in w(xO), then there is a sequence
{tj}, satisfying tj -+ +00 as j -+ +00, such that cp(tj, xO) -+ y as j -+
+00. Consequently, for any fixed t in (-00, +00), we have cp(t + tj, XO) =
cp(t, cp(tj, XO)) -+ cp(t, y) as n -+ +00 from the continuity of cpo This shows
that the orbit through y belongs to w(xO) which establishes the invariance
of w(XO).
We now show that w(xO) is connected. Suppose that w(xO) is not
connected so that there are two nonempty, disjoint, closed sets WI (XO)
and W2(XO) such that w(XO) = WI (XO) U W2(XO). Then there exist two dis-
joint open sets U1 and U2 with Wl(XO) C U1 and W2(XO) C U2. Let {tj}
be a sequence with cp(tj, XO) E U1, and {Tj} be another sequence with
cp(Tj, xO) E U2. We can choose these sequences so that Tj < tj < TjH.
Then there must exist a sequence {Sj} satisfying tj < Sj < TjH such that
cp(s;. XO) f/. U1 U U2. But there is a limit point of the sequence {cp(Sj,xon
which is not in w(xO) and this is a contradiction; see Figure 9.9. 0
With the terminology above, we now state the main theorem of this
section.
Theorem 9.22. (Invariance Principle) Let V be a real-valued function
and let U == {x E R2 : V(x) < k}, where k is a real number. Suppose
further that V is continuous on the closure U of U and is CIon U with
V(x) ::; 0 for x E U. Consider the subset S of U defined by
S == {x E U : V(x) = O}
9.4. An Invariance Principle 289
and let M be the largest invariant set in S. Then every positive orbit that
starts in U and remains bounded has its w-limit set in M.
Proof. For XO E U, let <p(t, XO) be the solution through XO and suppose
that it is bounded for t ~ o. Then V(<p(t, xO)) ~ 0 and thus V(<p(t, xO)) ~
V(XO) ~ k for all t ~ o. Consequently, <p(t, xO) E U for all t ~ O. Moreover,
V(<p(t, xO)) ---+ c, where c is a constant, as t ---+ +00. The continuity of V
implies that V(y) = c for any y E w(XO). Since w(xO) is invariant, we
have V(<p(t, y)) = c for all t E lR. Thus V(<p(t, y)) = 0 for all t E lR and
w(xO) C S. Now, the invariance of w(XO) implies that w(xO) C M. <>
For practical applications, we state several important consequences of
the Invariance Principle:
Corollary 9.23. If, in addition, every positive orbit is bounded, V is
positive definite, and M consists of the origin, M = {O}, then the origin
is asymptotically stable and all of U belongs to its basin of attraction. <>
Corollary 9.24. If, in addition, every positive orbit is bounded and V is
positive definite for x E U - {O}, then M = {O}, that is, the origin is
asymptotically stable and all of U belongs to its basin of attraction. <>
Theorem 9.25. If V : lR2 ---+ lR is a C l function such that V(x) ---+ +00
as Ilxll ---+ +00 and V(x) ~ 0, for all x E lR2, then every positive orbit is
bounded and has its w-limit set in M, the largest invariant set in {x E
lR2 : V(x) = O}. <>
so that the orbits of Eq. (9.18) lie on the level sets of this function; see
Figure 9.10. The parameter a, when a > 0, causes the system to lose
energy. Therefore, it is natural to compute V to see how this happens. A
simple computation yields
Let us now apply the corollaries of the invariance principle. Since V(x) ---+
+00 as I xii ---+ +00, for any k, each of the sets U in the statement of the
Invariance Principle is bounded. Moreover, it is clear that the set S is
contained in the xl-axis. Now, we determine the largest invariant set M
290 Chapter 9: Near Equilibria
Figure 9.10. Level sets of the Liapunov function V(Xl, X2) = ~ (xi + x~) +
~xt and the phase portrait of Eq. (9.18).
In order to bound the positive orbits of the damped pendulum, we put its
flow on the cylinder 8 1 X JR. As such, the flow of the damped pendulum has
two equilibria (0, 0) and (n, 0) on the cylinder. Recall from the discussion
of the flow of the undamped pendulum on the cylinder in Example 7.21
that the level sets of V are closed curves. A positive orbit of the damped
pendulum crosses a level curve of V from above to below if X2 > 0, and
from below to above if X2 < O. Consequently, all positive orbits of the
damped pendulum on the cylinder are bounded. Now, what we need is
an extension of the Invariance Principle to the cylinder; luckily, such an
extension is valid but we refrain from presenting the details. Therefore, we
conclude that the w-limit set of each orbit of the damped pendulum on the
cylinder is one of the two equilibrium points.
Finally, it is important to observe that the local phase portraits of the
undamped and damped penduli are qualitatively the same in a sufficiently
small neighborhood of the equilibrium point (n, 0). Globally, however, the
homo clinic loop of the undamped pendulum is now broken for the damped
pendulum. Breaking of homoclinic loops playa significant role in dynamics
and we will investigate it further at a later time. <:;
9.23. Consult a classic book: Decipher the equivalent statement and the proof of
Lemma 9.21 as given on page 198 of the classic book of Birkhoff [1927].
9.24. Where is Van der Pol's periodic orbit? Consider the Lienard form of Van
der Pol's oscillator:
where ,\ > O. The only equilibrium point is at the origin and it is unsta-
ble. We will show in Chapter 12 that this system has a unique periodic
orbit, encircling the origin, which attracts all other orbits except the origin.
Herein, we determine how far away this periodic orbit must lie from the
origin. Observe that reversing time, t f-+ -t, is equivalent to taking ,\ < o.
Take'\ < 0 and use the Liapunov function V(XI, X2) = xV2+x~/2 to show
that the basin of attraction of the origin contains the interior of the disk
xi + x~ < 3. Conclude that when ,\ > 0, the periodic orbit must lie in the
exterior of this disk.
292 Chapter 9: Near Equilibria
:i:t = X2
X2 = -2Xl - aX2 - 3x~,
Xl = -Xl
. 2 (9.21 )
X2 = X2 +x l .
This system has a unique equilibrium point at the origin, and its lineariza-
tion at this point is the canonical linear system
. (-1° 0)
x = 1 x,
X21
/--!-~WU(O,U)
~ '-.
lI \\
,/
/ U
(
I \
: • t 1 I
JL
\f
Figure 9.11. The local stable and local unstable manifolds of Eq. (9.21)
and its linearization near the origin.
where >'1 < 0, >'2 > 0, and the function g == (gl, g2) satisfies g(O) = 0,
Dg(O) = O. Notice that the local stable and unstable manifolds of the
linearization at the origin are, respectively, the Xl- and x2-axis. Let us
now suppose that our differential equation has been put into the normal
form (9.22). Then we have the following theorem:
Theorem 9.29. For the system (9.22), there is a 8 > 0 such that, in the
neighborhood U == {(XI, X2) : IXll < 8, IX21 < 8} the local stable and
local unstable manifolds of the equilibrium point at the origin are given by
We will not give a proof of this theorem because some of the necessary
technical details are beyond the intended scope of our book. Instead, we
will devise a constructive method for approximating, say, the function hs
to any desired accuracy in specific differential equations. To be of practical
use, such a method, unlike in Example 9.28, should not depend on the
explicit knowledge of the flow. We will accomplish this by deriving a scalar
differential equation whose solution satisfying hs(O) = 0 is the local stable
manifold of the equilibrium point at the origin. If we differentiate with
respect to t the defining equation X2(t) = hs (Xl(t)) given by the theorem
above, then we obtain
. dh s .
X2 = -Xl·
dXl
If we now substitute the expressions for Xl and X2 given by the differential
equation (9.22), and use hs for the X2 variable, then the solution of the
initial-value problem
(9.24)
296 Chapter 9: Near Equilibria
(9.25)
(9.26)
(9.27)
1 1
= 2b2x2 + 3fb3x2 + ... ,
2 3
hu (X2 )
thus recovering the same results we obtained earlier from explicit solu-
tions. <>
Let us now return to the beginning of this section and try to generalize
the concept of local stable and unstable manifolds of an equilibrium point
x. If we do not confine our attention to a local neighborhood U of x, then
we are led to the following definition of two invariant sets:
9.5. Preservation of a Saddle 297
Definition 9.31. The global stable manifold WB(x), and the global unsta-
ble manifold WU(x) of an equilibrium point x are defined, respectively, to
be the following sets:
WB(x) == {xO Em? : <pet, XO) -+ x as t -+ +oo},
WU(x) == {xO Em? : <pet, xO) -+ x as t -+ -00 }.
It is clear from this definition that in Example 9.28, we have, in fact,
determined the global stable and unstable manifolds of x = 0, because in
this case WB(X, U) = WB(x) n U and WU(x, U) = WU(x) n U. Unfor-
tunately, this is not true in general; global stable and unstable manifolds
can be very complicated and come arbitrarily close to themselves. This
excludes a global analog of Theorem 9.29, as seen in the example below.
Example 9.32. Global stable and unstable manifolds: Let us reexamine
Example 7.27 at >. = 0 in light of these remarks. The origin is an equi-
librium point of saddle type; thus, by Theorem 9.29, the local stable and
unstable manifolds are graphs over the local stable and unstable manifolds,
respectively, of the linearized equations. The global stable and global un-
stable manifolds of the origin, on the other hand, are not graphs because the
entire homoclinic orbit is part of both of these manifolds; see Figure 9.12.0
Overlapping of global stable and unstable manifolds usually leads to
nonlocal bifurcations. In higher dimensions, this is a cause of complicated
dynamical behavior. Despite the difficulties associated with these mani-
folds, unraveling their geometry is essential in the study of complicated
dynamical systems, as we shall see in later chapters.
We conclude this section with several remarks on the definition and
computational aspects of global stable and unstable manifolds. It is rather
difficult to determine these global manifolds using Definition 9.31 since it
appears that one must search through the entire plane for appropriate ini-
tial data. An alternative, but equivalent, way is to obtain the global mani-
folds from the corresponding local ones; to obtain WB(X), let the points in
WB(x, U) flow backward in time, and to obtain WU(x), let the points in
W U (x, U) flow forward in time:
WB(x) == U<pet, WB(X, U)),
t:50
WS{O)
x, x,
WS(O,U)
x,
Figure 9.12. Homoclinic orbit is part of both the global stable and global
unstable manifolds of the equilibrium point.
This method, coupled with the approximation procedure for local stable
and unstable manifolds we have presented earlier, provides a reasonably
practical way to compute the global stable and unstable manifolds.
Exercises - - - - - - - - - - - - - - - - - - - - .. 0.0
9.28. Consider the planar systems
(a) :b = -Xl, X2 = X2 + x~;
(b) Xl = X2, X2 = Xl + x~.
Show that the origin is a saddle point. Determine the stable and unstable
manifolds of the origin for the linearized as well as the nonlinear systems.
Sketch and compare the phase portraits of the linearized equations about
the origin and the nonlinear equations.
9.29. Find approximations for the local stable and unstable manifolds of all saddle
points of the following differential equations:
9.5. Preservation of a Saddle 299
has served as a model for interaction between local population growth and
global dispersion. A solution u(x, t) of this partial differential equation is
said to be a travelling wave solution with wave speed c > 0 if there is a
function v : IR ---+ IR such that u(x, t) = v(x - ct) for all x E IR and t E IR.
A travelling wave solution must satisfy the second-order ordinary differential
equation
v" + cv' + rv(1 - v) = 0,
-8u = -8
2u
+ u(1 - u)(u - a)
8t 8x 2 '
where 0 < a < 1/2, has a travelling wave solution which approaches zero as
t ---+ -00 and 1 as t ---+ +00. Also, show that it is the unique travelling wave
solution with these properties.
Suggestion: The ordinary differential equation for the travelling wave will
have the equilibrium points (0, 0) and (1, 0) as saddle points. You will
need to use the fact that stable and unstable manifolds depend continuously
on parameters and also observe how level sets of the conservative system
v" + u(1 - u)(u - a) = 0 are crossed.
where A1, A2 are positive numbers, and the function g satisfies g(O} =0
and Dg(O} = O. Let V be the quadratic function
V=Q V=Q
e
x,
f'
3. There is a point pe on the side e such that the solution <p(t, Pe) through
Pe remains in the square for all t ~ O. There is also a point Pe' on e'
with similar properties.
Hint: Consider two pieces of the side e such that solutions through
one piece leave the square via the side f, and the solutions through
the other piece leave the square via the side f'. Use continuity with
respect to initial data.
4. Show that <p(t, Pe} ---> 0 as t ---> +00.
Hint: Use the fact V(<p(t, Pe}} ---> 0 as t ---> +00.
5. The point Pe is unique on e. There is also a unique point Pe' on e' with
similar properties.
9.6. Flow Equivalence Near Hyperbolic Equilibria 301
Hint: If qe is any other such point distinct from Pe, then show that
V(Pe - qe) > 0 and V(cp(t, Pe) - cp(t, Pe)) > 0 for all t. Now, since
cp(t, Pe) -> 0 as t -> +00, conclude that cp(t, qe) eventually leaves the
square.
6. Show that the curve W = ,+(Pe) U {O} U ,+(Pe/) is a graph over the
xl-axis.
Hint: Take smaller squares.
7. Finally, establish that W above is 0 1 . Therefore, W is the local stable
manifold of the origin, that is, W = WI~c(O).
Hint: Consider the function V"1(x) = ~ (-xi! Al + 'lJx§I A2) depending
on'IJ > 0, and compute V"1 and let 'IJ -> o.
Exercises _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ .,.0. 0
9.33. Find two specific nonlinear planar differential equations with the following
properties: each has a unique equilibrium point, the linearized vector fields
at these equilibria are topologically equivalent, but the flows of the two
nonlinear vector fields on the whole plane are not topologically equivalent.
9.34. Try to prove the theorem of Grobman-Hartman in the special case where
f(O) = 0 and the eigenvalues of Df(O) have negative real parts.
Hint: Borrow ideas from Example 9.16 and the proof of Theorem 8.15.
Xl = A + 2XlX2
(9.28)
X2 = 1 + x~ - x~
depending on a parameter A.
Typical phase portraits of this system for three different values of the
parameter are depicted in Figure 9.14. For A = 0, there are two hyperbolic
equilibrium points at (0, 1) and (0, -1), both of which are saddle points.
Also, the X2-axis is invariant, that is, any orbit starting on the axis stays
on the axis. In particular, the orbit of any initial point lying on the X2-axis
and in between the two equilibria has its a-limit set as the saddle point
(0, -1) and its w-limit set as the other saddle point (0, 1). Thus, the system
(9.28) for A = 0 has a heteroclinic saddle connection.
For IAI =I 0, and small, there are still two saddle points, lying on
opposite sides of the x2-axis, but the saddle connection is no longer present.
Consequently, the bifurcation that occurs as A moves away from zero is
called breaking a saddle connection. <>
We will return to saddle connections in Chapter 13, and further explore
how they effect the global behavior of planar flows at large.
Exercises _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . 0 . <>
9.35. Breaking saddle connections: For the following systems show that, for k = 0,
there are saddle connections, and for k =1= 0, there are no saddle connections:
(a) :h = X2, X2 = - sinxi - kX2;
(b) Xl = ksinxi + Sinx2, X2 = - sin Xl + ksinx2.
A= -0.1
i 1\
--------- ---------~.
./
I \ , ----------~-==--------
--------~
)
~/
A = 0.0
/
---::.----- _..A/' ......._--
I.'-,.~ _ _ -
.. ----_.----
I i
I I
A = 0.1
presentation in Section 5.3, and the converse theorems of stability for spe-
cial types of vector fields. Lyapunov [1947], with a spelling variation, is
the original source; Cetaev's instability theorem is from Cetaev [1934]. La
Salle and Lefschetz [1961] contains an elementary exposition; Yoshizawa
[1966] has general converse theorems and historical references.
The Invariance Principle in Section 9.4 is due to La Salle [1960], al-
though similar ideas were developed independently by others; see, for exam-
ple, Krasovskii [1963]. This principle is used widely in applications and has
special prominence in infinite dimensional dynamical systems as explained
in Hale [1977], Henry [1981]' and Sell [1971].
A nice proof of Theorem 9.29 on the existence of stable and unstable
manifolds is in Irwin [1980]; see also Palis and de Melo [1982]. In the
exercises, we outlined a procedure for obtaining the existence of the stable
manifold of a saddle point by exploiting the fact that, if orbits exit on two
opposite sides of a box about the saddle point and enter on the other two
sides, then some solution remains in the box. This is a special case of a
principle formalized by Wazewski in 1947; precise formulations are in Cesari
[1963] and Hartman [1964]. An extension of this idea is the concept of an
"isolating block" of an invariant set, the existence of which is equivalent
to the property that an invariant set is isolated. Further generalizations
yield an index theory which is important in detecting when there are orbits
connecting two different invariant sets; see Conley [1978] and Rybakowski
[1987].
The linearization theorem near hyperbolic equilibria in Section 9.6 is
due to Grobman [1959] and Hartman [1964]. Another proof, which gener-
alizes to infinite dimensions, can be found in Palis and de Melo [1982].
10-----
In the Presence
of a
Zero Eigenvalue
10.1. Stability
In this section, we show how to determine the stability of a nonhyperbolic
equilibrium point of a planar vector field with one zero and one negative
eigenvalue. Since linear approximation is of no help in this pursuit, we will
have to examine how a particular nonlinear term of a vector field affects
the flow near such a nonhyperbolic equilibrium point. Let us begin with a
familiar example which encapsulates the essence of the general situation.
Example 10.1. Let k 2': 1 be an integer, a I:- 0 be a real number and
consider the product system
•
Xl = aXIk
X2 = -X2·
Regardless of the values of a and k > 1, the eigenvalues of the linearized
equation about the equilibrium point at the origin are always 0 and -1.
Consequently, to determine the stability type of the origin we need to
investigate the effect ofthe nonlinear term of the vector field. Since X2(t) ---;
o as t ---; +00, the stability properties of the equilibrium point x = 0 are
determined by the first scalar equation Xl = ax~. It is now evident that
the origin is asymptotically stable if a < 0 and k is odd, and unstable
otherwise; see Figure 10.1. 0
We now turn to the general setting. Let f be a given C k function, with
k 2': 1,
f : lR? ---; lR2 ; x 1-+ f(x),
satisfying
f(O) = 0, Df(O) =0 (10.1)
and consider the planar system of differential equations
Xl = h(xI, X2)
(10.2)
X2 = -X2 + h(xI, X2).
To bring the linear part of this system to the forefront, let us write it, for
a moment, in vector notation:
x = (~ ~1) x + f(x).
Notice that the linear part of the vector field about the equilibrium point
at the origin is in Jordan Normal Form with eigenvalues 0 and -1. In ap-
plications, the linearization of a vector field with one zero and one negative
eigenvalues may not always come in normal form [Eq. (10.2)]; however,
such a vector field can always be put into this form with a linear change of
coordinates and a rescaling of the independent variable t.
10.1. Stability 309
\ )
( '\
) ~
\ (
Figure 10.1. Phase portraits of Xl = axt X2 = -X2 near the origin for
a = -0.1 and a = 0.1.
implies that there is a constant {j > 0 and a unique C 1 function 7/J : {Xl
IX11 < 8} ---+ {X2 : IX21 < {j} such that
(10.3)
The latter relation in Eq. (10.4) follows from differentiating Eq. (10.3) with
respect to Xl and setting Xl = O.
The following theorem, which generalizes the observations in Exam-
ple 10.1, is not difficult to establish, except perhaps the arithmetic of "big
0" for which you may wish to consult the Appendix:
Theorem 10.2. Suppose that f = (h, fz) is a Ck+1 function with
(10.5)
Y1 = gl (Y1, Y2)
(10.6)
Y2 = -Y2 + g2(Y1, Y2),
where
gl(Y1, Y2) = h(Y1, 7/J(Y1) + Y2)
g2(Y1, Y2) = fz(Y1, 7/J(yd + Y2) - fz(Y1, 7/J(yd)
-7/J'(Y1)h(Y1, 7/J(yd + Y2).
The stability properties of the equilibrium point y = 0 of Eq. (10.6) are
the same as those of the equilibrium point x = 0 of Eq. (10.2).
Since the conclusions of the theorem concern a sufficiently small neigh-
borhood of the origin, we proceed, as you might suspect, to determine the
first several terms of the Taylor series of these functions about the origin.
Using Eqs. (10.5) and (10.4), we obtain, as Ilyll ---+ 0,
Theorem 10.4. Let 'l/J(xd be defined as in Eq. (10.3). Then the equi-
librium x = 0 of Eq. (10.2) is stable [respectively, asymptotically stable,
unstable] if and only if the equilibrium Xl = 0 of the scalar differential
equation
Xl = h(xt. 'l/J(XI)) (10.9)
is stable [respectively, asymptotically stable, unstable]. <>
Xl = /l(Xl, X2)
X2 = P,X2 + h(Xl, X2),
°
where f(O) = 0, Df(O) = 0, and P, "I- is a given constant. Discuss the
stability properties of the equilibrium at the origin.
Hint: Rescale the independent variable t, but be aware of the sign of p, when
interpreting flows.
10.4. Consider the planar system x = Ax + f(x), where f(O) = 0, Df(O) = 0,
and the matrix A has one zero and one nonzero eigenvalue. Describe how
you would reduce the discussion of the stability of the origin to the case
considered in the previous problem.
10.5. As a specific instance of the problem above, determine the stability type of
the origin of the system
10.2. Bifurcations
In this section, we extend the foregoing setting to systems of differential
equations that depend on parameters. Our goal is, of course, to investigate
the possible bifurcations of nonhyperbolic equilibrium points with one zero
and one negative eigenvalue. Initially, we will be content to account for
local variations in the number of equilibria and their stability types. Since
we are working in the plane, other changes in phase portraits are potentially
possible. We will address such issues in the next section.
Let F be a given Ok function, k ~ 1, depending on parameters A,
satisfying
F(O, x) = f(x) and f(O) = 0, Df(O) = o. (10.10)
Below, we consider the system of differential equations
satisfying
81/;
1/;(0,0) = 0, -8 (0,0) = 0 (10.12)
Xl
such that, for each X, with IIXII < AD, a point (Xl, X2), with IX11 < 8 and
IX21 < 8,_is an eql!.ilibrium point of Eq. (10.11) if and only if X2 = 1/;(X, Xl)
and F 1 (A, Xl, 1/;(A, Xl)) = O.
Proof. Apply the Implicit Function Theorem to find X2 = 1/;(A, Xl) as the
solution of -X2 + F2(A, Xl, X2) = O. <>
Definition 10.6. Let 1/; be as in Lemma 10.5. Then the Ok function
G(>., xd defined by
(10.13)
10.2. Bifurcations 315
(10.14)
Xl = A + xi
X2 = -X2,
where A is a scalar parameter. The equilibrium points of this system are
given by x = (Xl, X2) with X2 = 'ljJ(A, xd = 0 and Xl satisfying the bi-
furcation equation G(A, Xl) = A + xi = O. There are no solutions of the
bifurcation equation if A > 0, and two solutions Xl = ±R if A < O. The
corresponding equilibria of the scalar differential equation Xl = G()., xd
are hyperbolic, one of which is asymptotically stable and the other of which
is unstable. <>
In the case where the equilibria of the scalar differential equation
(10.15) are hyperbolic, it is not difficult to establish the stability types
of the corresponding equilibria of the planar system (10.11).
Theorem 10.8. Let 'ljJ(A, Xl) be as in Lemma 10.5 and x = (Xl, 'ljJe>.., xd)
is an equilibrium point of Eq. (10.11) with 11>-11 < AD, IX11 < 8, and
1'ljJ(>-, x1)1 < 8. Tben AD and 8 can be cbC?sen small enough so that
(i) x is a byperbolic stable node if 8G(A, x1)/8x1 < 0,
(ii) x is a saddle point if 8G(>-, xd/8x1 > O.
Proof. It is convenient to begin by making the transformation of variables
and (tiI, 112) = (Xl, 0) is an equilibrium point. The Jacobian of this vec~or
field evaluated at (Xl, 0) is given by the matrix (for the sake of brevIty,
variables are omitted)
aY2
@. )
-1 + ffi _ 01/; @.
aY2 lfih aY2
To determine the stability type of the equilibrium point (Xl, 0) it suf-
fices to discern the signs of the real parts of the eigenvalues of A(X, Xl)'
As we have shown in the exercises, the eigenvalues of A(X, xd vary con-
tinuously in X and Xl. Since the eigenvalues of A(O, 0) are 0 and -1, for
sufficiently small X and Xl, one eigenvalue of A(X, xI) is near 0 and the
other near -1. We now need to determine the sign ofthe eigenvalue near O.
The determinant of the Jacobian A(X, Xl) is
oFl
oA (0, 0, 0) =1= 0,
then there is a saddle-node bifurcation at A = 0, that is, when >..~ o;~ <
0, there are two hyperbolic equilibria, one saddle and the other an asymp-
totically stable node, and no equilibrium when >.. 0:;'1 0;~1 > O.
Proof: Compute the first several terms of the Taylor series of the bifurcation
function G(>", xd, then use the results from Section 2.3. 0
To make our exposition of the theory of bifurcation functions complete,
we should state a generalization of Theorem 10.8 when the equilibria of
Eq. (10.15) are not hyperbolic. It is considerably more difficult to prove
this extension, but here is the statement of the general situation.
10.2. Bifurcations 317
(10.16)
If the rod of the pendulum is rigid, then one can interpret the application
of the torque as pushing the pendulum with constant force that is perpen-
dicular to the rod. We would like to analyze the possible bifurcations of
the equilibrium positions of such a pendulum as a function of the value M
of the torque. To utilize the general results above, we have to transform
Eq. (10.16) into the normal form (10.11). For this purpose, we begin by
converting Eq. (10.16) into the first-order planar system
ill = Y2 (10.17)
Y2 = -Y2 - sinYI + M,
where YI = e and Y2 = 0. Since sinYI is periodic with period 271", we confine
our discussion to the values of YI in the interval [-71", 71"].
The equilibrium points of Eq. (10.17) are given by Y2 = 0 and sin YI =
M. For M > 1, there are no equilibrium points; when M = 1, there is a
single equilibrium point at (71"/2,0); for M < 1, there are two equilibrium
points. It appears that the equilibrium point at (71"/2, 0) undergoes a bi-
furcation at the parameter value M = 1. To analyze the behavior of the
orbits ofEq. (10.17) near M = 1 and (Xl, X2) near (71"/2, 0), let us use the
translation of variables
M=l+>..,
Zl = Z2 (10.18)
Z2 = -Z2 - cos Zl + 1 + >..
318 Chapter 10: In the Presence of a Zero Eigenvalue
To put this matrix into Jordan Normal Form, observe that its eigenval-
ues are 0 and -1 with the corresponding eigenvectors (1, 0) and (-1, 1).
Therefore, if we make the transformation of variables
z=Px, P= (1 -1)
0 1 '
x= (00 0) + p-l (
-1 x
0 )
-COS(Xl-X2)+1+>.'
or equivalently
Xl = - COS(Xl - X2) + 1 + >.
(10.19)
X2 = -X2 - COS(Xl - X2) + 1 + >..
The system (10.19) is now in the normal form (10.11) and the hypothe-
ses of Theorem 10.9 are satisfied. Therefore, at >. = 0 the system (10.19)
undergoes a saddle-node bifurcation; when>. < 0, there are two equilibria,
one an asymptotically stable node and the other a saddle, and there are no
equilibria when >. > o.
For future reference, as well as for practice, let us also compute the
bifurcation function for Eq. (10.19). From the second equation, we first
need to determine the function 'IjJ(>., Xl) satisfying
(10.20)
(10.21)
10.2. Bifurcations 319
From the form of the vector field (10.19) and Eq. (10.20) it is clear that the
bifurcation function is G(>", Xl) = 'l/J(>.., Xl). Therefore the scalar differential
equation (10.15) becomes
(10.22)
Exercises _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ .. ~. <:;
10.6. Draw the bifurcation diagrams for the equilibrium solutions of the following
systems:
(a) :h = -X2, X2 = AXI - X2;
(b)Xl=X2, x2=-x2+xi-A;
(c) Xl = 3Axl - 3Ax2 - xi - x~, X2 = AXI - X2·
10.7. Analyze the equilibrium points and their stability types of the system
Xl = AXI - xi + 2XIX2
X2 = (A - 1)x2 + x~ + XIX2,
Xl = fL - xr - XIX~ + XIX2
X2 = -X2 + x2xi - xi + A.
Find the approximate bifurcation curves near the origin in the (A, fL)-plane
for the equilibria near (0, 0). Also sketch the representative phase portraits
for each region of the parameter space.
10.9. Odd symmetry: Suppose A is a scalar parameter and the functions Hand
F2 in Eq. (10.11) satisfy, for i = 1, 2,
°
2. Show that the bifurcation function C (A, Xl) is odd in Xl·
3. If 8 2C(0, 0)/8A8xl i- and 8 3 C(0, 0)/8xr i- 0, show that there is a
pitchfork bifurcation at A = 0.
320 Chapter 10: In the Presence of a Zero Eigenvalue
:h = h(Xl, X2)
X2 = /1-X2 + VXl + h(Xl, X2),
where f(O) = 0, Df(O) = 0, and /1- i= 0 and v i= 0 are given constants. Dis-
cuss the stability properties of the equilibrium at the origin. First, be daring
and compute the bifurcation function without putting the linear part into
normal form. Then normalize the linear part and compute the bifurcation
function again. How do the two cases compare?
10.12. Bifurcation from a simple eigenvalue: In a neighborhood of the origin, obtain
those values a = a*(x~) such that the following system has an equilibrium
point on the line Xl = x~:
to use the method of bifurcation function to find an eigenvalue P,1 near p,~
for small>' =F O. A number p, = p,~ + v is an eigenvalue of A(>') if and only if
there is a nonzero vector x = (Xl, X2) such that [A(>') - p,~I - vI] x = 0,
which is equivalent to the pair of equations
.
Xl = aX3I (10.24)
X2 = -X2,
where a is a given real number. The origin is an equilibrium point for all
values of a and the linearization at the origin is
. (0 0) x.
x= 0 -1
if Xl < 0
if Xl = 0
if Xl > 0
Theorem 10.14. Let the vector field (10.2) be C k and consider a suf-
ficiently small neighborhood U of the origin in JR2 . Then there exists a
local center manifold we in U consisting of the graph of a C k function
h(XI) = X2. Moreover, there are positive constants Q and (3 such that, for
any solution x(t) with initial value x(O) E U, the estimate
(10.25)
(10.26)
. = ~
X2 oh (Xl ).Xl·
UXI
If we now substitute the expressions for Xl and X2 given by the differential
equation (10.2), then a solution of the partial differential equation
(10.27)
h(O) = 0, ~(O) =0
OXI
yields a center manifold defined by h(XI). The partial differential equation
(10.27) cannot, of course, be solved for h in most cases. Therefore, we o~t
for a sufficiently accurate approximate solution to facilitate a local analysls
324 Chapter 10: In the Presence of a Zero Eigenvalue
of the flow. To accomplish this, we expand h(Xl) into a power series in the
variable Xl as
(10.28)
{)h . 3
-h(xd = -() (XdaXl· (10.29)
Xl
satisfying
g(O) = 0, (){)g (0) = 0
Xl
which is C k , with k ~ 1, in a neighborhood of the origin, consider the
function M(g)(Xl) defined by
Then one can prove the following approximation result for any center man-
ifold:
10.3. Center Manifolds 325
In the comfort of this theorem, let us return to Figure 10.2 and deter-
mine the power series of the apparent center manifold resembling a cubic
curve.
Example 10.18. Continuation of Example 10.3: The power series solution
of the partial differential equation (10.27) in this case is given by
Thus the flow on the center manifold is determined by the scalar differential
equation
Xl = ax~ + xlh(Xl) = ax~ - xt + O(lxlI 5 ).
It is interesting to notice that the vector field of this scalar differential
equation has the same terms up through order four as the ones we have
obtained earlier using the method of bifurcation function. 0
We now generalize the theory of center manifolds to systems of differ-
ential equations (10.11) which depend on parameters, and investigate the
possible bifurcations near the origin for small ).. This extension may appear
formally to consist of insertion of a ). or two into the previous definitions
and theorems. For the sake of completeness, we will make such insertions.
From a geometric point of view, however, center manifolds become consid-
erably more complicated in the presence of parameters: for each small ).
there is a curve, and the collection of these curves form a surface--a center
manifold. Here is the precise definition:
Definition 10.19. A family ofCk curves WHO, U) in a neighborhood U
of the origin is said to be a local center manifold for Eq. (10.11) if
• Wf(O, U) is invariant under the Bow of Eq. (10.11), that is, ifx(t) is
a solution of Eq. (10.11) with the initial value x(O) E WHO, U), then
x(t) E WHO, U) as long as x(t) E U;
• WHO, U) is a graph of a Ck-function h()', Xl) = X2 and, for)' = 0, is
tangent to the Xl -axis at the origin, that is,
The existence theorem for an attracting center manifold for Eq. (10.11)
reads as follows:
Theorem 10.22. Let the vector field (10.11) be C k and consider a suf-
ficiently small neighborhood U of the origin in JR2 • Then, for 11)'11 small,
there exists a local center manifold W~ in U consisting of the graph of a C k
function h()', xd = X2. Moreover, for any solution x(t) with initial value
x(O) E U there are positive constants Q and f3 such that
(10.31)
Corollary 10.24. The omega limit set w(xO), if it exists, of any orbit of
Eq. (10.11) with initial value xO E U is an equilibrium point. 0
This corollary enables us to "fill in" the phase portraits of Eq. (10.11) near
the origin rather easily because it rules out the existence of, for example,
periodic orbits. Another practical implication, which is considerably more
than a mere corollary, of Theorem 10.22 is the following:
Theorem 10.25. An equilibrium point of Eq. (10.11) in U is stable [re-
spectively, asymptotically stable, unstable] if and only if the corresponding
equilibrium point of the scalar differential equation
(10.32)
. 8h (\ ) .
X2 = -8 1\, Xl Xl·
Xl
(10.34)
where the coefficients Cij are to be determined. We now substitute this
series into the partial differential equation (10.33) and determine Cij by
equating the coefficients of like terms.
Although center manifolds are not unique, the computational effective-
ness of the power series method above can again be demonstrated. Suppose
that
10.3. Genter Manifolds 329
satisfying
g(O, 0) = 0, og (0,0) = 0
OXl
Then one can prove the following approximation result for any center man-
ifold:
Theorem 10.26. Let h(A, xt} be a center manifold of Eq. (10.11). Sup-
pose that M(g)(A, Xl) = O((lAI + IXll)k) as (A, Xl) --+ 0, where k > l.
Then as (A, Xl) --+ 0,
The terms up through order two are the same as the ones we have obtained
in Example 10.11 with the method of bifurcation functions. Therefore,
using the center manifold theory it is now quite easy to construct the full
flow of the system of equations (10.19) on the (Xl, x2)-plane. To recover
the dynamics of the pendulum in the original coordinates (Yt. Y2), all that
330 Chapter 10: In the Presence of a Zero Eigenvalue
(10.36)
From Theorem 10.15, we also know that the same statement is true relative
to the equilibria of the equation
Exercises - - - - - - - - - - - - - - - - - - - - .~. 0
10.14. Many center manifolds: Find all center manifolds of the system
•
Xl = Xl2
X2 = -X2.
10.3. Center Manifolds 331
Xl = ,x+X~
X2 = -X2
in the (,x, Xl, X2)-space. Put ,x = 0 first.
0.16. No analytic center manifold: Consider the system
.
Xl = -Xl3
X2 = -X2+X~.
This vector field is, of course, analytic. Show that this system has no analytic
center manifold.
Hint: Let h(Xl) = I:T='; Cixl and determine that C2 = 1, Ci = 0 for i odd,
Ci+2 = iCi for i even .
.0.17. Show that the equilibrium point at the origin of the system
10.20. On the machine: Plot on the computer, using PHASER, for example, some
representative phase portraits of the damped pendulum with torque and
rotated pendulum to observe the bifurcations.
II~I
11 _ _ _ __
In the Presence of
Purely Imaginary
Eigenvalues
11.1. Stability
As we have seen in Example 9.9, when the eigenvalues of the linearized vec-
tor field at an equilibrium point are purely imaginary, the local dynamics
about the equilibrium point cannot be determined by the linear approxi-
mation. Indeed, depending on the nonlinear terms, the equilibrium can be
unstable, stable, or even asymptotically stable. Consequently, we need to
investigate the effects of the nonlinear terms in each particular situation.
In this section, we show how to carry out such an investigation by reducing
the dynamics in the neighborhood of a nonhyperbolic equilibrium point
with purely imaginary eigenvalues to the dynamics of a 27r-periodic scalar
differential equation.
Let us begin by recalling briefly the dynamics of Example 9.9.
Example 11.1. Consider the planar system
Xl = X2 + aXI(xI + x~)
(11.1)
X2 = -Xl + aX2(xI + x~),
where a is a given real number. Regardless of the value of the constant a,
the origin is an equilibrium point and the eigenvalues of the linearization
at the origin are ±i. If we introduce polar coordinates (r, ()) defined by
(11.3)
This is a rather special product system. Since iJ > 0, the orbits spiral
monotonically in () around the origin. Therefore, the stability type of the
origin of Eq. (11.1) is the same as that of the equilibrium point r = 0 of
the radial equation r = ar3. Of course, r = 0 is asymptotically stable if
a < 0, stable at a = 0, and unstable for a > 0; see Figure 11.1. <>
Unlike the example above, planar systems for which the linearization
near an equilibrium point has purely imaginary eigenvalues do not always
turn out to be product systems when transformed into polar coordinates.
However, with some care, we can still pursue the line of reasoning in this
example and reduce the problem to the analysis of a 27r-periodic, rather
than autonomous, scalar differential equation. To be specific, let f be a
given Ok function, k ::::: 2,
Figure 11.1. For Eq. (11.1), origin is always an equilibrium point with
purely imaginary eigenvalues; however, it is asymptotically stable for a =
-0.5, stable for a = 0.0, and unstable for a = 0.5.
336 Chapter 11: In the Presence of Purely Imaginary Eigenvalues
satisfying
f(O) = 0, IIDf(O)11 < 1, (11.4)
where the norm of the Jacobian matrix IIDf(O)1I is a nonnegative real num-
ber such that IIDf(O)xll ~ IIDf(O)llllxll for all x E JR2 • We should remark
that if we were interested in only stability of equilibria it would be suffi-
cient to prepare the differential equation by appropriate transformations
to reduce it to the case f(O) = 0, Df(O) = 0 so that f contains terms of
degree two and higher. We have chosen to impose the lesser restriction on
the norm of the derivative of f for the purpose of bifurcation studies which
we will undertake in the next section. Now, consider the planar system of
differential equations
To bring the linear part of this system to the forefront, let us write it, for
a moment, in vector notation:
Notice that the linear part of the vector field about the equilibrium point at
the origin is in Jordan Normal Form with eigenvalues ±i. In applications,
the linearization of a vector field with purely imaginary eigenvalues may
not always come in normal form [Eq. (11.5)]; however, such a vector field
can always be put into this form with a linear change of coordinates and a
rescaling of the independent variable t.
Since the linear part has rotational symmetry, it is reasonable to in-
troduce polar coordinates (11.2) to investigate the dynamics of the system
(11.5) in a sufficiently small neighborhood of the origin. As a result of a
short calculation, we see that, in polar coordinates, Eq. (11.5) becomes
r= ~(r, 0)
(11.6)
iJ = 1 + 8(r, 0),
where
~(r, 0) =!I(rcosO, -r sin 0) cosO - h(rcosO, -rsinO) sinO
1
8(r, 0) = - - [!I (r cos 0, -r sin 0) sin 0 + h(r cos 0, -r sin 0) cos 0]
r
°
(11. 7)
when r =f. 0, and at r = we define
dr
dO = R(r, 0), (11.9)
where
lR(r, 0)
R(r, 0) = 1 + 8(r, 0)
R(O, 0) = o. (11.10)
The solutions of Eq. (11.9) give the orbits of Eq. (11.5). We can also
recover the solutions of Eq. (11.5) as a function of time from the solutions
of Eq. (11.9) by following the steps below:
• Fix ro and find the solution r(O, ro) of Eq. (11.9) satisfying the initial
value r(O, ro) = roo The orbit of Eq. (11.5) through the point xO =
(ro, 0) is then given by
• The solution x(t) of Eq. (11.5) through the point xO = (ro, 0) is then
given by
Xl(t) = r(O(t), ro) cosO(t) (11.13)
X2(t) = -r(O(t), ro) sinO(t).
338 Chapter 11: In the Presence of Purely Imaginary Eigenvalues
Figure 11.2. For Eq. (11.14), the origin is an asymptotically stable equi-
librium point with purely imaginary eigenvalues.
The procedure used in the example above can easily be extended to the
general case of Eq. (11.5). All that is needed is the transformation theory
from Section 5.2 to convert the scalar equation dr / dO to the equation dp/ dO
for which the lowest order term in the Taylor expansion of the vector field
has a nonzero constant coefficient. For Eq. (11.9), it is possible to show
that this lowest order term must always be odd (see the exercises). With
these remarks, the following result is immediate.
Lemma 11.3. Suppose that f = (/1, h) is a C 2 k+2 function in Eq. (11.5)
with the corresponding transformed scalar equation
. {o
r -
-
2
_e- 1 / r sin(l/r)
ifr =°
otherwise
£1 = 1.
The Taylor series of dr/dB at the origin is identically zero. It is easy to see
that the origin is stable; yet, it is not a center. To wit, notice that there
are infinitely many concentric periodic orbits encircling the origin with
amplitudes, or radii, (k7r)-l, where k is any positive integer. The periodic
orbits with amplitudes [(2k + 1)11"]-1 are asymptotically stable while the
ones with amplitudes [(2k + 2)11"]-1 are unstable. Consequently, the origin
is stable but not asymptotically stable. <.;
This concludes the stability analysis of an equilibrium point whose
linearization has purely imaginary eigenvalues. The method of polar co-
ordinates and reduction to the scalar equation dr/dB also provides an ef-
fective tool for studying periodic orbits of Eq. (11.5) about the origin. In
the remaining part of this section, we study such periodic orbits. The
observations below are almost self-evident.
Lemma 11.6. There is a bounded neighborhood U of the origin in JR2
such that each periodic orbit r of Eq. (11.5) lying in U encircles the origin;
also, ifxo = (ro, 0) E r with ro > 0, then the solution r(B, ro) of Eq. (11.9)
satisfying the initial value r(O, ro) = ro is 211"-periodic in B. Conversely, if
r(B, ro) is a 211"-periodic solution of Eq. (11.9), then the orbit r(xO) with
xO = (ro, 0) of the planar system (11.5) is a periodic orbit. The minimal
period T of such a r is the first value of t for which the solution B(t) of
Eq. (11.12) satisfies
B(T) = 211". (11.18)
The reduction of the discussion of the orbits of Eq. (11.5) near the ori-
gin to the discussion of solutions of Eq. (11.9) has important consequences
for limit sets of orbits of Eq. (11.5). The result below is a special case of the
Poincare-Bendixson Theorem, a fundamental result which we will explore
more fully in the next chapter:
11.1. Stability 341
Xl = X2
X2 = -Xl - x~.
field in your computations. Also, show that there is a first integral. Find
one and draw the level curves near the origin.
11.4. Discuss the stability and instability of the equilibrium points of the system
Xl = -X2 +X~
X2 = -Xl + X~.
Note especially the equilibrium point at (1, 0).
11.5. Always odd power: Suppose that the transformation theory of Section 5.2 is
applied to Eq. (11.9) to obtain dp/d8 = a,l +o(lplj) with a i= O. Show that
j is an odd integer.
Hint: If j is even, then the origin is unstable for Eq. (11.5). Replacing 8 by
-8, one obtains the same result. Show that this is a contradiction.
11.6. Coo vs. analytic: Find a Coo, but not analytic, example of Eq. (11.5) so
that the transformation theory applied to dr/d8 leads to dp/d8 = 0 yet the
origin is unstable.
11.7. Symmetry and center: Consider the second-order differential equation x+
X2 + X = 0, or the equivalent system
Xl = X2
. 2
X2 = -Xl - X2·
has the form yet) = csin(ct + d), where c and d are constants. Notice
the different periods of different periodic solutions. Discuss the stability
properties of each solution as well as the stability properties of the orbits on
the plane.
11.11. Hyperbolicity and asymptotic phase: Let r be an orbitally asymptotically
stable periodic orbit corresponding to a 21r-periodic solution 'IjJ(O) of the
scalar equation (11.9). If 'IjJ(O) is hyperbolic, show that r is orbitally asymp-
totically stable with asymptotic phase as follows:
1. Without loss of generality, take 'IjJ(O) = 0, that is, the periodic orbit
corresponds to the zero solution of Eq. (11.9).
2. Prove that, for ro small, there are positive constants a and k such that
Ir(O, ro)1 ::; ke- a9 for 0 ~ O.
3. If e(t) = O(t) - t, then e(t) = f:(t) n(r(O, ro), 0) ~ dO, where Idt/dOI ::;
2, if ro is small.
4. There is a constant K such that In(r(O, ro), 0) ~ I ::; K e- a9 , for 0 ~ O.
5. For any c > 0, there is a To > 0 such that le(t) - e(r)1 < c if t, r ~ To.
Thus, there is a v > 0 such that e(t) - v --+ 0 as t --+ +00.
Then, in any neighborhood U ofthe origin in rn.2 and any given 'xo > 0 there
is a Xwith IXI < 'xo such that the differential equation x = A(X)x+F(X, x)
has a nontrivial periodic orbit in U.
It is remarkable that the essential hypotheses of the theorem concern
only the linear part of the vector field. The requirement that the vector
11.2. Poincare-Andronov-Hopf Bifurcation 345
Xl = X2 + F(A, r2)xI
(11.20)
X2 = -Xl + F(A, r2)x2'
r= F(A, r2)r
e = 1.
(11.21 )
Xl = X2
(11.22)
X2 = -Xl + 2AX2 - X~X2.
----x
(a)
(b)
-~-----~-------x
(c)
Figure 11.3. Bifurcation diagrams of Eq. (11.20) for three different func-
tions F: (a) for F(A, r2) = A is degenerate; (b) for F(A, r2) = A - r2 is
supercritica1; and (c) for F(A, r2) = _(r 2 - C)2 + c2 + A is subcritical.
periodic orbit near the origin for some small values of >., as seen in Figure
11.4. To gain insight into the dynamics of Eq. (11.22), let us perform some
detailed computations which are also indicative of the general situation.
Using polar coordinates and the transformation theory presented in the
previous section, we will show below that, as the eigenvalues cross the
348 Chapter 11: In the Presence of Purely Imaginary Eigenvalues
A" -0.1
A" 0.1
A" 0.3·
imaginary axis, the origin gives up its stability to a periodic orbit. More
specifically, we will demonstrate that, for each small A > 0, there is a unique
nontrivial periodic orbit r>. near the origin which is orbitally asymptotically
stable. Moreover, r>. --t 0 as A --t O.
Let Fl (A, x) = 0 and F2(A, x) = 2AX2 - X~X2. Then there is a AO > 0
such that
IIDF(A, 0)11 < 1 for IAI < AO
and thus the conditions (11.4) for reduction to a 27r-periodic scalar equation
dr / dfJ are satisfied as long as IAI < AO. Indeed, for the oscillator of Van der
Pol, Eq. (11.9) is given by
da 1 ()
dfJ = '8 cos 4fJ ,
(11.25)
Notice that the p2 term is not present. The Poincare map IT of Eq. (11.25)
is the map
(11.26)
350 Chapter 11: In the Presence of Purely Imaginary Eigenvalues
To find the periodic solutions of Eq. (11.25) near the origin, we need to
locate the fixed points of its Poincare map (11.26); equivalently, solve the
equation
[A + O(A2)]pO - [i
+ O(A)]pg + O(P6) = O.
The solution Po = 0 of this equation corresponds to the equilibrium
point at the origin. The other solution satisfies the equation
or
Po = v'8J:\ + O(A). (11.28)
The latter equation (11.28) corresponds to a fixed point of the Poincare map
and the derivative of the Poincare map at this fixed point is 1- 2A + 0 (A 2 ).
For small positive A, this derivative has absolute value less than one which
implies that the fixed point, hence the corresponding periodic solution, is
asymptotically stable.
The results of these computations on the oscillator of Van der Pol
can be summarized most conveniently in the bifurcation diagram depicted
in Figure 11.5, where the approximate radius Po of the periodic orbit is
plotted as a function of the parameter A. The equilibrium point at the
origin is included as a periodic orbit as well. The solid curve represents
the asymptotically stable periodic orbit and the dashed line the unstable
solution. As A is increased through A = 0, the origin gives up its stability
to the periodic orbit which grows in amplitude proportional to J:\. <>
We now turn to a proof of the Poincare-Andronov-Hopf bifurcation
theorem. To make the subsequent computations manageable, it is conve-
nient to put the linear part of the vector field in a simpler form. Using a
linear change of variables, we may transform the matrix of the linearized
vector field at the origin to the matrix
a(A) f3(A)) .
(
-f3(A) a(A)
From the assumption f3(0) =I- 0, we can change the time variable so that
f3(A) = 1 for IAI small. Also, the assumption (da/dA)(O) =I- 0, in conjunc-
tion with the Inverse Function Theorem, implies that there is a one-to-one
11.2. Poincare-Andronov-Hopf Bifurcation 351
Figure 11.5. The bifurcation diagram of Van der Pol's oscillator: approxi-
mate radius of the periodic orbit, which is asymptotically stable, as a func-
tion of the parameter A.
correspondence between a(>.) and >.. This permits us to use a(>.) as the
parameter rather than >.. As a result of all these transformations, we may
assume that the linear part of the vector field at the equilibrium point is
of the form
satisfying
F(>', 0) = 0, DxF(>., 0) = 0, (11.29)
• For IAI < AO and IT - 27l"1 < 80 , every T-periodic solution x(t) of
eq. (11.30) satisfying Ilx(O)11 = a and Ilx(t)11 < ao must be given by
the function x*(t, a), except for a possible translation in phase.
Proof. Using the technique of reduction from the previous section, we
compute that the 27l"-periodic scalar equation (11.9) for the system (11.30)
has the form
dr
d(} = Ar + peA, r, (}), (11.31)
with
peA, 0, (}) = 0, DrP(A, 0, (}) = O. (11.32)
From Lemma 11.6, we need to investigate the 27l"-periodic solutions of
Eq. (11.31). If rCA, (), a) is a solution of Eq. (11.31) with initial value
rCA, 0, a) = a, then rCA, () + 27l", a) = rCA, (), a) for all () if and only if
rCA, 27l", a) = a. From the variation of the constants formula, the solutions
of Eq. (11.31) satisfying rCA, 0, a) = a are given by
(1- e- 27rA ) a + io r
27r
e- AS peA, rCA, s, a), s) ds = O. (11.34)
Using the Implicit Function Theorem, we will show that the values of
a and A satisfying this equation is a curve in the (a, A)-plane and that this
curve is a graph over the a-axis expressed by A as a function of a. From
Eq. (11.32), a = 0 satisfies Eq. (11.34) because the integrand vanishes. This
trivial solution corresponds to an equilibrium solution. To find nontrivial
periodic solutions, let us consider the function h(a, A) given by
h(a, >.*(a)) = O.
Now, with this >.*(a), the function r*(O, a) == r(>.*(a), 0, a) is a 271"-
periodic solution of Eq. (11.30). Consequently, the orbit through the point
xO(a) = (a, 0) given by
O*(T*(a), a) = 271".
In particular, we have T*(O) = 271". If we now define
then it is not difficult to see that x* satisfies the conditions of the theo-
rem. <>
The stability type of the periodic orbits in the theorem above can be
inferred from the derivative of the function >.*(a) when this derivative is
not zero; see Figure 11.6. More specifically, we have the following theorem
with a somewhat technical proof:
Theorem 11.16. Let >.*(a) be the function given in Theorem 11.15 and let
ra be the corresponding periodic orbit of Eq. (11.30). Then, for sufficiently
small a = ii, the periodic orbit r a is orbitally asymptotically stable if
d>.*(a)jda > 0, and unstable if d>.*(a)jda < O.
Proof. In this proof we will continue to use the notation developed in the
proof ofthe previous theorem. For fixed a, let A= >.*(a). Using the solution
a a
...........
"
--------------~-----------A ------~----------A
Supercritical Subcritical
Figure 11.6. Two typical graphs of the function ).*(a). Stability type of a
periodic orbit r a with amplitude a can be inferred from the bifurcation di-
agram: ra is orbitally asymptotically stable if d).*(a)jda > 0, and unstable
ifd).*(a)jda < 0.
(11.37)
To make the notation manageable in subsequent computations, it is con-
venient to rewrite this equation as
or
1- e- 27rA *(a) + g()..*(a), a) = O. (11.39)
The existence of such a function 9 follows from Eq. (11.32). Now, differen-
tiating Eq. (11.38) with respect to a and then putting a = a yields
+ da
d (ag()..,
- I
a)) a=a + da
d (ag()..*(a), I
a)) a=a = O.
Combining the first and the third "terms" in the equation above in con-
junction with Eq. (11.36), and combining the second and the fourth terms
in conjunction with Eq. (11.39), we arrive at
Now, from the relations (11.40) and (11.41), the desired conclusion
is self-evident. <:;
As a reward of this somewhat intricate proof, you should reexamine
the earlier bifurcation diagrams in this chapter. In particular, you should
Q.
differentiate the bifurcation curve (11.27) of the oscillator of Van der Pol.
Xl = Xl + X2
X2 = (A - 2)XI + (A - 1)x2 - xf - X~X2.
In this example, you should first put the linear part for A = 0 in Jordan
Normal Form.
11.15. Center in Hamiltonian systems: Consider a Hamiltonian function of the
form
H(XI, X2) = ~ (x~ + x~) + 0 ((lxll + IX21)2)
.
XI=-
aH
aX2
.
X2=--
aH
aXI
has a center at the origin. This is easy to show using the method of Liapunov
functions. Establish the existence of a center from the Poincare-Andronov-
Hopf bifurcation theorem.
Hint: Consider the one-parameter system
for A in a neighborhood of zero. Now verify that this system satisfies the
hypotheses of the bifurcation theorem and that there are no periodic orbits
except for A = O. For further information on this approach, see Schmidt
[1978] and his exposition in Marsden and McCracken [1976].
11.3. Computing Bifurcation Curves 357
dp
dO ~ [Ck(>', 0) - 80(>',
= L...J 8bk 0) ] p.
k
k=l
Now, choose the functions bk so that
~: (>., 0) = Ck(>', 0) - Ck(>'), (11.44)
Ck(>') = -
1 1271'
Ck(>', s) ds.
211' 0
With these choices of bk , the differential equation (11.42) formally reduces
to the following differential equation with constant coefficients:
+00
dp = L Ck(>') pk. (11.45)
dO k=l
Here is an important practical fact about this differential equation:
358 Chapter 11: In the Presence of Purely Imaginary Eigenvalues
Lemma 11.17. The formal power series (11.44) contains only the odd
powers of p, that is,
+=
dp ,,_ (') 2k+l (11.46).
de = ~ C2k+l /\ P .
k=l
. _ (a(A) {3(A))
x - -{3(A) a(A) x + F(A, x), (11.47)
dp _ [a' (0) 2]
de - {3(0) A + O(A) p -
[C ] 3 4
{3(0) + O(A) p + O(p ), (11.48)
where a'(O) = (dajdA)(O) and c is some constant. From the form of the
Poincare map of Eq. (11.48) and Theorem 11.16, the following result on the
bifurcation curve A(a), where a is the approximate amplitude, ofEq. (11.47)
should be evident:
11.3. Computing Bifurcation Curves 359
a a
-- , ,
......
\
------r-------- A --A
a a
/ ..... ... --
I
/
~---~-A
Exercises _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ .. y>. 0
11.16. FitzHugh Neuron Model: In 1961, Fitzhugh proposed the following system
as a model of nerve impulse transmission:
where 0 < a < 1, I' > 0, b > 0, and q are constants; H(t) is the Heaviside
function H(t) = 0 for t < 0 and H(t) = 1 for t 2: O. In this model, q
represents the stimulus, Xl represents the response (instantaneous turning
on of sodium permeability through the nerve membrane), and X2 represents
a recovery variable (turning on of potassium permeability). Suppose that
I' ::; 3(a 2 - a + 1)-1 and show that there is a critical value qo of q such that
a Poincare--Andronov-Hopf bifurcation occurs at qo. Discuss the stability
of the resulting periodic orbit. For further information on this important
model, see FitzHugh [1961] and Jones and Sleeman [1983].
11.17. Chemical instabilities and sustained oscillations: Consider the chemical re-
action
A-+X
B+X-+Y+D
2X +Y -+ 3X
X-+E,
11.3. Computing Bifurcation Curves 361
where the initial and the final concentrations ofthe chemicals A, B, D, and
E are constant, and the concentrations of X and Y satisfy the differential
equations
x = a - (b + l)X + X2y
Y=bX_X2y
for some positive constants a and b. If you have not studied chemistry, just
begin with this system of differential equations and explore their dynamics.
1. Show that (a, b/a) is an equilibrium point which is stable for a2 + 1 > b
and unstable for a 2 + 1 < b.
2. Fixing a and using b as a parameter, show that the conditions of the
Poincare-Andronov-Hopfbifurcation theorem are satisfied at b = a2 +1
for the differential equations Xl = X - a and X2 = Y - b/a.
3. For b = a 2 + 1, determine the stability properties of the solution
(Xl, X2) = (0, 0).
4. Find the approximate formula of the bifurcation curve for the periodic
orbits.
For further information on this problem, see Lefever and Nicholis [1971].
11.18. A predator-prey model: Consider the system
. = rX1 (1X
Xl - -l ) - -
!3X1X2
--
k 0+X1
X2 = SX2(1- ~),
VX1
where all the parameters are positive. In this model, the predator X2 be-
comes satiated during periods of abundance and whose carrying capacity is
proportional to the amount of prey, Xl, available. For more information on
these equations, see May [1973].
(i) Show that there is exactly one equilibrium point x at which the preda-
tor and prey coexist, that is, both coordinates of the equilibrium point
are positive.
(ii) Show that there is a value of the parameter s at which the equilibrium
point x undergoes a Poincare-Andronov-Hopf bifurcation.
(iii) If you try to determine the stability properties of the resulting periodic
orbit, be aware. The computations are very lengthy; this may be a
good place to explore symbolic manipulations on the machine.
11.19. A generic bifurcation: Show that in the system below there is a generic
Poincare-Andronov-Hopf bifurcation from the equilibrium point (1, 1) at
A = 0:
Xl = Xl [1 + ~A2 - HX1 - 1 - A)2 - X2]
X2 = X2( -1 + Xl).
Determine whether the bifurcation is subcritical or supercritical.
11.20. A nongeneric bifurcation: Consider the differential equation, in polar coor-
dinates, given by
r = r( _2A2 + 3Ar 2 - r 4 )
9 = 1.
362 Chapter 11: In the Presence of Purely Imaginary Eigenvalues
x + (,\ + x)x + x + x 2 = 0,
where the damping coefficient changes sign only once. For a more general
discussion of such equations, see Obi [1954].
and the references therein for this important application of the Method of
Liapunov-Schmidt to the study of bifurcations of periodic orbits.
Poincare--Andronov-Hopf bifurcation has appropriate generalizations
for partial differential equations which can be found in, for example, Henry
[1981]' Kielhofer [1979], or Marsden and McCracken [1976].
12-_ _ __
Periodic
Orbits
(12.1)
12.1. The Poincare-Bendixson Theorem 367
where>. is a scalar parameter. There are two equilibrium points; the one
at (0, 0) is a saddle for all values >., and the other at (2/3, 0) is a source
(unstable) if >. < 0 and a sink (stable) when>. > O. Notice that at >. =
0, these equations essentially coincide with the Hamiltonian system-the
fish-whose Hamiltonian is given in Eq. (7.34). Indeed, the perturbation
terms are chosen so that the homoclinic loop of the fish remains intact
under the perturbation. To see this, we compute the derivative of the
function
V(Xl, X2) = x~ - x~ + x~
along the solutions of Eq. (12.1) to obtain
Thus, for all values of the parameter >., the curve xi -x~ +x~ = 0 is invariant
for the flow; this curve is, of course, the homoclinic loop containing the
origin.
Let us now examine the flow inside the homo clinic loop. Using the
Invariance Principle, it is not difficult to show the following: when >. <
0, the w-limit set, and when>. > 0, the a-limit set of any point inside
the homoclinic loop, except the equilibrium point (2/3, 0), is part of the
homoclinic loop. With a little more work, using some of the ideas to be
presented in Section 12.3, one can show that these limit sets are, in fact, the
entire homoclinic loop. Typical phase portraits of Eq. (12.1) depicting these
situations are shown in Figure 12.1. If you try to locate the homoclinic loop
as a limit set of an orbit on the computer, for any finite time the "computed
limit set" looks like a periodic orbit. However, if you watch carefully, you
will notice that the orbit moves ever so slowly as it passes closer to the
origin, and it takes a long time to make one round about the loop. <>
Let us now turn our attention to periodic orbits and isolate an impor-
tant special case of Theorem 12.1 which suggests a way to determine the
existence of a nontrivial periodic orbit that is not necessarily close to an
equilibrium point.
Theorem 12.5. (Poincare--Bendixson) If w(xO) is a bounded set which
contains no equilibrium point, then w(xO) is a periodic orbit. <>
In order to use the Poincare--Bendixson theorem to exhibit the exis-
tence of a nontrivial periodic orbit, one could attempt to construct an open
bounded set 1) in JR.2 which contains no equilibrium point and such that
any solution that begins in 1) remains in 1) for all t 2': 0, that is, 1) is an
open and bounded positively invariant set. Next, for any xO E 1), one also
shows that w(XO) contains no points on the boundary of 1). Then, since
1) contains no equilibrium points, w(xO) must be a periodic orbit. Let us
illustrate these remarks on a concocted example.
368 Chapter 12: Periodic Orbits
,= -1
,=0
),=5
Figure 12.1. Phase portraits of Eq. (12.1). Notice that the homoc1inic
loop is an w-limit set when>. < 0 and an a-limit set when>. > O.
12.1. The Poincare-Bendixson Theorem 369
Figure 12.2. Partial phase portrait of Eq. (12.2) near the boundary of
the annulus ~ < xi + x~ < 1.
Xl = X2
(12.2)
X2 = -Xl + x2(1 - x~ - 2x~).
Observe that the origin is the only equilibrium point of Eq. (12.2). There-
fore, we will attempt to construct an annular region 'D with the desired
properties mentioned above. To accomplish this, we compute the derivative
of the function V(Xl' X2) = (xi + x~)/2 along the solutions of Eq. (12.2):
Since V(Xl' X2) 2': 0 for xi + x~ < ~, and V(Xl' X2) :::; 0 for xi + x~ > 1,
any solution which starts in the annulus ~ < xi + x~ < 1 remains in this
annulus for all t 2': O. Since the origin is not in the closure of this annulus,
from the Poincare-Bendixson theorem, there exists at least one periodic
orbit of Eq. (12.2) in the annulus; see Figure 12.2. <)
In general, there may be several difficulties in utilizing the theorem of
Poincare-Bendixson to locate periodic orbits of specific differential equa-
tions. First, unlike the example above, it is usually a nontrivial task to
construct a region V with the desired properties. Second, to determine the
number of periodic orbits in V one often has to uncover special properties
of the differential equation.
One particularly important case is when we can ascertain that there is
at most one periodic orbit in V, then there will be exactly one, say r, and
w(XO) = r for all XO E V. Since 1'+ (XO) spirals with increasing time toward
r on one side of rand r belongs to the interior of 'D, it follows that points
xO on both sides of r have their positive orbits approaching r. Thus, r is
370 Chapter 12: Periodic Orbits
Xl = X2
(12.3)
X2 = -Xl + A(l - X~)X2
has a nontrivial periodic orbit for all values of the scalar parameter A.
Proof. We will consider the case A > O. When A = 0, the equation becomes
the linear harmonic oscillator. The case A < 0 can be reduced to the
first case by reversing time. To show the existence of a periodic orbit
we will invoke the Poincare-Bendixson theorem. For this purpose, we will
construct a positively invariant region bounded by a closed curve encircling
the origin. Since the origin is the only equilibrium point and the eigenvalues
of the linearized equations at the origin have positive real parts, no orbit,
except the origin itself, can have the origin as its w-limit set. This implies
that there must be a periodic orbit inside the positively invariant region.
We now begin the construction of a simple (without self-intersections)
closed curve K which will form the outer boundary of a positively invariant
region. The idea for constructing the curve K is to begin at a point A
on the negative X2-axis and use the special properties of the vector field
to obtain a curve lying in the left half-plane which intersects the positive
x2-axis at a point E and such that the angle between the tangent vector
to this curve and the vector field (12.3) is in the interval (0, 71"). Since
Eq. (12.3) is symmetric with respect to the origin, we can also define the
reflection of this curve through the origin and obtain points A' and E'. If
A' > E', then the curve AEA' E' A will be a suitable curve K.
The precise construction of K consists of piecing together various curve
segments. First, we draw an auxiliary curve Q,
(12.4)
To construct the first piece of our curve K, we take a point A = (0, xg)
on the negative x2-axis sufficiently far away from the origin, and follow the
orbit of the system
Xl = X2
(12.5)
X2 = .x(1 - XDX2
passing through the point A. An easy integration shows that this orbit
intersects the line Xl = -1 at the point B = (-1, -2.x/3 + xg). Along the
arc AB we have
-Xl + .x(1 - XDX2 _ .x(1 - X~)X2 = _ Xl < 0;
X2 X2 X2
(12.6)
emanating from B until the orbit, which is a circular arc with its center
at the origin, hits the component of the curve (12.4) in the upper left
quadrant. Let C denote the point of intersection; such a point always
exists if the point A is taken sufficiently far from the origin. Along the
curve BC we have
-Xl + .x(1 -
X~)X2
--=--~--=..:--=- + -Xl -_ /\'(I - 2)
Xl < O.,
X2 X2
372 Chapter 12: Periodic Orbits
When x = -1, the left-hand side of Eq. (12.8) is positive, and when Ixi is
sufficiently large the left-hand side is negative; hence, there exists a solution
x~ of Eq. (12.8). Among the solutions of Eq. (12.8) we take the one nearest
to -1. When the point A is sufficiently far from the origin, the point D
lies to the right of C. It is clear that the orbits of Eq. (12.3) are crossing
the segment of the curve (12.4) between C and D from left to right.
To continue our curve K, we follow the orbit of Eq. (12.7) starting
from point D until it hits the x2-axis at a point which we will denote by
E. Since
-Xl + ..\(1 - Xi)X2
X2
the orbits of Eq. (12.3) cross the curve DE from left to right.
The first half of our curve is now constructed and it is ABCDE. To
construct the other half, we observe that Eq. (12.3) is symmetric with
respect to the origin. Let A' B' C' D' E' be the reflection of ABC D E with
respect to the origin. Since D, hence E, is fixed, we may ensure that A'
lies above E by taking A far from the origin. Also, observe that orbits
of Eq. (12.3) cross the curve segment EA' from left to right. From the
symmetry, it is now clear that the region encircled by the closed curve K =
ABCDEA' B'C' D' E' A is positively invariant for the flow of the oscillator
of Van der Pol.
In the proof above, we used a point of tangency D of the curve Q and
the vector field (12.7). The proof could have been completed by choosing
any other point D on the same component of Q. However, the choice in
the proof is optimal in the sense that it allows one to construct the smallest
region containing periodic orbits. More precisely, taking E as fixed by the
construction above, one can vary A so that either the point C also becomes
a point of tangency for Q and the vector field (12.6) or the point E = -A.
This is the optimal K by the method of construction given in the proof. <)
After developing a little more mathematics, we will prove in the next
section that the equation of Van der Pol has in fact a unique nontrivial
periodic orbit which is stable when ..\ > 0, and unstable when ..\ < O.
12.1. The Poincare-Bendixson Theorem 373
then we have
Again, this expression is not of one sign, but do not despair. Notice that
the line X2 = -1 is invariant under the flow, and the vector field crosses
the line Xl = -1 in the same direction. Thus if there is a periodic orbit
it must lie entirely in one of the four regions separated by these two lines.
However, the function div Bf keeps one sign on anyone of these four re-
gions. Consequently, from Dulac's Criterion, our quadratic system has no
periodic orbits. 0
We end this section with the statement of a geometric theorem on the
symbiotic relationship of a periodic orbit with an equilibrium point.
Theorem 12.11. Let r be a periodic orbit enclosing an open set U on
which the vector field is defined. Then U contains an equilibrium point. 0
Exercises - - - - -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ "0" 0
12.1. Determine the phase portraits of the system
Xl = X2
X2 = Xl - 2x~ + oXx2(xi - x~ + x~),
for all values of the scalar parameter oX. What are the possible limit sets?
Hint: Use V(XI, X2) = xi - x~ + x~ and the invariance principle.
12.2. A nonconvex periodic orbit: Show that, when oX is sufficiently large, say,
oX > 10, a periodic orbit of Van der Pol's equation is not a convex closed
curve.
Hint: At the points (-1, X2) with X2 > 0 show that dX2/dxI > 0 and
d2x2/dx~ > o.
Xl = Xl + x~ + x~
X2 = -Xl + X2 + X2X~
has no nontrivial periodic orbit.
12.2. Stability of Periodic Orbits 375
Xl = Xl - XIX~ + X~
X2 = 3X2 - X2X~ + xi
Xl = (2 - Xl - 2X2)XI
X2 = (2 - 2XI - X2)X2
Definition 12.12. The Poincare map or first return map II near a periodic
orbit r is defined to be the map
The points on the transversal section L", have a natural ordering: two
points xO = p + aov and Xl = P + a1 v satisfy xO ;::: Xl if and only if
ao ;::: a1. Using this ordering, the Poincare map II is said to be monotone
if XO ;::: Xl on Lo implies II(xO) ;::: II(xl).
Some of the most basic properties of the Poincare map are listed in
the theorem below:
Theorem 12.13. The Poincare map possesses the following properties:
(i) The Poincare map II near the periodic orbit r is a monotone C1 map.
(ii) The orbit 'Y(xO) of a point xO E Lo is a periodic orbit if and only ifxo
is a fixed point of the Poincare map, that is, II(xO) = xO.
(iii) The periodic orbit r, with pEr, is orbitally asymptotically stable if
II'(p) < 1, and unstable ifII'(p) > l.
Proof. We indicate a graphic reason for the monotonicity of II. The re-
maining assertions are geometrically self-evident from our earlier studies of
scalar maps; therefore, we refrain from giving formal details.
Let xO and Xl be on Lo with XO ;::: Xl. Consider the simple closed
curve Cx ! consisting of the part of the orbit 'Y(xl) between the points xl
and II(Xl) together with the line segment on L", between xl and II(Xl); see
Figure 12.5. Then, from the Jordan Curve Theorem, Cx ! has an interior
and an exterior. Since the orbits cross L", in the same direction, and the
orbits cannot intersect each other, we have II(xO) ;::: II (Xl ) which implies
the monotonicity of the Poincare map. <>
12.2. Stability of Periodic Orbits 377
Recall from Lemma 8.21, the Liouville's formula, that the fundamental
matrix solution X(t) with X(O) = I of the T-periodic linear system (12.10)
satisfies the relation
and cp(t, p) = cp(t + T, p). Since cp(O, p) = cp(T, p) = f(p) and cp(t, p) =
X(t)f(p), we have
X(T)f(p) = f(p). (12.11)
II'(p)v =
aT
aa (p) f(p) + X(T)v. (12.12)
From Eqs. (12.11) and (12.12), it follows that the matrix of X(T) in the
basis {f(p), p} is
X(T) =
(
° -8T(p)/aa)
1
II'(p) .
r= r(l- r2)
(12.14)
iJ = 1,
the periodic orbit is given by r(t) = 1, and the transversal section L be-
comes
L = {(r, ()) E m.+ x Sl:r > 0, () = O}.
12.2. Stability of Periodic Orbits 379
Since the system (12.14) is a product system, it is easy to obtain the general
solution:
The return time for any point on L is 271". Thus the Poincare map is given
by
II(ro) = [1 + (ro2 - 1) e-47r] -1/2 .
This map has a fixed point at ro = 1, which, of course corresponds to the
periodic orbit. The derivative of the Poincare map at this fixed point is
easily seen to be
II'(l) = dII I = e- 47r < 1.
dro ro=l
We will accomplish this by proving that the value of the integral is negative.
Consider the function
V(X1' X2) = .x(1 - xi)x~ = -2.x(1 - xi) [~xi - V(Xb X2)] . (12.15)
Figure 12.6. Phase portrait of Eq. (12.16). The unit circle is an unstable
and nonhyperbolic periodic orbit.
and its periodic solution (Xl(t), X2(t)) = (cost, sint). It is easy to see,
using formula (12.9) in Theorem 12.15, that the periodic orbit is nonhyper-
bolic. The global phase portrait of Eq. (12.16) can easily be determined by
transforming the system to polar coordinates Xl = r cos () and X2 = r sin ():
r = r(l - r2)2
8= 1.
Xl = X2
X2 = 0: (1 - xi - xn X2 - Xl
has an orbitally asymptotically stable periodic orbit for the parameter value
0: = 0.05. This system is used in Franke and Selgrade [1979] as a test case
of a rigorous numerical procedure for locating an orbitally asymptotically
stable periodic orbit of a planar system.
382 Chapter 12: Periodic Orbits
12.8. Oscillators with hyperbolic periodic orbit: Using reasoning similar to the one
in the proof of Theorem 12.18, establish the following more general result:
Consider the second-order equation x + f(x)x + g(x) = 0, where f and g,
are, say, C 1 • Suppose that
(i) f(x) < 0 for Xl < X < X2, and f(x) > 0 for X < Xl or X > X2, where
Xl < 0 < X2;
(ii) xg(x) > 0 for X l' 0;
(iii) G(X1) = G(X2), where G(x) = fox g(x) dx.
Show that every periodic orbit is hyperbolic with n ' (p) < 1. Thus, if there
is a periodic orbit, it is unique. For assistance, see Coppel [1965], p. 86.
Under mild additional assumptions it can be proved that a periodic orbit
actually exists, see Levinson and Smith [1942].
12.9. Hyperbolicity and finiteness: Consider a planar differential equation :ii: =
f(x) with the property that the vector field f is transversal to the unit disk
D and that D is positively invariant. Suppose further that all equilibrium
points and all periodic orbits are hyperbolic and there is no orbit connecting
saddles. Prove that the number of periodic orbits in D is finite using the
following suggested steps:
1. There are only a finite number of equilibria in D.
2. If there were infinitely many periodic orbits, then there must be a
nested sequence of them: (i) /1 ~ /2 ~ ... , or (ii) /1 C /2 C ....
3. Let 9j = /j U (interior ofrj) and suppose that (i) is satisfied and define
S = nj9j. Show that the boundary of S is invariant.
4. S is either a closed orbit or contains an equilibrium point. Show that
this leads to a contradiction.
5. How do you handle case (ii)?
solution cp(A, t, xo) of Eq. (12.17) satisfies cp(A, T(A, xO), xO) E Lt:. There-
fore, we define the Poincare map depending on parameters as II(A, xO) =
cp(A, T(A, xO), XO) mapping Lli into Lt:. The Poincare map II(A, xO) will
be monotone for the same reason that II(O, xO) was in Theorem 12.13. Of
course, periodic orbits near r o correspond to fixed points of II(A, xO).
Now, the general results in Section 3.3 on fixed points of monotone
maps can be applied to the Poincare map II(A, xO). For example, if ro is
hyperbolic, then for each A with IAI small, there is a unique periodic orbit
r>. near r 0 and r>. is also hyperbolic.
When ro is nonhyperbolic, the bifurcations near the periodic orbit r o
are determined from the bifurcations of the Poincare map II(A, xO). The
example below corresponds to a saddle-node bifurcation of the nonhyper-
bolic fixed point of the Poincare map at, for instance, xO = (1, 0).
Example 12.20. A saddle-node bifurcation of periodic orbits: Consider
the planar system
xi + x~ = 1 - v'tanA,
which is a circle of radius less than 1. For the parameter value A = 0, this,
of course, is Example 12.9 which has a single nonhyperbolic (unstable)
periodic orbit at xi
+x~ = 1. If A < 0, then system (12.19) has no periodic
°
orbits because r > and all the solutions, except the origin, go to infinity
as t ---+ +00; see Figure 12.7.
The system (12.19) undergoes other bifurcations at the parameter val-
ues, for example, A = 7f / 4 or A = 7f /2 which you might like to investigate.
If you need assistance, consult Andronov et al. [1973], pp. 212-217. (;
384 Chapter 12: Periodic Orbits
A= -0.1
;\=0
;\ = 0.1
Exercises _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4\;).0
.2.10. A quadratic system with two limit cycles: Consider the quadratic planar
system depending on a parameter >..:
Notice that the parameter A rotates the vector field. For A = 0.8, there is a
stable limit cycle surrounding the equilibrium point (0,0), and an unstable
limit cycle around the equilibrium point (2, 2). Locate these limit cycles
using a computer. To find the unstable one, you should run the solutions
backward with a negative step size. This system is contained in the library
of PHASER under the name hilbert2.
(12.20)
c= -0.1
c = 0.1
and compute its derivative along the solutions of Eq. (12.20) to obtain
For -4/27 < c < 0, using the Invariance Principle, one can see that there
is an orbitally asymptotically stable periodic orbit lying on the curve x~ -
xI -
+ x~ c = 0, with Xl > 0. As c --t 0, the periodic orbit approaches
the homoclinic loop through the origin. For c > 0, the homo clinic loop is
broken and also there is no periodic orbit. This sequence of bifurcations is
illustrated in Figure 12.8. <>
When a homoclinic loop is broken, the birth of a (unique) periodic
orbit can be established under fairly general assumptions; however, we
refrain here from such a formulation. Homoclinic loops play a significant
role in bifurcation theory and we will encounter them again in Chapter 13
when we consider planar flows at large.
:h =X2
X2 = Xl - x~ + AX2 + aXlX2·
Observe that the origin is a saddle point. Fix the value of a at a positive
value. Then experiment on PRASER to convince yourself that there is a
negative value of A at which the system has a homoclinic loop. Now, change
A a little to break the homoclinic loop, and search for the unique periodic
orbit in the proximity of the original homoclinic loop.
Bibliographical Notes _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ~
The details of the central result of Poincare and Bendixson can be found
in many sources; see, for example, Hale [1980], Hartman [1964], and Hirsch
and Smale [1974]. A proof of the Jordan Curve Theorem is in New-
man [1953]. A generalization of the Poincare-Bendixson theory for two-
dimensional surfaces other than the plane are given in Schwartz [1963].
A comprehensive reference on the search for limit cycles, Dulac func-
tions, etc., on the plane is Yeh [1986]. The proof of the existence of a
limit cycle of the oscillator of Van der Pol is from Ye [1986] and that of its
hyperbolicity is from Coppel [1965].
388 Chapter 12: Periodic Orbits
However, the CO distance yields neighborhoods that are 'too large.' For
instance, two vector fields that are CO close may not have the same number
of hyperbolic equilibria; see Figure 13.1. To avoid this undesirable situa-
tion, we introduce the C1 distance by requiring the functions as well as
their derivatives to be close at all points of 1J:
In this definition, we view the derivatives as linear functions and use any
norm on lR4. With the C1 distance, we define the 0 neighborhood of f to be
the set of all vector fields g in Xk(1J), with k ~ 1, satisfying Ilf - gill < O.
The resulting topology on Xk(1J) is called the C 1 topology.
We also will have occasion to employ cr distances by imposing close-
ness conditions on higher-order derivatives,
and use the cr topology on the set of vector fields Xk(1J); we omit further
details.
13.1. Structurally Stable Vector Fields 391
Figure 13.1. Two (scalar) functions that are close in the CO topology but
not in the C 1 topology may have different number of zeros.
Despite the difficulties posed by specific vector fields, the general situ-
ation is less of a concern as "most" planar vector fields in Xk(D) turn out
to be structurally stable.
13.1. Structurally Stable Vector Fields 393
Exercises _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ .~. ()
13.1. A structurally stable system: Show that the planar system
w(U) = n
7;:::0
I'+(cp(r, U)),
where the overline is the closure of the positive orbits of x = f(x) through
all x O E U.
Asymptotic behavior of a dissipative system can be localized to the
study of a flow on a subset of the plane with many desirable properties.
Definition 13.12. A set A is said to be a global attractor of a dissipative
system if A is compact, connected, invariant, and w(U) c A for every
bounded set U.
Theorem 13.13. A dissipative system has a unique global attractor. <>
Using the attractor, one can show the existence of an appropriate
global Liapunov function with the following properties: The Liapunov func-
tion vanishes on the attractor, the level curves surround the attractor, and
19.2. Dissipative Systems 395
the vector field points inside the level curves except the zero level curve.
This is how a proof of Theorem 13.10 goes; however, the existence of the
Liapunov function is just that and one does not generally expect to be able
to construct it.
These discouraging words aside, let us now consider some specific ex-
amples of dissipative systems. We can, of course, recall Van der Pol again,
in which case the global attractor is the limit cycle together with its interior.
We now offer another example of a dissipative system.
Example 13.14. The global attmctor of a damped conservative system:
Consider the second-order equation x + x - x + X3 = 0, or the equivalent
first-order system
Xl = X2
X2 = -X2 + Xl - x~ .
We will show that the global attractor of this system, as seen in Figure 13.2,
is the unstable manifold of the origin together with the equilibrium points
(-1,0) and (1, 0) :
A = WU(O, 0) U {(-I, O)} U {(I, O)}.
We first establish the existence of the global attractor by using the
Invariance Principle, Theorem 9.25, and the Liapunov function
Since V(x) = -x~ ~ 0 and the level curve V-I(c) for c large is a closed
curve, each orbit is bounded. From the Invariance Principle, for any
XO E JR2, the omega-limit set w(xO) is one of the three equilibria (-1, 0),
(0, 0), or (1, 0). The system is therefore dissipative and there exists a global
attractor.
We now determine the structure of the attractor. The equilibrium
points (-1, 0) and (1, 0) are asymptotically stable while (0, 0) is a saddle
point. The set WU (0, 0) U { (-1, 0) } U { (1, 0) } is a compact invariant set
and thus must belong to the global attractor. To establish that this set is
indeed the entire global attractor, we need to show that the alpha-limit set
a(xO) for any point xO E A is an equilibrium point.
396 Chapter 13: All Planar Things Considered
for all nand t E [0, 1]. Taking the limit as t -> +00, we ~ave V('P(t, y)) =
V(y) for t E [0, 1] and, consequently, for t E JR. Thus, V('P(t, y)) = 0 for
t E JR and y is an equilibrium point. <)
This concludes our brief diversion into dissipative systems and we now
return to our main task in this chapter, the generic bifurcations of one-
parameter planar vector fields.
w(U) = U w(xo).
xOEU
Using an example from the previous exercise, show that w(U) =1= w(U).
To motivate our search for generic bifurcations, let us begin our ex-
position with a simple analogy: in three dimensions, for example, a curve
cannot avoid intersecting a surface while going from one side to the other;
a curve, on the other hand, is not likely to intersect another curve or a
point while moving around in three dimensions. A one-parameter fam-
ily of vector fields in Xk(1J) defines a curve in this infinite dimensional
space of vector fields. Suppose that there is a hypersurface----{;odimension-
one submanifold--consisting of structurally unstable vector fields and that
structurally stable vector fields on the two sides of the hypersurface are
topologically different. While varying the parameter, if the curve of vector
fields moves from one side of the hypersurface to the other, then the curve
cannot avoid hitting the hypersurface, but in all likelihood it would miss
subsets of smaller dimensions.
Let us proceed by making the notion of a codimension-one submanifold
of the infinite dimensional space Xk(1J) precise.
Definition 13.15. A subset S of Xk(1J) is said to be a C r submanifold of
codimension one if there is an open subset U of Xk(1J) and a C r function
H : U ~ JR, for some r ~ 1, such that S = {f E U : H (f) = O} and that
DH(f) =f 0 for all fEU.
With the intent of identifying certain codimension-one submanifolds
consisting of structurally unstable vector fields, we will single out various
subsets of Xk(1J). In particular, we would like a vector field on one of these
submanifolds to be structurally stable with respect to the perturbations
along the submanifold, while the vector field is structurally unstable with
respect to perturbations in Xk(1J).
Definition 13.16. A structurally unstable vector field f is said to be first-
order structumlly unstable if there is a neighborhood of f in the subset
of structurally unstable vector fields with the induced topology, such that
every structurally unstable vector field in this neighborhood is topologically
equivalent to f.
We will identify co dimension-one submanifolds of Xk(1J) consisting of
first-order structurally unstable vector fields by violating the conditions for
structural stability as listed in Theorem 13.6 in as mild a way as possible.
In the case of equilibria, we consider two types of nonhyperbolic equilibria.
The first kind is when zero is a simple eigenvalue of the linearization and
the scalar vector field on the center manifold, or the bifurcation function,
starts with a quadratic term. This kind of nonhyperbolic equilibria was the
subject of Chapter 10; see, in particular, Definition 10.6, Theorem 10.10,
and Theorem 10.15.
Definition 13.17. A nonhyperbolic equilibrium point x off is called an
elementary saddle node if zero is a simple eigenvalue of Df(x) and the
398 Chapter 13: All Planar Things Considered
a #0.
The second kind of nonhyperbolic equilibrium point is the one with
purely imaginary eigenvalues. Following the setting of Chapter 11, in polar
coordinates, we will assume that the radial equation dp/dt in normal form
starts with a cubic term; see, in particular, Lemma 11.3. Recall that the
quadratic term is necessarily zero.
Definition 13.18. A nonhyperbolic equilibrium point x off is called an
elementary composed focus if the eigenvalues of Df(x) are purely imaginary
and in polar coordinates the radial equation in normal form is given by
a #0.
(a) (b)
8 2 : The set of vector fields which have only one elementary composed
focus as nonhyperbolic equilibrium point, have all other equilibrium
points and periodic orbits hyperbolic, and do not have any saddle
connections.
8 3 : The set of vector fields which have only one quasi-hyperbolic periodic
orbit, have all other periodic orbits and equilibrium points hyperbolic,
and do not have any saddle connections. Moreover, there may not be
two saddle point separatrices going to the quasi-hyperbolic periodic
orbit, one for t -+ -00 and the other for t -+ +00 (see Figure 13.4).
8 4 : The set of vector fields which have an elementary homoc1inic loop
as the only saddle connection, and have all equilibrium points and
periodic orbits hyperbolic. Moreover, a separatrix of a saddle point
may not go to the elementary homoc1inic loop for t -+ -00 or t -+ +00.
8 5 : The set of vector fields which have only one saddle connection from one
saddle point to another, and have all equilibrium points and periodic
orbits hyperbolic. <)
The collection of the sets of first-order structurally unstable vector
fields described in the theorem above forms a co dimension-one submanifold
of Xk('D).
Theorem 13.23. The set U~=l 8 i is a codimension-one C k - 1 submanifold
of Xk('D), with k ~ 4, and it is open in the set of all structurally unstable
vector fields (in the induced topology). <)
To uncover the bifurcations of first-order structurally unstable vector
fields lying on one of these codimension-one submanifolds, we need to de-
scribe a neighborhood off E 8 i in the set of vector fields Xk('D). Since 8 i is
a codimension-one submanifold of Xk('D), a vector field f E 8i has a neigh-
borhood Wi C Xk('D) consisting of three disjoint sets, Wi = Ui U Si U Vi,
400 Chapter 13: All Planar Things Considered
Figure 13.4. Partial phase portrait of a vector field that is not first-order
structurally unstable. When the nonhyperbolic periodic orbit disappears, a
saddle connection appears.
such that Ui and Vi are open subsets of Xk(1J) containing structurally sta-
ble vector fields, and Si is an open subset of Si. Moreover, the vector fields
in each component of Wi are topologically equivalent. We will momentarily
describe the dynamics of vector fields in each component of a neighborhood
Wi. First, however, let us explain the consequences of such knowledge for
bifurcations.
A one-parameter vector field is a curve in Xk(1J). Let us suppose that
such a curve of vector fields crosses one of the codimension-one subman-
ifolds Si transversally for some value of the parameter; see Figure 13.5.
Here, transversal crossing means that at the point of contact the curve is
not tangent to the codimension-one submanifold; this is the typical situa-
13.3. One-pammeter Generic Bifurcations 401
tion. Now, the knowledge of the flows of the vector fields in each neighbor-
hood Wi yields a characterization of the typical bifurcations of first-order
structurally unstable vector fields. We should emphasize that while the list
of bifurcations is for local variations of the parameter near a bifurcation
value, the consideration of the flows are global in V.
We now proceed with the enumeration of neighborhoods of first-order
structurally unstable vector fields. In each description, we only indicate
the changes near the nonhyperbolic equilibria or periodic orbits and saddle
connections; see Figure 13.6.
Theorem 13.24. (All codimension-one bifurcations of first-order struc-
turally unstable vector fields) Each first-order structurally unstable vector
field f E Si has a neighborhood Wi = Ui U Si U Vi such that, in each
component, vector fields have the following dynamics:
• Near SI: There are two cases to consider. (1) Saddle-node bifurcation
of an equilibrium: A vector field f E SI has an elementary saddle node
at. an equilibrium point x and no separatrix of the saddle node forms
a loop; there are no equilibrium points of f near x if f E Ul ; there is
a hyperbolic saddle and a hyperbolic node near x if f E VI.
(2) Saddle-node bifurcation on a loop: A vector field f E SI has an
elementary saddle node at an equilibrium point x and a separatrix of
the saddle-node forms a loop; there are no equilibrium points off near
x iff E Ul, no separatrix loop but a hyperbolic periodic orbit near its
neighborhood; there is a hyperbolic saddle and a hyperbolic node near
x and a connecting orbit from the saddle to the node near the original
saddle-node loop if f E VI.
• Near S2: Generic Poincare-Andronov-Hopf bifurcation: A vector field
f E S2 has an elementary composed focus at an equilibrium point x;
there is a hyperbolic equilibrium point but no periodic orbit off near x
iff E U2; there is a hyperbolic equilibrium surrounded by a hyperbolic
periodic orbit near x iff E V2·
• Near S3: Saddle-node bifurcation of a periodic orbit: A vector field
f E S3 has a quasi-hyperbolic periodic orbit 'Y which is stable from
one side and unstable from the other; there is no periodic orbit of f
near 'Y iff E U3; there are two hyperbolic periodic orbits, one orbitally
asymptotically stable and the other unstable, near 'Y if f E V3·
• Near S4: Elementary homoclinic loop bifurcation: A vector field f E S4
has an elementary homoc1inic loop r at an equilibrium point x; there
is no periodic orbit of f near r if f E U4; there is a unique hyperbolic
periodic orbit near riff E V4.
• Near S5: Breaking a saddle connection: A vector field f E S5 has a
saddle connection; there is no saddle connection iff E U5 or in V5 • <:)
We hope that the list in the theorem above and the illustrations in Fig-
ure 13.6 make our efforts of almost a dozen chapters a bit more meaningful.
402 Chapter 13: All Planar Things Considered
U; S; V;
~>
"--
•
~
",----
~
cD ~ ~
~
G ~ @
2g ~ )>QJ
¥++H
Figure 13.6. Generic bifurcations of first-order structurally unstable pla-
nar vector fields depending on one parameter. Only partial phase portraits
near orbits undergoing changes are shown.
13.4. Bifurcations in the Presence of Symmetry 403
Exercises - - - - - - - - - - - - - - -_ _ _ _ _ ..
13.8. Saddle node on a loop: Consider the system
Q. 0
In polar coordinates Xl = r cos () and X2 = -r sin (), show that the system
becomes
r = r(1 - r2)
8= I+A+rcos(}.
Now, verify the following dynamics as the parameter A is varied: The circle
r = 1 is always invariant under the flow and all solutions except the origin
approach this circle in forward time. On the circle r = 1, there are two
equilibria for A < 0, and IAI small, corresponding to the zeros of I+A+cos () =
0. Observe that one of the equilibria is a saddle and the other is a node. For
A > 0, there are no equilibria and the circle r = 1 is a periodic orbit.
from Example 9.37. As we saw there, this system has two saddle points
(0, 1) and (0, -1). Also, there is a saddle connection between them, part of
the x2-axis, so that the system (13.1) is structurally unstable. For example,
the perturbation
Xl = A + 2XIX2 (13.2)
X2 = 1 + xi -
x~
404 Chapter 13: All Planar Things Considered
for IAI sufficiently small, leaves the topological type of Eq. (13.1) unchanged.
Of course, the perturbation (13.2) does not posses the symmetries of
the original system (13.1) above. (;
Example 13.26. Transcritical bifurcation reconsidered: The vector field
•
Xl = Xl2
X2 = -X2
has an equilibrium point at the origin which is quasi-hyperbolic of saddle-
node type. As we saw in the previous section, this system belongs to a
codimension-one submanifold of type 51 of Theorem 13.3.
Now, let us consider the one-parameter perturbations of this vector
field in such a way that for all parameter values the origin remains an equi-
librium point with no restriction on its stability type. The one-parameter
vector field
Xl = AXI + x~
(13.3)
X2 = -X2,
for instance, satisfies this requirement. At A = 0, we have the original
vector field. For nonzero A there are two hyperbolic equilibrium points;
the origin is asymptotically stable and the other unstable when), < 0, the
origin is unstable and the other is asymptotically stable when), > 0. This,
of course, is a transcritical bifurcation, not a saddle-node bifurcation.
We infer now that evidently the vector field (13.3) does not cross the
codimension-one submanifold 51 transversally as the parameter), is varied,
and thus the resulting bifurcation is not one of the generic codimension-one
bifurcations; see Figure 13.7. (;
Restricting the set of allowable perturbations is an important practical
consideration in many contexts. For instance, a vector field may possess
certain symmetry properties, such as being conservative or gradient, that
must simply be preserved when perturbed. Before we investigate the dy-
namics and bifurcations of such vector fields in the following chapter, we
take a short diversion and examine a couple of planar vector fields depend-
ing on two parameters.
13.5. Local Two-parameter Bifurcations 405
s,
:h = AX1 + 2X1X2 + xr
. 1 2 2
X2 = 4 - Xl - X2
:h = AX1 - xr
X2 = -X2
exhibiting a pitchfork bifurcation has odd symmetry, that is, the vector field
satisfies f( -x) = -f(x).
Rather than pursuing such a general setting, the goal of this modest section
is to introduce several prominent two-parameter examples of planar vector
fields.
The vector fields in the examples below concern the dynamics in the
neighborhood of an equilibrium point of planar vector fields whose lin-
earization has two zero eigenvalues but one eigenvector, that is, the Jordan
Normal Form of the linearized vector field has the form
where >'1 and A2 are two parameters near zero. Any linear system in a suf-
ficiently small neighborhood of the matrix A is topologically equivalent to
A(Al' A2) for some values of the parameters. Below, we want to generalize
this idea of unfolding to nonlinear vector fields in the neighborhood of an
equilibrium point whose linear part is the matrix A.
Example 13.27. Unfolding a double zero eigenvalue after Bogdanov and
Takens: Consider a planar vector field depending on two parameters such
that the vector field has an equilibrium point for some fixed values of the
parameters. Also, suppose that this equilibrium point is nonhyperbolic
with two zero eigenvalues but one eigenvector. Moreover, we assume that
our vector field is "in general position" in the set of all planar vector fields
with the above yroperties. This last condition is a bit too technical to state
here precisely; it is a certain generic transversality condition met by almost
all vector fields, and is obtained from the theory of normal forms.
Now, the remarkable fact is that any vector field possessing the proper-
ties above is locally topologically equivalent to one of the two two-parameter
vector fields
Xl =X2
(13.4)
X2 =Al + A2Xl + xi ± XlX2,
depending on the ± sign. More specifically, there are a change of pa-
rameters (with nonzero Jacobian at the fixed values of the parameters)
and a homeomorphism (depending continuously on the parameters) of a
sufficiently small neighborhood of the equilibrium point which maps the
orbits of any planar vector field with the properties above to the orbits of
Eq. (13.4) in the neighborhood of the origin while preserving the sense of
directions of the orbits. In this sense, the vector field (13.4) is said to be
an unfolding of the nonhyperbolic singularity A.
13.5. Local Two-parameter Bifurcations 407
2 c 3
o
o
A
:h =X2
(13.5)
X2 =AlXl + A2X2 + xf ± x~,
depending on the ± sign. Notice that the linear part of the vector field
(13.5) is the unfolding of A in the set of linear vector fields that we have de-
termined in Section 8.4. The distinguishing feature of the vector field (13.5)
is that the origin is an equilibrium point for all values of the parameters.
This two-parameter vector field has a similar bifurcation diagram to that
of the previous example, except that there is no saddle node-bifurcation.
Further information on Eq. (13.5) with the + sign is contained in the ex-
ercises. 0
Example 13.29. Unfolding a double zero eigenvalue with odd symmetry:
Consider the two-parameter vector fields
Xl =X2
(13.6)
:h =AlXl + A2 X 2 ± xf - Xf X 2,
depending on the ± sign. Again, the linear part of this vector field is the
unfolding of A in the set of linear vector fields. This time, however, the
vector field has the odd symmetry, that is, f(A1, A2, -x) = -f(Al' A2, x).
Consequently, the nonlinear terms are third degree. We refrain from saying
more about this system except to invite you to ponder about the rather
complicated bifurcation diagram for the - sign depicted in Figure 13.9. 0
Here we conclude Chapter 13. Despite the title of this chapter, we
continue with the prevailing theme of vector fields with special properties
and turn next to the dynamics and bifurcations of conservative and gradient
systems.
Exercises _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ .. CV. 0
13.11. On the Bogdanov-Takens example:
(a) Draw the phase portrait of Eq. (13.4) with + sign for the parameter
values (AI, A2) = (0, 0).
13.5. Local Two-parameter Bifurcations 409
@@
4 --
c
@9
E-
®} B
A 6
(b) Calculate the bifurcation curves on which the planar system undergoes
saddle-node and Poincare-Andronov-Hopf bifurcations.
(c) This system is stored in the library of PHASER under the name dzero1.
Experiment on the machine and try to detect the bifurcations numeri-
cally.
3.12. Scaled double zero eigenvalue with origin fixed: It not easy to see the dy-
namical features of the system (13.5) in computer simulations. To facilitate
such an exploration, as well as for a theoretical proof, it is convenient to
410 Chapter 13: All Planar Things Considered
:b =X2
X2 =Xl - xi + (b + dXl)X2,
where band d are two parameters. For the choice of the scaling, see Chow
and Hale [1982]' p. 445. Use the initial data (1.5, 0), (1.3, 0), (1.1, 0), and
(-0.1, 0.25) and plot the four orbits for the following parameter values:
fix b = -0.1, and vary d through 0.1, 0.103, 0.105, and 0.115;
fix b = 1, and vary d through 1, 1.101, 1.105, 1.1, and 1.15;
fix b = 3, and vary d through 3, 3.05, 3.17, 3.23, 3.255, and 3.26.
Assistance: These equations are stored in the library of PHASER under the
name dzero2.
13.13. Scaled double zero eigenvalue with odd symmetry: For numerical investiga-
tions, it is convenient to scale Eq. (13.6) to obtain
Xl =X2
X2 =Xl - X~ + (b + dxi)x2,
where band d are two parameters. Use the initial data (0.1, 0) and (1.5, 0)
forward in time, and (2, 0) backward in time, with the following parameter
values: fix b = -0.1, and vary d through 0.05,0.1,0.105,0.109,0.125,0.126,
0.135, and 0.139.
Assistance: These equations are stored in the library of PHASER under the
name dzero3.
13.14. Poincare-Andronov-Hopf bifurcation: If an equilibrium point has purely
imaginary eigenvalues but is not an elementary composed focus, then the
radial equation in normal form must begin with terms of degree of at least
five. Suppose that
and show that the following properties hold near the origin:
(i) Any C k perturbation, with k ~ 6, of the vector field can have at most
two periodic orbits.
(ii) Give a specific perturbation for which there are exactly two periodic
orbits.
if, for any E > 0 there is a fj > 0 such that Ilf - gill < fj implies that there
exists an E homeomorphism h : D -> D which maps the orbits of f to the
orbits of g while preserving the sense of direction of time. By the way,
an E homeomorphism moves no point more than E. Peixoto [1959] showed
that this definition of structural stability is equivalent to the one given in
the text. With this result, the openness of the set of structurally stable
systems became self-evident.
The statement of the theorem on the characterization of structurally
stable vector fields on a disk was first given by Andronov and Pontrjagin
[1937]. The appropriate generalization of this result to orient able compact
two-manifolds and the density of structurally stable systems are due to
Peixoto [1962]; see also Palis and de Melo [1982]. In dimensions greater
than two, structurally stable systems are not dense, as shown by Smale
[1966].
A general exposition of dissipative systems is given in the recent mono-
graph by Hale [1988]; see also Temam [1988].
The restriction to a disk is a nontrivial assumption. In applications, it
is not easy to establish that a given system is dissipative. In the case of a
nondissipative system, there can be bifurcations that can only be observed
by the global considerations of the vector field on the entire plane. Such
bifurcations are called bifurcations at infinity, For polynomial vector fields
on the plane, bifurcations at infinity are studied by Sotomayor [1985] and
Sotomayor and Paterlini [1987].
The study of dynamics and bifurcations in the presence of symmetry
is currently an exciting area of our subject. A comprehensive exposition is
available in Golubitsky and Schaeffer [1985] and Golubitsky, Stewart, and
Schaeffer [1988].
The general ideas on the unfolding of nonlinear vector fields near an
equilibrium point are discussed by Arnold [1983]. Example 13.27 was inves-
tigated by Bogdanov [1981] and Takens [1974]. The proof of the versality
of the unfolding is quite difficult; full details are in Bogdanov [1981]. Ex-
ample 13.28 is analyzed in Carr [1981] and Chow and Hale [1982]. The
system with odd symmetry in Example 13.29 is studied in Takens [1974],
with further essential computations in Carr [1981].
14-_ _ __
Conservative
and
Gradient Systems
(14.2)
of the energy function. A simple, but practically important, observation
about the level curves of an energy function is that they are symmetric
with respect to the Xl-axis.
It is preferable to extract as much information as possible from the
scalar potential function, rather than the energy function. With this intent,
we proceed with an examination of the role of the extreme values of a
potential function.
14.1. Second-order Conservative Systems 415
as x -+ o. If V"(O) > 0, then the level curves of E near the origin are
ellipses. Since the origin is an isolated equilibrium point, each orbit near
the origin is a closed curve, that is, the origin is a center. <>
Unstable manifolds of saddle points playa key role in the shapes of
phase portraits of conservative systems. Here are some facts on such un-
stable manifolds.
Lemma 14.3. Suppose x = (Xl, 0) is a saddle point of Eq. (14.1) with
stable and unstable manifolds WS(x) and WU(x), respectively. Then
(i) the reBection of WS (x) through the Xl-axis is WU(x);
(ii) if 1'+(Y) for y E WU(x) is bounded and V(z) i- V(Xl) for all saddle
points (z, 0) different from x, then I'(Y) is a homoclinic orbit with
a(y) = w(y) = x.
Proof. To prove the first part of the lemma, we observe that the value of
the energy on WB(X) and WU(x) is equal to E(x). Also, the direction of
the motion along orbits is reversed as the orbit crosses the Xl-axis. This is
sufficient to obtain the conclusion in the first part.
For the second part, we notice that the Poincare-Bendixson Theorem
implies that the w-limit set of y E WU(x) cannot be a periodic orbit and
therefore must contain equilibrium points and orbits connecting them. The
416 Chapter 14: Conservative and Gmdient Systems
Xl = X2
. 2
X2=Xl-Xl·
Xl = X2
. 3
X2=Xl-X l •
Figure 14.1. The potential function V(Xl) = -xi/2 + xf!3 and phase
portrait of the fish in Example 14.4.
Xl = X2
X2 = -XI(1- xd(~ + Xl)'
The vector field has a center at (0, 0) corresponding to the nondegenerate
minimum of the potential function at O. Also, there are two saddle points,
(-~, 0) and (1, 0), corresponding to the nondegenerate critical points at
_ ~ and 1. To complete the phase portrait, we need to look at the unstable
manifolds of the saddle points. Since V(l) > V( -~), the unstable manifold
418 Chapter 14: Conservative and Gradient Systems
Figure 14.2. The potential function V(xt} = -xi!2 + xi!4 and phase
portrait of the equation of Dufling in Example 14.5.
of the saddle point (1, 0) is above the unstable manifold of (-~, 0) on the
upper plane. Also, the right piece of the unstable manifold of (-~, 0) is a
homo clinic loop encircling the center. Now, the rest of the phase portrait
can readily be filled in, as shown in Figure 14.3. <>
Example 14.7. Consider the potential function
Xl = X2
X2 = -xI(l - XI)(~ + Xl)'
14.1. Second-order Conservative Systems 419
Figure 14.3. The potential function V(Xl) = ~xi + ~xr - ~xi and phase
portrait of Example 14.6.
Figure 14.4. The potential function V(Xl) = ~x~ - ix~ - ~xt and phase
portrait of Example 14.7.
(14.3)
Xl = X2
X2 = (1 - xt}e- X1
has a center at (1, 0). As long as the value of the potential is negative, the
orbits are periodic. When the potential is positive, however, each orbit is
unbounded, as shown in Figure 14.5.
14.1. Second-order Conservative Systems 421
Exercises--------------------.~. 0
14.1. Determine the potential functions of the second-order conservative systems
below. From these functions, construct the phase portraits:
(a)x+x+x3=0; (b)x+x-x 3 =0;
(c) x + x - x 2 = 0; (d) x + x(l - x)(O.l - x) = O.
14.2. Prove that nondegenerate critical points of a potential function are isolated.
14.3. A minimum for the potential does not imply a center: Find a potential
function V(X1) that has a minimum at :h and yet there is no neighborhood
of (Xl, 0) in which all orbits of Eq. (14.1) are periodic. Can this happen if
we require the function V(X1) to be analytic?
14.4. A maximum for the potential does not imply instability: Find a potential
function V(X1) that has a maximum at Xl and yet the equilibrium point
(Xl, 0) is stable. Can this happen if we require the function V(X1) to be
analytic?
14.5. Unstable manifolds say it all: Formulate and prove a result to the effect
that "the unstable manifolds of the saddle points determine the structure
of the flow of a generic potential function." Even if you are apprehensive
about this difficult and somewhat vague problem, use the fact to draw phase
portraits from generic potentials.
14.6. Hamiltonian flows preserve area: The result of this problem explains the
absence of sinks or sources in the phase portraits of Hamiltonian, hence
conservative, systems. Recall from Section 7.4 that, for a given C 1 function
H : lR? ---+ lR, the planar system
. oH(q, p)
q = op
. oH(q, p)
p= - oq
A(t) = r
iD(t)
div f(x) dx.
(c) Prove now that the flow of a Hamiltonian system preserves area.
14.7. Find a Hamiltonian for the system
Xl = X2
X2 = A- xi
changes as the parameter A is varied from negative to positive values.
A < 0: The potential function is generic. There are two critical points
of the potential function both of which are nondegenerate. The minimum
at M corresponds to a center; the maximum at - Mcorresponds to a
saddle point and it has a homoclinic loop encircling the center. The phase
portrait is shown in Figure 14.6a.
A = 0: As we have observed above, the potential function is not
generic. The critical point at 0 is degenerate and corresponds to an unsta-
ble equilibrium point and its stable and unstable sets form a cusp. These
sets as well as the complete flow are depicted in Figure 14.6b.
A > 0: Again, the potential function is generic. Indeed, there are no
critical points of the potential function and thus no equilibrium points of
the flow. The complete flow is shown in Figure 14.6c.
In summary, as A is varied from negative to positive values, the non-
degenerate maximum and minimum of the potential function merge into a
degenerate critical point and disappear. In the phase portrait, the homo-
clinic loop shrinks to a point and disappears. The bifurcation at A = 0 is
called a saddle-center bifurcation. <>
Example 14.12. Potential with equal maxima: Consider the potential
function
V(A, Xl) = ~(1 + A)xi - ~AX~ - :txt
depending on a scalar parameter A in the range -1 < A < 1.
426 Chapter 14: Conservative and Gmdient Systems
Xl = X2
X2 = -Xl (1- xI)(l + A + Xl)
changes with A as depicted in Figure 14.7.
There is clearly a bifurcation in the dynamics of the equation at A = o.
As A --> 0 from negative values, the homo clinic orbit defined by the unstable
manifold of the saddle point at (-1 - A, 0) becomes larger, coinciding
eventually with parts of the stable and unstable manifold of the saddle
point (-1,0) to form heteroclinic orbits between the two saddle points.
For A --> 0 from positive values, a phenomenon similar to that for negative
A occurs but with the role of the saddle points interchanged. <>
The bifurcations that we have encountered in the two preceding ex-
amples represent typical situations for one-parameter families of potential
functions. In fact, with only one parameter to vary, one does not expect
more than two maximal values of the potential to coincide, or to have
more than two nondegenerate critical points merge to a degenerate critical
point. In lieu of making this statement precise, we now present a potential
function depending on two parameters.
Example 14.13. A two-parameter potential: Consider the potential func-
tion
V(A, J.L, Xl) = AXI + ~J.Lxi - ixi
depending on two real parameters A and J.L. We now analyze the bifurcations
of the corresponding conservative vector field
Xl = X2
X2 = -A - J.LXI + x~
for small values of the parameters near zero.
Since the critical points of V are the zeros of a cubic polynomial, there
can be at most three critical points. In fact, we showed in Chapter 2 that
the bifurcation curve in the (A, J.L)-space for the critical points of V is the
cusp 4J.L3 = 27A 2 , as shown in Figure 14.8.
14.2. Bifurcations in Conservative Systems 429
B
A
If ('\, J.l) lies below this cusp, there is only one critical point of V which
is a nondegenerate maximum and the flow is just a hyperbolic saddle. For
('\, J.l) = (0, 0), the only critical point of V is Xl = 0 and it is degenerate;
the flow has only an unstable equilibrium point.
For the parameter values on the cusp, the potential function has two
critical points, one of which is a nondegenerate maximum, and the other
of which is degenerate. As the parameter values cross into the cusp, the
degenerate critical point splits into a nondegenerate maximum and a nonde-
generate minimum, as in Example 14.11. Inside the cusp, the critical points
are always two nondegenerate maxima and a nondegenerate minimum, but
the flows are not equivalent for all parameter values. Indeed, there are pa-
rameter values at which the two maxima have the same maximum values;
such parameter values are exactly the positive J.l-axis. As parameter values
cross the positive J.l-axis, the homo clinic orbit turns into two heteroclinic
orbits and then back to a homo clinic orbit, as in Example 14.12. <>
This concludes our study of dynamics and bifurcations of second-order
conservative systems. We now turn our attention to gradient systems,
another class of vector fields defined in terms of a function yet with different
dynamics.
.C).
432 Chapter 14: Conservative and Gradient Systems
Exercises _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 0
14.10. Perturbations of bounded potentials: For A a real parameter, discuss the
bifurcations in the flows for the following potential functions which are
bounded in one or both directions:
(i) V(Xl) = A - xle- X1 ;
(ii) V(xd = 1 - A - COSX1.
14.11. Small perturbations at infinity may change the flow: Given a potential func-
tion Vo(z) which approaches 00 as JzJ ----> 00 and V~(z) > 0 for JzJlarge, show
that there is a perturbation V>.(z) such that V>.(z) ----> 00 as JzJ ----> 00 and
also has two critical points which approach 00 as A ----> O.
14.12. Rotating pendulum: Consider a pendulum of mass m and length l con-
strained to oscillate in a plane rotating with angular velocity w about a
vertical line. If u denotes the angular deviation of the pendulum from the
vertical and I is the moment of inertia, then
ft - (cosu - A) sinu = 0,
where A = g/(w 2 1). Discuss the flows for each A > 0 paying particular
attention to the bifurcations in the flow.
Help: Consult Hale [1980], p. 178.
14.13. Show that the following equations are Hamiltonian and discuss the bifurca-
tions in the flow for nonnegative values of the parameters:
(i) Xl = xl(a - bX2), X2 = -X2(C - dXl).
Hint: Let Xl = eq and X2 = eP.
(ii) Xl = Xl(1 - Xl)(a - bX2), X2 = -x2(1- X2)(C - dXl).
Hint: Let Xl = eq /(1 + e q ) and X2 = eP/(1 + eP).
14.14. Discuss the bifurcations in the flows of the following equations depending
on two parameters, A and jt:
(i) Xl = X2, X2 = A + f-LXl - xi;
(ii) Xl = X2, X2 = -(Xl - f-L)(1 - xl)(1 + A + xd.
the solutions of a gradient system and the Invariance Principle from Chap-
ter 9. We then explore the dynamics and bifurcations of several specific
gradient systems, some of which are old favorites. Using these examples as
a guide, we identify a subclass of structurally stable gradient systems that
are also generic in the space of gradient vector fields.
Definition 14.14. If F : lR,2 ____ lR, is a C 2 function, the gradient vector
field is
-Y'F(x) == - (~F(X),
aXl
~F(X))
aX2
and the corresponding gradient system of differential equations is
x = -Y'F(x). (14.4)
The points at which the gradient of F vanishes correspond to the
equilibria of the gradient system (14.4). To underline the significance of
such points, we introduce the following definition:
Definition 14.15. A point x is said to be a critical point of F ifY' F(x) =
o. A critical point x is called nondegenerate if the eigenvalues of the Hessian
at x, the matrix of the second partial derivatives
a2F _ a2F _)
8x l 8x l (x) 8x l 8x2(x) ,
(
a2 F _ a2 F _
8X28xl (x) 8x 28x2(x)
are nonzero.
In the case of nondegenerate critical points, precise information on the
dynamics of the corresponding equilibria can be obtained from the local
geometry of the graph of the function F.
Lemma 14.16. An equilibrium point of a gradient system (14.4) is hyper-
bolic if and only if the corresponding critical point of F is nondegenerate.
Ifx is a hyperbolic equilibrium of (14.4), then
• x is an unstable node if and only if F has an isolated maximum at x;
• x is asymptotically stable if and only if F has an isolated minimum at
x;
• x is a saddle point if and only if F has a saddle at x. <:;
The key observation for the verification of this lemma is that the ma-
trix of the linear variational equation about an equilibrium point of the
gradient system (14.4) is the Hessian matrix of F evaluated at that point.
The eigenvalues of the Hessian matrix are real because it is a symmetric
matrix. Consequently, nondegenerate critical points correspond to hyper-
bolic equilibrium points.
The most remarkable aspect of the dynamics of gradient systems is
that equilibrium points are the only possible limit sets.
434 Chapter 1.1.: Conservative and Gmdient Systems
Now, the Invariance Principle of Section 9.4 implies that w(x(O)) belongs
to the set of equilibria. If the equilibria are isolated, then w(x(O)) is a single
point because the limit set is connected. The statement on the boundedness
of positive orbits is a consequence of the fact that F(x(t)) :::; F(x(O)). <>
This theorem and its proof have several noteworthy implications. The
first corollary is that the solutions of a gradient vector field cross the level
sets of the function F orthogonally and inward, except at critical points.
Thus, from the geometry of the level sets of F, one can infer much about
the phase portrait of a gradient system. The second implication, which is
the important one in the sequel, is that a gradient system cannot have any
periodic or homoclinic orbits.
With these general facts at our disposal, it is now time to investi-
gate the dynamics of specific examples of planar gradient systems. Before
proceeding with the more substantial examples below, you may wish to
reconsider the linear gradient system in Example 7.14.
Example 14.18. Reaction-diffusion: The planar differential equations
Figure 14.9. The function F and the corresponding gradient flow of the
reaction-diffusion gradient system in Example 14.18.
Xl = -x~ - bXIX~ + Xl
X2 = -X~ - bx~x2 + X2,
xi b 2 2 X~ x~ x~
F(Xb X2) = 4 + 2XIX2 + 4 - 2 - 2'
whose graph is given in Figure 14.lOa.
Let us now outline the construction of the phase portrait of this gra-
dient system for the parameter range 0 < b < 1. To locate the equilibria,
we have drawn in Figure 14.lOb the zero set of the first component of the
vector field with dashed curves, and the zero set of the second component
with solid curves. There are nine equilibrium points corresponding to the
intersections of these two cubic curves. All equilibria are nondegenerate
and labeled with SN, S, and UN to designate, respectively, stable node,
436 Chapter 14: Conservative and Gradient Systems
(c)
Xl = -),+x~
X2 = -X2
and that, for)' = 0, the origin is a degenerate critical point of F where the
Hessian has a simple zero eigenvalue. 0
As we have indicated above, a gradient system cannot have periodic
or homoclinic orbits; thus, no need to be concerned with their bifurcations.
However, the possibility of a heteroclinic saddle connection is real, as we
can cite Example 9.17.
Example 14.21. Breaking a heteroclinic saddle connection: Let us recon-
sider the one-parameter system
from Example 9.17. As we saw there, for all parameter values near zero
there are always two hyperbolic saddle points. However, for ), = 0, there
is a heteroclinic saddle connection between these two saddle points, but it
is broken for any nonzero value of the parameter. Now, the relevance of
this example in the context of the present section is the fact that this is a
gradient system with the function
Exercises - - - - - - - - - - -_ _ _ _ _ _ _ _ _ 60' 0
14.15. Linear gradient systems: Show that a linear system x= A x is gradient if
A is a symmetric matrix.
14.16. Always gradient: Show that any product system
is a gradient system.
14.17. Many degenerate critical points: Consider the function F(Xl, X2) = xix~.
Find the critical points of F and show that they are all degenerate. Draw
the graph of the function. Finally, construct the phase portrait of the cor-
responding gradient vector field.
14.3. Gradient Vector Fields 439
4.18. Show that each of the systems below is a gradient system, determine the
values of the parameters for which the vector field is generic, and discuss
the bifurcations in the flows:
(i) Xl = Xl + (3X2, X2 = X2 + (3Xl for (3 E JR.
(ii) Xl = J1(X2 - xd + xl(l - xi), X2 = -J1(X2 - xd + x2(1 - x~) for
J1 > o.
4.19. Analyze the flow of the gradient system
(a) Show that the w-limit set of every solution exists and is an equilibrium
point.
(b) Show that this system cannot be a gradient system.
Hint: Compute the eigenvalues of the linear variational equation at the
equilibrium point (1, 0) .
.4.23. Unfolding a function and its gradient vector field: There is a theory of un-
folding a function which is one of the cornerstone ideas of catastrophe theory.
For example, the universal unfolding of the function F(Xl' X2) = ~(x~ +x~)
is the three-parameter family of functions
Thorn calls the catastrophe, the set in the three-dimensional parameter space
for which F changes its number of preimages, associated with this function
the hyperbolic umbilic. Explore the phase portrait of the corresponding
gradient vector field. One could also unfold this gradient vector field in
the set of gradient vector fields. Comparison of the two unfoldings raises
interesting issues, as discussed in the second reference below.
References: General facts about catastrophe theory including the hyperbolic
umbilic can be found in, for example, Poston and Stewart [1978]. The
gradient vector field above is discussed by Guckenheimer in Peixoto [1973] .
.4.24. Unfolding the elliptic umbilic: Another entry in Thorn's famous list of seven
elementary catastrophes is the unfolding of the elliptic umbilic
440 Chapter 14: Conservative and Gradient Systems
Explore the phase portrait of the gradient vector field of this function.
Help: Information about the elliptic umbilic is contained in Poston and
Stewart [1978]. The gradient vector field of the elliptic umbilic is stored in
the library of PHASER under the name gradient.
and Palis and Smale [1970] for gradient vector fields on a compact man-
ifold of any dimension: gradient systems with only hyperbolic equilibria
and transversal intersection of stable and unstable manifolds are struc-
turally stable, and structurally stable gradient systems are open and dense
in the set of all gradient systems.
15-_ _ __
Planar
Maps
X t-+ f(x).
In this section, after several brief general remarks, we explore the geometry
of the orbits of maps in the case f is a linear function.
Most of the necessary notation and many of the concepts from the
theory of scalar maps as expounded in Chapter 3 are easily generalized to
planar maps. Since it is quite likely that you have studied that chapter a
When both the positive and negative orbits exist, the orbit, of XO is the
union of the two: ,(xO) = ,+(xO) U ,-(xO).
The most notable positive orbit of a map is one consisting of a single
point that is fixed under all iterates of the map.
X f-+ Ax (15.2)
° ° ° }
." +(x 0) -_ { x, A x, ... , An x, ....
A= (0.°9 0.80) .
We first need to find the fixed points of the linear map x f-+ A x, that is,
determine the solutions of the linear system (A - I)x = o. Since A - I is
446 Chapter 15: Planar Maps
Xl
Figure 15.1. A single orbit, and the phase portrait of the hyperbolic sink
in Example 15.5.
invertible, the origin x = 0 is the only fixed point of the linear map. The
powers of A are given by
It is evident that An approaches the zero matrix as n ----t +00. Thus, the
origin is an asymptotically stable fixed point, as seen in Figure 15.1. To
infer how a positive orbit approaches the origin, notice that the eigenvalues
of A are 0.9 and 0.8 with corresponding eigenvectors VI = (1, 0) and
v 2 = (0, 1), respectively. For any initial value xO = (x~, xg), we have
15.1. Linear Maps 447
Consequently, the positive orbits approach the origin faster in the direction
of v 2 than in the direction of v l . 0
When examining static pictures of phase portraits of planar maps you
should keep in mind that an orbit is just a sequence of discrete points and
not a continuous connected curve. As a result, it could at times be difficult
to interpret certain phase portraits. Here is an example of this sort.
Example 15.6. A hyperbolic sink with reflection: Consider the linear map
with the coefficient matrix
Following the notations and the computations in the previous example, for
any initial vector Xo = (x~, xg), we have
AnxO = (O.9)nx~vl + (-1)n(o.8)nxgv 2 •
Again, A nxO -+ 0 as n -+ +00 for every xO. However, due to the presence
of the negative eigenvalue, the positive orbit through xO jumps back and
forth across the Xl-axis; see Figure 15.2. If we try to fit smooth curves
through the points on an orbit, there would usually be a piece above the
Xl-axis and another one below, the two forming a cusp at the origin. 0
Example 15.7. A hyperbolic source: Consider the coefficient matrix
A = (\J 1
1~2)'
whose eigenvalues are greater than one. For any initial value xO = (x~, xg),
we have
AnxO = (1.1)nx~vl + (1.2)nxgv 2 .
It is evident that the origin is an unstable fixed point; see Figure 15.3.
Furthermore, by considering the iterates of the inverse of this map, A -n ,
it is easy to deduce that the a-limit set of any point is the origin. 0
Example 15.8. A hyperbolic saddle: Consider the coefficient matrix
1.1
A= (
o
with one eigenvalue greater and the other larger than one in absolute value.
Since for any initial vector xO = (x~, xg), we have
AnxO = (1.1)nx~vl + (0.9)nxgv2 ,
th~ origin is unstable. However, unlike a source, w(O, xg) = (0,0) and
a(x~, 0) = (0, 0); see Figure 15.4. 0
448 Chapter 15: Planar Maps
...... ,
Figure 15.2. A single orbit, and the phase portrait of the hyperbolic sink
with reflection in Example 15.6.
0.9
A= (
°
whose eigenvalues are 0.9 and 1. Observe now that, in addition to the origin,
every point on the x2-axis is a fixed point. Moreover, since the iterates of
an initial vector xO = (x~, xg) are
we have w(x~, xg) = (0, xg). It is evident that the origin is a stable fixed
point; see Figure 15.5. 0
15.1. Linear Maps 449
Figure 15.6. A single orbit of the nonhyperbolic linear map with reflec-
tion in Example 15.10.
0: = ACOSW, f3 = Asinw,
15.1. Linear Maps 451
where ). = J a 2 + (32 and -7r < W ::; 7r, the coefficient matrix becomes
A =). (co~w sinw). (15.4)
-smw cosw
A = 0.996
A = 1.000
,
'-.
!
/
A = 1.001
Figure 15.7. A single (!) orbit ofthe linear map (15.4) with complex
eigenvalues for A = 0.996, A = 1.000, and A = 1.001.
Exercises _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ .0.0
15.1. Linear Maps
15.1. Describe the dynamics of the linear maps with the coefficient matrices below,
453
a. (0.°2 0).
2 '
b. c.
d. (~ ~} e. ( 1/v'2 1/v'2)
-1/v'2 1/v'2 ' f. (-1° 0)
1 .
Notice that for all values of the parameter A, the eigenvalues of the coefficient
matrix are 1. At A = 0, it is the identity map so that the origin is a stable
fixed point. Try to determine the phase portrait of this linear map for
nonzero, both positive and negative, values of the parameter A. Is the origin
still stable for nonzero A? Do the same problem for the nonhyperbolic linear
(-1° A)
map
x f-+ -1 x.
The difficulty with these examples is the fact that 1, or -1, is not a simple
root of the minimal polynomial; for more information, see, for example, La
Salle in Hale [1977].
15.3. Show that the w-limit set of a point is invariant.
15.4. First integmls for maps: The orbits of maps are sets of discrete points.
However, it is often advantageous to know the equations of the curves on
which positive orbits may lie. Such curves are usually the level curves of a
function-first integral-that is constant along orbits (see Section 7.4). The
notion of a first integral for maps is defined as follows:
A function H : lR? -+ IR is called a first integml for a map f if
(H 0 f)(x) = H(x)
f(x) = ( co~w
-smw
sinw) x
cosw
454 Chapter 15: Planar Maps
Xl = Yn-!,
X n1 +1 -- xn
2
(15.7)
(~ i),
which has the eigenvalues 0 and A. Consequently, the origin is asymptoti-
cally stable if 0 < A < 1, and unstable if A > 1.
The Jacobian matrix at the other fixed point (1 - 1/ A, 1 - 1/ A) is
456 Chapter 15: Planar Maps
(15.8)
We will point out later in this chapter the significance of this seemingly
irrelevant observation on the bifurcations of the nonhyperbolic fixed point
(1-1/>',1- 1/>.) as the parameter>' is varied near >. = 2.0
. .
,..-J/f':
Figure 15.8. Asymptotically stable fixed point of the delayed logistic map
(15.7) at (1 - 1/)" 1 - 1/),) for the parameter value). = 1.98. This is a
single orbit! The shadow of sixth root of unity is visible in the presence of
six strands.
We now turn our attention to the fine structure of the flow of a map
near a hyperbolic fixed point. The counterpart of the results from differ-
ential equations that we presented in Chapter 9 can easily be formulated
for maps when a map is differentiable with a differentiable inverse, that
is, when it is a diffeomorphism. For the remainder of this section, let us
assume that f is a diffeomorphism, thus excluding zero as an eigenvalue.
15.2. Near Fixed Points 457
The global stable manifold W 8 (x) and the global unstable manifold
WU(x) can be constructed from the corresponding local ones; to obtain
W8(X), let the points in W8(X, U) flow under backward iterates, and to
obtain WU(x), let the points in WU(x, U) flow under forward iterates:
(15.9)
a = 1.4, b= 0.3.
At these parameter values there are two fixed points and their coordinates
are
_ -(1-b)±J(1-b)2+4a
Xl = 2a '
15.2. Near Fixed Points 459
(a)
Figure 15.9. (a) Local stable and unstable manifolds of a saddle point of
the Henon map; and (b) global stable and unstable manifolds of the same
saddle point with many intersections-transversal homoc1inic points.
Linearization about the fixed points reveals that the fixed point
Xl = 0.63135448 ... , X2 = 0.18940534 ...
is a saddle point with the eigenvalues
J.Ll = 0.15594632 ... , J.L2 = -1.92373886 ...
The slope of eigenvector of J.Ll is 1.92373886 ... and the slope of eigenvector
of J.L2 is -0.15594632 ....
For a crude approximation of the unstable manifold of the saddle point,
we can take a sufficient number of initial points lying on the eigenvector
of J.L2 and iterate forward. For the stable manifold, we iterate points on
the eigenvector of J.L2 using the inverse of the map. The result of such
an experiment, with further refinements, is depicted in Figure 15.9, where
intersection points of the stable and the unstable manifolds are visible.
The global dynamics of the Henon map are quite complicated and
the mathematical details are still fragmentary for many parameter values,
including the ones chosen by Henon. However, we should note that for
a = 1.4 and b = 0.3, Henon discovered through numerical experiments that
in a region of the plane almost all solutions eventually get attracted to a
set-the Henon attractor-that is neither a fixed point nor a periodic orbit.
For a picture of this strange attractor, see Figure 15.10. It is interesting to
observe that the attractor appears to be the global unstable manifold of the
saddle point. We will provide further information on the Henon map later
in this chapter, including a novel reason why a numerical analyst might
also be interested in this map. <)
460 Chapter 15: Planar Maps
..........
~,~.-"
Figure 15.10. The "strange attractor" of the Henon map at the param-
eter values a = 1.4 and b = 0.3. The picture is obtained by iterating the
initial value (0.63135448, 0.18940634) and plotting the iterates 1000-5000.
The second picture is an enlargement of a region near the initial point.
Exercises - - - - - - - - - - - - - - - - - - - . 0 . 0
15.6. Linearization and nonhyperbolicity: Find two planar maps with the follow-
ing properties: both have a fixed point and the linearization at the fixed
points are the same linear map but the fixed point of one of the maps is
asymptotically stable while the fixed point of the other map is unstable.
15.7. Nonhyperbolic fixed point of the delayed logistic map: The delayed logistic
map is stored in the library of PHASER as a planar map under the name
dellogis. Using the machine, investigate the following questions:
(a) At the parameter value a = 1, the only fixed point of the delayed logistic
map is the origin. Since one of the eigenvalues is equal to 1, the fixed
point is not hyperbolic and thus linearization yields no information
15.2. Near Fixed Points 461
The interesting observation about the new form of the map is that when
b = 0, the first component is the logistic map. Consequently, the Henon
map contains all the complexity of the logistic map, and more. You should
experiment numerically on PHASER using very small nonzero values of b.
15.12. The Lozi map: This map is a piecewise linear version of the Henon map
obtained by replacing the square term with absolute value:
where a and b are real parameters. The map of Lozi is stored in the library
of PHASER under the name lozi. Take a = 1.7 and b = 0.5; plot the iterates
1000-3000 of the initial vector (0.11, 0.21). The resulting picture is the Lozi
462 Chapter 15: Planar Maps
attractor. Try several other initial points. Can you determine a trapping
region?
Notes: It is often the case that mathematical analysis tends to be easier for
piecewise linear maps than smooth nonlinear ones. In fact, unlike the case
of the Henon map, it has been proved that the Lozi map has a hyperbolic
strange attractor; see Lozi [1978] and Misiurewicz [1980].
15.13. Liapunov functions for maps: The stability type, more importantly, the
basin of attraction, of a fixed point of a planar map can also be determined
using an appropriate modification of Liapunov functions that we explored
in Part II.
A real-valued function V : IR? ---t IR is called a Liapunov function centered
about a fixed point x of a planar map f if
(i) Vex) > 0 for x '" x;
(ii) Vex) = 0;
(iii) (V 0 f)(x) ::::: Vex).
Using the quadratic function veX!, X2) = x~ +x~, analyze the stability and
the domain of attraction of the fixed point (0, 0) of the planar map
where a and b are two real parameters. You should consider several cases
depending on the values of the parameters; a2 = b2 = 1 requires special
care.
References: For more information on this example, Liapunov functions, and
the Invariance Principle for maps at large, see the expository article by La
Salle in Hale [1977], or his books (La Salle [1976 and 1987]).
A= (~l ~2)'
where both Al and A2 are negative so that the equilibrium point at the
origin is asymptotically stable. The eigenvalues of 1 + hA are 1 + hAl and
1+hA2' both of which have modulus less than 1 if the step size h is assumed
to be sufficiently small and positive. In this case, the fixed point at the
origin of the planar linear map (15.11) is asymptotically stable also.
(ii) Saddle preserved: If Al > 0 and A2 < 0, then one of the eigenvalues
of 1+hA must have modulus greater than 1 and the other less than 1 when h
is small and positive. Consequently, the saddle character of the equilibrium
is reflected in the phase portrait of the approximate linear map about the
corresponding fixed point at the origin.
(iii) Center destroyed: Since nonhyperbolic equilibria are very sensitive
to small perturbations, we have reason to fear that numerical approxima-
tions will destroy a center. To validate that such fears are indeed well-
founded, let us consider the planar linear differential equations with the
coefficient matrix
.( h) ~ Yn+l - Yn-l
Y n ~ 2h . (15.13)
(15.14)
15.3. Numerical Algorithms and Maps 465
11-2 = -h + Vh 2 + 1.
When h > 0, one eigenvalue, 11-1, is smaller than -1 and the other is between
o and 1. Consequently, the fixed point (1, 1) is a saddle point of the map
(15.14).
Now, the disappointing conclusion is that no matter how small a pos-
itive step size we use, it is impossible to capture the asymptotic stability
of the equilibrium point y = 1 with the central difference algorithm. <)
We now come to our final example in this section; we trust you will
find it exciting. The numerical approximation scheme we are about to
introduce may not have a wide practical appeal from the viewpoint of
numerical analysis, but the resulting planar map is certainly a famous
one-the Henon map.
Example 15.25. The Logistic equation, a mixed difference scheme, and
the Henan map: We saw above that while Euler's algorithm was capable
of capturing the asymptotic stability of the fixed point y = 1 of the logistic
equation (15.12), the central difference algorithm failed hopelessly. To see
how things might degrade from good to bad, let us now use a weighted
"average" of the two methods to approximate y at t = nh:
the stability types and some of the bifurcations of these fixed points in the
exercises. For now, let us manipulate the form of the map (15.16) a bit
more.
If we make the affine transformation
Zl
>. +-
= -
2h
h[h - >.
--Xl
1 + >.
+ 1]
'
Z2 = ->. 2h+-h[h--X2
- >. + 1]
1- >.
and put
we obtain
(~~) ~ (1 + X:X~ aX~) . (15.17)
This, of course, is the map we have been seeking-the Henon map. The
parameter values a = 1.4 and b = 0.3 chosen by Henon correspond approx-
imately to >. = 0.538 and h = 1.9. <>
After this interlude on numerical analysis, we now return to basics and
investigate bifurcations of fixed points of planar maps.
. _(-1.5 2)
x- 0 -1 x.
What is the largest safe step size in the numerical integration of the system
using Euler's algorithm?
15.16. Central difference and hyperbolic equilibria: Consider the linear scalar dif-
ferential equation
if = ay,
where a is a nonzero real number so that the origin is a hyperbolic equilib-
rium. Approximate the derivative of y with the central difference
.( h) ~ Yn+l - Yn-l
y n ~ 2h '
Show that the origin is a saddle point of this map. It is evident from this
example that using the central difference one cannot preserve the stability
of a hyperbolic equilibrium point of a scalar differential equation for either
t --> +00 or t --> -00.
5.17. A center-preserving algorithm: Here, we describe an algorithm that captures
the qualitative feature of a linear center. Consider the second-order scalar
differential equation
y+y=o
whose phase portrait, when written as a first-order system, is a center.
(a) Show that approximating the second derivative of y by the centml dif-
ference
"(t) R;j y(t + h) - 2y(t) + y(t - h)
Y h2 '
(b) Set xf = Yn and xr; = Yn-l, and convert the second-order difference
equation to the equivalent planar linear map
and verify that both are complex and have unit moduli.
(d) Show that each orbit of the planar linear map lies on an ellipse and the
ellipses are concentric. Determine the equations of the ellipses.
(e) We captured the qualitative character of the center, but not the quan-
titative one because the orbits of the differential equation lie on circles
in the (x, x)-plane. Can you choose a better coordinate system for the
corresponding map so as to capture the quantitative character of the
center?
15.18. A hyperbolic limit cycle and Euler: Consider our old-time favorite example
of a planar differential equation:
Xl = -X2 + Xl (1 - x~ - x~)
X2 = Xl + x2(1 - x~ - x~),
where a and b are some constants. To compute the numerical value of this
integral for given values of a and b, consider the planar map
where
JL -=f 1, g(O, 0) = 0, Dxg(O, 0) = O.
The fixed points of the map (15.18) are the solutions of the pair of equations
(
Xl)
X2
I--t (A + Xl + AX2 + ~!)
0.5X2 + AXl + Xl
,
where A is a scalar real parameter ranging in a small neighborhood of o.
Simple computations yield X2 = 'lj;(A, Xl) = 2x!+2AX1, and the bifurcation
equation (15.19) for the fixed points of the map becomes
A + 2A2Xl + (1 + 2A)Xr = o.
If A < 0, there are two zeros of the bifurcation function. From part
(i) of Theorem 15.26, the fixed point with the smaller Xl coordinate is
asymptotically stable, while the other fixed point is a saddle. When A = 0,
these two fixed points coalesce at the origin which is a nonhyperbolic fixed
point. If A > 0, there are no zeros of the bifurcation function and thus
no fixed points either. This, of course, is a saddle-node bifurcation for the
nonhyperbolic fixed point at the origin as the parameter A passes through
zero. <>
Example 15.28. Saddle node in the Henon map: Let us consider the
Henon map from Section 15.2:
_ -(1-b)±y'(1-b)2+4a
Xl =
2a '
At a = -(1 - b)2/4, there is a unique fixed point. Linearization at this
fixed point shows that the fixed point is nonhyperbolic with 1 as an eigen-
value; the other eigenvalue is less than one in absolute value. Close to this
bifurcation value, there are no fixed points when a < -(1- b)2/4, and two
hyperbolic fixed points when a > -(1 - b)2/4. One of these fixed points is
unstable and the other stable. <>
We refrain from presenting a detailed statement of period-doubling
bifurcation when one eigenvalue passes through -1. Instead, we will be
content to demonstrate this important bifurcation in the Henon map.
15.4. Saddle Node and Period Doubling 471
Exercises _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ It Q, <)
.5.21. A period-7 orbit for Henon: For the parameter values a = 1.24 and b = 0.3,
the Henon map (15.20) has an attracting periodic orbit of period 7. Find it
on the computer by using appropriate initial data, for example, XO = (0, 0).
Also, try other initial data. While leaving b fixed, increase a gradually until
this periodic orbit doubles its period. Can you double the period again?
Hint: When plotting orbits, leave out the transients; that is, do not plot,
say, the first 1000 iterates.
l5.22. A pitchfork bifurcation: Consider the following planar map depending on a
scalar parameter A:
(a) Notice that the origin is a fixed point for all values of A. Show that the
origin undergoes a pitchfork bifurcation near A = 3/2~._-:_
(b) There are additional fixed points at (±JA - 3/2, ±JA - 3/2). At A =
3, compute that the eigenvalues at these fixed points are -1 and -1/2.
Analyze the bifurcation as A passes through 3. In your computations
you should translate the fixed point to the origin and then put the
linear part into Jordan Canonical Form.
l5.23. Bifurcation in mixed-difference algorithm: Consider the map (15.16) from
the previous section. Fix the step size h and, while varying A, show the
following:
(a) One of the fixed points of the map undergoes a pitchfork bifurcation
at A = h/2;
(b) determine the value of A at which the fixed point (1, 1) loses its stability
to an asymptotically stable periodic orbit of period 2.
472 Chapter 15: Planar Maps
5.24. A two-parameter map: Consider the following map depending on two real
parameters, >\1 and A2:
(
Xl)
X2
f-+ (AI + Xl + A2X2 + ~f)
0.5X2 + A2Xl + Xl
.
Analyze the bifurcation of fixed points near the origin for (AI, A2) small in
norm. What is happening near the curve A~ - Al = O?
Hint: The bifurcation equation is Al + 2A~Xl + (1 + 2A2)xf = O.
The origin is a fixed point of this map for all values of >... The linear
part of this map is the same as the linear map (15.4) that we examined in
Example 15.11: as the parameter>" is increased through 1, the eigenvalues
cross the unit circle from the inside to the outside, and thus the stability
type of the origin changes from one of asymptotic stability to that of in-
stability. To see what else might be happening near the origin, we need
to investigate the effects of the nonlinear terms. For this purpose, it is
convenient to transform the map (15.21) to polar coordinates to obtain
(15.22)
It is now evident that for>.. > 1, the map (15.22) has an invariant circle
of radius r = JI=1. Moreover, the omega-limit set of every positive orbit,
except the origin, is contained in this circle. The flow on the invariant circle
is simple rotation with rotation number w. For the bifurcation diagram of
Eq. (15.21), see Figure 15.13. <>
474 Chapter 15: Planar Maps
-----------'----------------
°
with 1J..L(Ao)I = 1;
(iii) d~ IJ..L (A) I > at A = AO;
(iv) ttk(AO) =I- 1 for k = 1, 2, 3, 4.
Then there is a smooth A-dependent change of coordinates bringing F into
the form
F(A, x) = F(A, x) + 0 (1IxI15)
and there are smooth functions a(A), b(A), and W(A) so that in polar coor-
dinates the function F(A, x) is given by
( r) f-t ( Itt(A)lr-a(A)r 3 ).
(15.23)
e () + W(A) + b(A)r2
If a(AO) > 0, then there is a neighborhood U of the origin and a {j >
such that, for IA - Aol < {j and x O E U, the w-limit set ofxo is the origin
°
15.5. Poincare-Andronov-Hopf Bifurcation 475
Then the magic coefficient a of the cubic term in Eq. (15.23) in polar
coordinates is equal to
where
and
"Re" in formula (15.25) represents the real parts of those complex numbers,
and all the partial derivatives are evaluated at the fixed point at the origin.
Admittedly, the formula above appears rather formidable, but rest
assured that we will not attempt to derive it. Instead, let us demonstrate
its utility on one of our earlier maps.
Example 15.32. Delayed logistic map continued: Here, we continue with
the analysis in Example 15.17 near the parameter value AO = 2 and estab-
lish, as consequence of the theorem above, that the fixed point (1/2, 1/2)
undergoes a supercritical Poincare-Andronov-Hopf bifurcation.
From our earlier computations, it is evident that the hypotheses (i)-
(iv) of Theorem 15.31 are satisfied. The only remaining hypothesis to verify
is the computation of the coefficient a(2) of the cubic term in normal form
[Eq. (15.23)]. To accomplish this, we set A = 2 and put the delayed logistic
map
(15.26)
into the form (15.24). For this purpose, we first translate the fixed point
(1/2, 1/2) to the origin by applying the transformation
Next, we put the linear part of the map into Jordan Normal Form by
making one further change of variables:
where
0
p- (
../3/2
15.5. Poincare-Andronov-Hopf Bifurcation 477
( Xl) (1/2
f X2 = ..;3/2
This map is now in the form (15.24) and we can use formula (15.25)
with the nonlinear terms gl(X1. X2) = -2X1X2 - 2x~ and g2(X1. X2) = o. A
short computation yields
and
a= -~+7 >0.
Consequently, the bifurcation is supercritical: for >. > 2 and >. - 2 suf-
ficiently small, the delayed logistic map (15.26) has an attracting closed
curve encircling the fixed point (1 -1/>., 1- 1/>.); see Figure 15.14.
We should emphasize that the smooth invariant curve is guaranteed
for values of the parameter that are sufficiently near the bifurcation value.
What happens "far away" from the bifurcation value may be rather com-
plicated and perplexing, as shown in Figure 15.15. <:;
We now once again return to the general setting of Theorem 15.31 and
indicate how to arrive at the normal form (15.23). During our normal form
computations, the necessity of the hypothesis (iv) will become self-evident.
To simplify the notation, let us suppose that, by a >.-dependent transforma-
tion, the linear part of our map is already in normal form [Eq. (15.24)]. Of
course, a, {3, g1. and g2 would depend on >.; however, let us leave>. out of
our formulas. With these simplifications, our difficult computational task
is made feasible by introducing the complex change of variables
(15.27)
P
-1
=
(1 i)
1 -i '
1
P = 2i
(i i)
1 -1
478 Chapter 15: Planar Maps
"
~
(~'1
...' .•.
.cfifi·
Figure 15.15. Far away from the bifurcation value, at oX = 2.265, the
invariant "circle" of the delayed logistic map (15.6) folds onto itself and
becomes nonsmooth. For emphasis, a piece of the invariant curve near its
tip is enlarged.
Now, a short calculation yields that the function 'ljJ has the form
480 Chapter 15: Planar Maps
the other function 1jj is simply the conjugate of 'I/J and has the linear term
pz. Thus it suffices to consider only one of the component functions, say,
'I/J, which is really a function of a scalar complex variable z. Indeed, this is
the main advantage of working in complex coordinates: a real planar map
becomes a scalar map of a complex variable.
Our next task is to simplify, or to eliminate, as many of the higher-
order terms as possible in the Taylor series of 'I/J using successive complex
transformations. The following lemma indicates the possible simplifica-
tions:
Lemma 15.33. If I-£k i:- 1 for k = 1, 2, 3, or 4, then there is smooth change
of variables bringing 'I/J into the form
(15.28)
~u ~02
')'11 = 1-£(1 _ p) , ')'02 = ---.
1-£ - p
2
These choices are possible because of our assumption that 1-£ i:- 1 and 1-£3 i:-
I, the latter one being necessary for the third choice.
15;5. Poincare-Andronov-Hopf Bifurcation 481
Xl = X2 = ! (-1 + v'4>. + 1) .
When the parameter passes through>. = 3/4, it follows from linearization
that this fixed point changes from a sink to a source while the eigenvalues
leave the unit circle through ±i. Consequently, the nonresonance condition
in the Poincare-Andronov-Hopf bifurcation theorem is not satisfied. Do
we still get an invariant smooth closed curve encircling this fixed point as
the eigenvalues cross the unit circle?
The answer to this question in the case of our example is delightfully
simple. As >. increases through>. = 3/4, two orbits of period 4 bifurcate
from the fixed point: a sink
482 Chapter 15: Planar Maps
::W~f:(;;i;;::>:;"'<
. it·, "
...<
and a saddle
where
P± = !(1±V4,X-3), q= ~ (-1 + V4,X + 1) .
The four horizontal and vertical line segments connecting the points of
the periodic sink contain the saddle points, and form an invariant square
which bifurcate from the fixed point. Thus, there is still an invariant closed
curve-a square-encircling the fixed point, but the curve is no longer
smooth; moreover, the dynamics on the square is not simply a rotation.
This invariant square is illustrated in Figure 15.16. Further information on
this system is contained in the exercises. <>
15.5. Poincare-Andronov-Hopf Bifurcation 483
Exercises _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ .. <:J. 0
5.25. Verify that the canonical example (15.21) satisfies all the hypotheses of the
Poincare-Andronov-Hopf bifurcation theorem. Using the elaborate formula
(15.25), revalidate the stability type of the invariant circle.
5.26. A subcritical Poincare-Andronov-Hopf bifurcation: Analyze the bifurcations
of the map
-sinw)
cosw
(Xl).
X2
Suggestions: Do this problem two ways. First, transform the map to po-
lar coordinates and analyze its dynamics directly. Second, verify all the
hypotheses of Theorem 15.31 and compute the sign of the cubic term in
cartesian coordinates using formula (15.25) .
.5.27. No Hopf in Henon: Show that the fixed points of the HellOQ map (15.20)
do not undergo a Poincare-Andronov-Hopf bifurcation for any values of the
parameters .
.5.28. On the invariant square: Consider the map (15.29):
L5.29. A discrete predator-prey model: Consider the following planar map depend-
ing on two parameters a and b:
detDf(x) = 1 (15.30)
then in a suitable complex coordinate system the map f can be put into
the normal form
is a real polynomial in Izl2 and the function 9 vanishes with its derivatives
up to order q - 1 at z = z = O. The square brackets denote the largest
integer in q/2. <>
Equipped with this normal form result, we are now ready to state our
main stability theorem.
Theorem 15.38. (Stability of an Elliptic Fixed Point) Let f be an area-
preserving planar map with an elliptic fixed point at the origin satisfying
the conditions in the lemma above. If the polynomial a (lzI2) does not
vanish identically, then the origin is a stable fixed point. <>
In the stability criterion above, the typical situation is that q = 4,
s = 1, and al =I OJ it is interesting to notice that this is exactly the same
nonresonance condition as in the Poincare-Andronov-Hopf theorem of the
previous section.
As we will not validate this stability theorem, let us explore its ramifi-
cations on a substantial example. In order not to burden you with extensive
normal form computations, we will be content to explore the dynamics of
the example below through numerical simulations.
Example 15.39. Cremona map: Consider the quadratic planar map
A map of the form (15.33) is called a twist map because such a map leaves
each circle of constant radius invariant and the orbits are simple rotations
on these circles. We will impose the further restriction
d"t :f: 0
dr
in the annulus so that the angle of rotation is not constant and depends on
the radius.
From our investigations in Chapter 6 on circle maps, it is easy to
describe the dynamics of a twist map (15.33): on each circle for which the
rotation number is a rational number the orbits are periodic, otherwise
they are dense on the circle. It is a remarkable fact that under small area-
preserving perturbations some of these invariant circles persist.
Theorem 15.40. (Twist Theorem) Consider in polar coordinates the fol-
lowing area-preserving perturbation of a twist map
(15.34)
488 Chapter 15: Planar Maps
Figure 15.18. In the vicinity of the origin of the Cremona map (15.32)
as the parameter is increased by and by past .A = 211"/3. At .A = 211"/3, the
origin is unstable.
15.6. Area-preserving Maps 489
w -
I-27f pi- 2': clql-
q
5/2
(15.35)
r(7) = G1(e, 7)
(15.36)
0(7) = 7 + G 2(e, 7)
with Gland G 2 of period 27f in 7, which is invariant under the map (15.34).
The positive orbits on the curve (15.36) are given by the simple rotation
7 f--+ 7 + w. <>
~©l
Q(~
~---~...•...... ~~/ .............. :
(
Figure 15.19. Disintegration of invariant curves of the Cremona map
away from the origin.
Exercises - - - - - - - - - - - - - - -_ _ _ _ _ ,. \/.0
15.31. Fixed points of Cremona: Show that the Cremona map (15.32) has another
fixed point (Xl, X2), where
Xl = 2tan(A/2),
Can you determine the stability type of this fixed point? This map also
has many periodic orbits; up to period 4, they can be found explicitly; see
Henon [1969].
Aid: This map is stored in the library ofPHASER under the name cremona.
15.32. Inverse of Cremona: Verify that the planar quadratic map
is the inverse of Cremona map (15.32). To follow full, not just positive,
orbits of an invertible map on the computer an explicit expression for the
inverse map is needed. You may wish to use the inverse map above while
investigating the dynamics of the Cremona map.
Aid: This map is stored in the library of PHASER under the name icremona.
15.33. Consider the cubic planar mapping
2XI + X2 = Xl + m
Xl + X2 = X2 + n,
where m and n are integers.
(c) Determine that the origin is a saddle, and compute its stable and unsta-
ble manifolds. You may want to do this on the plane and then identify
the squares with integer coordinates. Do the stable and unstable man-
ifolds intersect?
(d) Show that the point (1/2, 1/2) is a periodic point of period 3. Find
some other periodic points with different periods. In fact, show that
its periodic points are dense in the unit square.
Notes: This famous map is stored in the library of PHASER under the name
anosov. Also, see Arnold [1968 and 1983], Devaney [1986], and Smale [1966]
for "several important occasions in which anosovappears.
Some of the specific maps described in the text have been the subject of
many studies. The delayed logistic map is described in Maynard-Smith
[1968] from a biological viewpoint. The finer details are elucidated in
Pounder and Rogers [1980], and particularly in Aronson et al. [1980 and
1982]. The mathematical literature on the Henon map is very extensive
and still growing; while Henon [1976] is, of course, the place to start, you
should consult Benedicks and Carleson [1991] for an exciting new develop-
ment. The Cremona map, suggested by Siegel and Moser [1971], is studied
extensively by Henon [1969].
The details of the local theory of hyperbolic fixed points, including
the Hartman-Grobman theorem for maps (see Section 2.6), as well as sta-
ble and unstable manifolds are given in Hartman [1964]' Irwin [1980], and
Nitecki [1971]. The center manifold theorem for maps is described by Carr
[1981] and Lanford [1973]. Poincare [1892] remarked on the dynamical
complexity associated with homo clinic points. Smale [1965 and 1967] and
Sil'nikov [1967] showed the depth of this complexity; see also Newhouse's
exposition in Guckenheimer et al. [1980]. A discussion of structural sta-
bility for diffeomorphisms, in the spirit of Chapter 13, is available in Palis
and de Melo [1982].
In his review article, Ushiki [1986] describes some of the recent trends
in numerical analysis of differential equations from the viewpoint of dynam-
ical systems. Several representative references for "safe" approximations of,
for example, hyperbolic periodic orbits, are Braun and Hershenov [1977],
Beyn [1987], Eirola and Lorenz [1988], and Kloeden and Lorenz [1986].
The effects of floating point arithmetic in the Henon map are addressed in
Curry [1979] and Hammel et al. [1988]. Methods for computing invariant
curves of maps are examined in van Veldhuisen [1988a].
The Poincare-Andronov-Hopf bifurcation theorem for maps is of more
recent origin than the one for differential equations. Indeed, contrary to
the usual practice, it would be more appropriate to call it "invariant cir-
cle," or "Neimark-Sacker" bifurcation; see Neimark [1959], Sacker [1965],
and Ruelle and Takens [1972]. The exposition in Lanford [1973] is par-
ticularly accessible and contains most of the necessary details; a general
reference is Iooss [1979]. An account of the stability formula is given in
Wan [1978]. Strong resonances were first investigated by Arnold [1983]
and Takens [1974]; an exposition is contained in Whitley [1983]. The case
of fourth roots of unity still remains unresolved.
The study of area-preserving maps was initiated by Poincare and stud-
ied extensively by Birkhoff [1927]. The main reference for a proof of the
Twist Theorem is Moser [1973]; see also Moser [1962 and 1967] and Siegel
and Moser [1971]. More recent developments are described in a review by
Moser [1986]. A comparable theorem was formulated for analytic Hamilto-
494 Chapter 15: Planar Maps
1111 ..
In the final part of our book, we break the barrl", of dimen-
sion two and venture into higher dimensions. Dynamical
diversity in such dimensions is truly bewildering. Con-
.. sequently, to bring our book to a conclusion in a finite
number of pages, we attempt here to convey the current
excitement of our subject with mere thumbnail sketches
of several prominent examples. This chapter consists of abbreviated geo-
metrical descriptions of two classical examples from the theory of forced
oscillations: Van der Pol and Duffing. These nonautonomous planar sys-
tems contain a term that is a periodic function of time--hence the title
of the chapter. Because of the time periodicity of the nonautonomous
terms, the qualitative dynamics of these equations are studied most con-
veniently in the space lR? x 8 1 . Indeed, since in this space both of these
equations possess global Poincare maps, the results from the previous chap-
ter become the natural mathematical backdrop. In the chaotic behavior
of Duffing's equation, the decisive role is played by transversal homoclinic
points of its Poincare map. We expound on this important connection by
including a description of the dynamics of planar maps near such points.
498 Chapter 16: Dimension Two and One Half
Xl = X2
(16.1)
X2 = -Xl + (1 - XDX2 + )..f(t) ,
where f is a T-periodic function of the independent variable t, and ).. is a
real parameter. The term )..f(t) is called the forcing function. When).. = 0,
there is no forcing and the system (16.1) is the oscillator of Van der Pol.
As we pointed out in Chapter 4, it is necessary to examine the tra-
jectories (Xl. X2, t) of the nonautonomous system (16.1) in lR2 x lR rather
than the orbits in lR2 • Equivalently, we may consider the orbits of the
three-dimensional autonomous system
Xl = X2
In this setting, let us first re-examine the case of ).. = 0 : the limit cycle, the
isolated periodic orbit, of the unforced oscillator of Van der Pol becomes a
cylinder; that is, topologically it is homeomorphic to 8 1 x lR. This cylinder
is an invariant manifold in the sense that any solution starting on the
cylinder remains on it for all positive time. Moreover, this invariant cylinder
attracts all nearby solutions. For)" = 0, the invariant cylinder is filled with
a family of periodic solutions, as seen in Figure 16.1a. Of course, the
cylinder under the projection lR2 x lR -+ lR2 simply becomes the limit
cycle.
Now, let us consider the case of periodic forcing with small amplitude,
that is, 1)..1 small. In this case, there is still a cylinder in ill? x lR close
to the invariant cylinder of the unforced oscillator. This new cylinder is
again an invariant manifold of solutions of the forced equation (16.1) and
attracts all nearby solutions. However, the flow on the invariant cylinder
of the forced equation can be quite different from the one of the unforced
oscillator. We have plotted in Figure 16.1b several solutions of the forced
oscillator (16.1) with the forcing function 0.2 cos 3t.
Next, we would like to view the solutions of Eq. (16.1) in a different
setting which offers several distinct advantages. Since the forcing function
16.1. Forced Van der Pol 499
Figure 16.1. (a) Solutions of the unforced oscillator of Van der Pol are
attracted to an invariant cylinder in lR? x lR. (b) The invariant cylinder
persists under periodic forcing with small amplitude. The forcing term is
O.2cos3t.
CUT PLANE:
xmin: x max:
-4.00000 4.000000
ymin: ymax:
-4.00000 4.000000
z min: z max:
-10.0000 10.00000
MAPPOINC:
xmin: x max:
-4.00000 4.000000
.. ymax:
ymin:
'
-4.00000 4.000000
CUT PLANE:
x min: x max:
-4.00000 4.000000
y min: y max:
-4.00000 4.000000
z min: z max:
-10.0000 10.00000
MAP POINC:
xmin: x max:
-4.00000 4.000000
y min: y max:
-4.00000 4.000000
Figure 16.2. An orbit and its Poincare section of the oscillator of Van der
Pol: (a) Unforced and (b) with forcing term 0.3 cos 3t.
manifold of the forced oscillator (16.3) and attracts all nearby solutions.
For A = 0, the flow on this torus is qualitatively a parallel flow which we
studied in Section 6.2. If A =I- 0 with IAI is small, the torus still persists
but the flow on it usually is not a parallel flow and may have orbitally
asymptotically stable and unstable periodic orbits.
The qualitative dynamics of Eq. (16.3) that we have just described can
be recast in terms of a Poincare mapj see Figure 16.2. To this end, we take
the plane L consisting of the points (Xl, X2, 0) as our cross section which
solutions pierce transversally in view of the last component of Eq. (16.3).
The Poincare map II is then defined by following the solution through
(Xl, X2) E L for time T when the solution next intersects L, that is,
II: L -+ Lj
16.2. Forced Duffing 501
where ip(T, Xl, X2, 0) is the solution of Eq. (16.3) through (Xl, X2, 0) at t =
O. Now, the global dynamics ofEq. (16.3) have the following interpretation.
For 1>'1 small, the equilibrium point at the origin becomes an unstable fixed
point of the Poincare map II. This fixed point is very near the origin. The
iterates of all other points converge to an invariant closed curve encircling
the fixed point; see Figure 16.2. This invariant curve is, of course, a cross
section of the invariant torus. Consequently, the asymptotic dynamics
of periodically forced oscillations with small amplitude forcing term can
be captured in terms of the dynamics of a circle diffeomorphism that we
have studied in Chapter 6. When >. = 0, the circle diffeomorphism is a
simple rotation with no fixed points. For 1>'1 small, however, it can have
asymptotically stable and unstable fixed points.
If the amplitude of periodic forcing is large, then the dynamics of the
periodically forced Van der Pol's oscillator (16.1) gets rather complicated
.0.0
as the map on the invariant closed curve ceases to be a diffeomorphism.
We refrain from delving further into these intricacies.
Exercises _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
16.1. Forced Van der Polan PHASER: The oscillator of Van der Pol with the
forcing term>' cos wt is stored in the 3D library of PHASER under the name
Jorcevdp. Study large amplitude forcing and observe the dynamics on the
invariant curve of the Poincare map. Also, investigate the effect of the
forcing frequency w.
Xl = X2
X2 = Xl - x~ + >.f(X3) (16.5)
X3 = 1 modT.
502 Chapter 16: Dimension Two and One Half
CUT PLANE:
x max:
@
x min:
@:2
-1.50000 1.500000
ymin: y max:
-1.50000 1.500000
z min: z max:
-10.0000 10.00000
MAP POINC:
."
....
. x max:
x min: ,.,
... ..'') 1.500000
-1.50000 /" (,
y min:
".
". , •• J Y max:
-1.50000 1.500000
o
CUT PLANE:
x lOtin:
-9.59999 1.599999
Y Min: Y Max:
-1..99001iJ 1..99999111
Z Min:
-UJ.9999 19.09999
MAP POIHC:
x Min:
-9.59999 1.599990
Y Min: Y Max:
-1.00990
CUT PLANE:
-1..59999
Y Min: Y Max:
-1.59999 1.590990
Z Min:
-19.9990 19.09999
HAP POINC:
x Min:
-1.59099 1.500909
Y Min: Y Max:
-1..50990 1..599999
of the origin and the dynamics near the homoclinic loops of the unforced
equation becomes rather complicated, as illustrated in Figure 16.5. In order
to describe this complexity in a bit more detail, we now turn to a general
exposition of orbits near a transversal homoclinic point of planar maps.
whose entries are chosen from the symbol set A. The set S can be endowed
with a metric to make it a topological space. Indeed, it can be verified that
d( ) +00
'""' 8n h 8 { 0 if Sn = sn
S, S = ~~' were n = 1 if Sn -# sn,
n=-oo
defines a distance between two bi-infinite sequences S and s. In words, two
sequences are "close" if they agree on a sufficiently long central block.
We next define a map 0' : S - S by
O'(s)n = Sn+1.
This map is a homeomorphism of S and it is called, appropriately, the shift
map because it simply shifts the entries of a sequence by one place to the
right.
For the convenience of labeling the orbits of 0', let us agree to use
an overline to denote the repeating segment of a bi-infinite sequence; for
example, { ... 010101.01010101 ... } is denoted by {01.01}. With this con-
vention, it is evident that 0' has two fixed points {O.O} and {TI}. Also,
it is easy to see that 0' has many periodic orbits; for example, here is a
periodic orbit of period 2:
{01.01} f-+ {1O.1O} f-+ {01.01 }.
Indeed, the shift map 0' has an amazing variety of orbits, as listed in the
following lemma.
16.4. Forced and Damped Duffing 505
Lemma 16.1. The shift map on the space of bi-infinite sequences with
two symbols has
• a countably infinite number of periodic orbits, including periodic orbits
of arbitrarily high period;
• an uncountably infinite number of nonperiodic orbits, including count-
ably many homoclinic and heteroclinic orbits;
• a dense orbit. <>
Now, we state the main result of this section which asserts that the
dynamics in a neighborhood of a transversal homoclinic point is at least as
complicated as that of the shift map.
Theorem 16.2. Let n : IR? -+ JR2 be a planar diffeomorphism with a
transversal homoclinic point q. Then, in any neighborhood of q, the map
n has a hyperbolic invariant set on which the iterate nn, for some positive
integer n, is topologically equivalent to the shift map on two symbols. <>
Since topological equivalence preserves qualitative features of orbits,
on the invariant set, there are many fixed points, periodic, nonperiodic,
homoclinic, and heteroclinic orbits. An equally important fact is the hy-
perbolicity of the invariant set. Although we have not defined hyperbolicity
of a general invariant set, the implication is that the invariant set persists
under small perturbation of the map.
The actual dynamics near a transversal homo clinic point is in fact
more complicated than that of the shift map on two symbols. Further
generalizations of shift maps on bi-infinite sequences with infinite number
of symbols are needed to account for the full dynamics. Instead of pur-
suing such extensions, we now turn to an investigation of the creation of
transversal homoclinic points in a specific system.
W"(P(~ WU{p)
p WS{p) p
I WS{p)
homoclinic transversal
tangency homoclinic
points
Let us first quickly note that the dynamics of Eq. (16.6) with damping
but no forcing, that is, when J-l > 0 and A = O. In this case, the w-limit set of
each orbit is one of the three equilibria (-1, 0), (0, 0), or (1, 0). Following
our previous convention, let us now view the solutions of Eq. (16.6) as the
orbits of
Xl = X2
X2 = Xl - xr - J-l X 2 + Af(X3) (16.7)
X3 =1 mod T
16.4. Forced and Damped Duffing 507
CUT PLANE:
x min: x max:
-1.70000 1.700000
Y min: y max:
-1.70000 1.700000
z min: z max:
-10.0000 10.00000
MAP POINC:
x min: x max:
-1.70000 1.700000
y min: y max:
-1.70000 1.700000
CUT PLANE:
x Min:
-1.79999 1. .799999
Y Min: Y Max:
-1.79999 1.799999
Z Min:
-HL9999 19.99999
HAP POINC:
)( Min:
-1.79999 1.799999
y Min: Y Max:
-1..79909 1.71iJ9999
Exercises _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4Q. 0
16.5. Damped Duffing is dissipative: Prove that the equation of Duffing is dissipa-
tive if there is positive damping but no forcing. Identify the global attract or .
16.6. Duffing on PHASER: The equation of Duffing with damping and periodic
forcing term A cos wt is stored in the 3D library of PHASER under the name
forceduf. In Figure 16.7, we implied that for f1, = 0.2, there is a homoclinic
tangency for some value of A in the interval [2.0, 2.5]. Try to locate the value
of A for which the orbit through the origin first becomes complicated. Also,
investigate the effect of the forcing frequency on the dynamics.
16.7. Two attractors: For the parameter values f1, = 0.3 and A = 0.15, the forced
damped Duffing has both a large stable periodic orbit and a more compli-
cated attractor. Find them on PHASER. Try the initial data (1.8, 0.0, 0.0)
for the periodic orbit.
PHASE PORT:
x l'IIin:
-10.0999 HI,9""""
y Min: y Max:
-19.9909 HJ.91iJ"liJg
z Min:
-19.9999 19.99999
MOP POINe:
x Min:
-HJ.liJ999 HJ.99999
y ,dn:
-1.9,9999 l.IiJ.9liUiUiJ9
PHASE PORT:
x .... in:
-HI. 99liJ9 19.09999
y Min:
-Hl.991iJ1iJ UJ.991iJ1iJIiJ
z ",in:
-10.9""" 19.9"999
MOP POINe:
x .dn:
-HL999' 19.99999
y Min:
-19. ",,99 HJ.9QI1UiJliJ
PHASE PORT:
)( Min:
-l.IiJ.liliCiUitliJ HiJ.91i11i1"1iI
Y Min: Y Max:
-HJ."""9 19.99999
::z; Min:
-1.9."""9 19,"""90
MAP POINe:
)( Min:
-H!J.CiJIiJQIiI 19.999""
Y Min: Y Max:
-19.9999 HJ.liUiJIiJ99
X2 = Xl + 0.2X2 (17.1)
X3 = 0.2 + X3(XI - A)
depending on the scalar parameter A. It is, of course, theoretically not
possible to locate a periodic orbit, let alone a Poincare map, of this system.
Numerically, however, it is a routine matter to see that there is an orbitally
asymptotically stable periodic orbit for A = 2.2. For A = 3.1, this periodic
orbit becomes unstable and a new stable periodic orbit of twice the period
appears. As the parameter A is increased further, a periodic orbit of period
4 appears, and the process continues. In Figure 17.1, we have plotted the
periodic orbit and successive period-doubling bifurcations. <)
Xl + Xl [X3 + d(1.0 -
=(A - b)XI - CX2 X~)]
X2 =CXI + (A - b)X2 + X2 [X3 + d(1.0 - X~)] (17.2)
X3 =AX3 - (xi + X~ + x~),
where A is a parameter, and b, c, and d are constants which we will fix
below. These equations represent an unfolding of the equilibrium at the
origin with a pair of purely imaginary and a zero eigenvalue. Here, we will
not attempt to elucidate the dynamics of this system in its full generality;
rather, we will be content to numerically illustrate only the birth of an
invariant torus. For this purpose, we set b = 3.0, C = 0.25, and d = 0.2,
while varying the parameter A.
For A > 0 and small, there is an asymptotically stable equilibrium
point, with a positive x3-coordinate near the origin. At approximately A >::::
1.68, this equilibrium point becomes unstable and undergoes a Poincare-
Andronov-Hopf bifurcation. The resulting periodic orbit is hyperbolic and
orbitally asymptotically stable.
At A = 2.0, the periodic orbit is no longer hyperbolic, but is still
orbitally asymptotically stable due to the nonlinear terms. For A > 2.0, the
periodic orbit becomes unstable and an attracting invariant torus appears
near the periodic orbit. As the parameter is increased past 2.0, the invariant
torus grows quite rapidly. This sequence of bifurcations is illustrated in
Figure 17.2. <>
Here, we will set a = 0.3375 and e = 0.633625, while varying the remaining
parameter b.
There is an equilibrium point p = (-lie, 0, 0) and the eigenvalues of
the matrix of linearization at p are approximately 0.4625 and -0.4 ± 1.1i.
Therefore, the first set of conditions for the theorem of Sil'nikov listed
above are satisfied. The last hypothesis on the existence of a Sil'nikov-
type homo clinic orbit is usually very difficult to establish. Ho:vever, for
some value of the parameter b near b ~ 2.16, it is possible to venfy for our
518 Chapter 17: Dimension Three
"
Figure 17.3. A Sil'nikov type homoc1inic orbit in Example 17.3. The ini-
tial values are (-1.576,0.0, 0.0) and b = 2.16.
Exercises - - - - - - - - - - - - - - - - - - - - . 0 . 0
17.4. On Example 17.3: This example is stored in the library of PRASER under
the name silnikov. Use it to recreate the figures in this section and for
further exploration of the dynamics.
17.5. Piecewise linear vs. smooth: The statement of the theorem of Sil'nikov is
really for smooth vector fields. With appropriate modifications, it is also
applicable to the piecewise linear example in the text. Now, consider the
following smooth version of our vector field:
Xl =X2
X2 =X3
X3 = - X2 - aX3 + bXl(1.0 - xI).
17.4. The Lorenz Equations 519
Figure 17.4. A complicated orbit of Example 17.3 with the same initial
data as in the previous figure but with b = 0.82.
(a) Take a = 0.4 and b = 1.6064. Verify that the linearization of the equi-
librium point at the origin satisfies the requirements on the eigenvalues.
(b) As is often the case with smooth vector fields, it does not appear to be
possible to establish the existence of a homo clinic orbit to the origin.
However, experiment numerically to convince yourself that there may
be such a Sil'nikov orbit. Use initial data near the origin close to the
unstable manifold.
(c) Study the parameter values b = 1.0232 and b = 0.872.
Assistance: These equations are stored in the library of PHASER under the
name silnikov2. Also consult Arneodo, Coullet, and Tresser [1982J.
Exercises _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ .. Q. (;
17.6. Lorenz is dissipative: Using the quadratic Liapunov function
17.4. The Lorenz Equations 521
-22. 9911U1I99
show that the Lorenz equations are dissipative for the parameter values set
in the text.
Reference: Consult Appendix C of Sparrow [1982] for the details of the
necessary computations and other possible Liapunov functions.
17.7. Elementary bifurcations in Lorenz: Fix b = 8/3 and (j = 10.0, and establish
the following bifurcations as the parameter r is varied:
(a) For 0 < r < 1, the origin is the global attractor.
(b) At r = 1, the origin undergoes a pitchfork bifurcation. To explain the
presence of pitchfork, rather than saddle-node bifurcation, verify that
the differential equations are invariant under the reflection symmetry
(Xl, X2, X3) f-+ (-Xl, -X2, X3).
(c) For r > 1, many global things happen, but they are hard to identify
theoretically or numerically. For example, at r = 13.962 ... one piece
of the unstable manifold of the origin forms a homo clinic loop. Conse-
quently, for values of r larger than this one, the dynamics get noticeably
complicated.
(d) For r = 24.74 ... , two of the equilibrium points undergo Poincare-
Andronov-Hopf bifurcation. It is subcritical, in case you look for it
numerically.
17.8. Period doubling in Lorenz: Fix, as usual, b = 8/3 and (j = 10 for the
following numerical experiments:
(a) For r = 100.5, use the initial data (0.0, 5.0, 75.0) to locate an asymptot-
ically stable periodic orbit. Watch its projection on the (Xl, x3)-plane.
Make sure to discard transients.
(b) Observe period doubling in the parameter range 99.524 < r < 100.795.
Try, for example, the initial data above with r = 99.65.
522 Chapter 17: Dimension Three
Bibliographical Notes - - - - - - - - - - - - - - - - - I f ®
Example 17.1 displaying a sequence of period-doubling bifurcations is taken
from Rossler [1976]. Period-doubling sequences occur frequently in systems
that eventually become chaotic. In case you have missed them, see the
exercises for the forced damped oscillator of Duffing in the previous chapter,
or the Lorenz equations.
Example 17.2 exhibiting an invariant torus is from Langford [1985].
The unfolding of a purely imaginary and a zero eigenvalue is a codimension-
two bifurcation. This important bifurcation, "interaction of periodic and
steady state mode," is studied in Langford [1979]; see also Spirig [1983].
The dynamics near a homoclinic orbit in three dimensions as described
in the text is due to Sil'nikov [1965]; generalizations to higher dimensions
are in Sil'nikov [1970]. Example 17.3 is from Arneodo, Coullet, and Tresser
[1982]. Recently, homoclinic orbits of ordinary differential equations have
attracted much attention; several representative sources are Glendinning
and Sparrow [1984], Guckenheimer and Holmes [1983], and especially Wig-
gins [1988]. Sil'nikov-type homo clinic orbits have been found in applica-
tions such as nerve axon equations; see, for example, Evans, Fenichel, and
Feroe [1982], and Hastings [1982].
The Lorenz equations command vast mathematical and experimental
literature. The original paper Lorenz [1963] is must reading. The book by
Sparrow [1982] and the references therein should provide sufficient resources
for further exploration.
1 8 - -_ __
Dimension
Four
fi) •
~
This is the final chapter! At the same time, it is the begin-
ning of a new geometric adventure into dimension four-
the hyperspace. Arguably, the most natural differential
equations residing in dimension four are the Hamiltonian
systems with two degrees of freedom. Hence, we have cho-
sen them as the subject of this chapter. Following a rapid
introduction to the setting of Hamiltonian systems, we outline a topolog-
ical program for the study of a small class of Hamiltonians---completely
integrable systems-that can be analyzed successfully. From this contem-
porary viewpoint, we then study the flow of a pair of linear harmonic
oscillators. Here, the term bifurcation gains yet another meaning in the
context of level sets of the energy-momentum mapping. Our success with
completely integrable systems is somewhat overshadowed by their rarity.
Indeed, a satisfactory analysis of a general Hamiltonian system in four
dimensions-unlike the case of the plane, one degree of freedom-is cur-
rently beyond reach. To hint at this complexity, we conclude the chapter
with an example of a Hamiltonian that, in all likelihood, is nonintegrable.
524 Chapter 18: Dimension Four
H: IR4 -+ IR;
where Xi are the position and Yi are the generalized momentum variables.
We will assume in the sequel that the Hamiltonian is at least a C 1 function.
The time evolution of the system is then governed by the vector field XH
given by the following system of four first-order differential equations of
Hamilton:
for i = 1, 2. (18.1)
°
Example 18.4. Pair of linear harmonic oscillators: Consider the follow-
ing family of Hamiltonian functions with m > and n > 0:
(18.3)
This is the sum of the energy functions of two harmonic oscillators with
frequencies m and n. The corresponding linear system has eigenvalues ±im
and ±in. It is easy to verify that the function
DH(x, y) + >'DL(x, y) = 0
for some real number >. -I=- 0, which is a Lagrange multiplier problem.
In case (i), the only critical point of H or L is the origin (0, 0, 0, 0) and the
corresponding critical value is 0. In case (ii), the solution of the Lagrange
multiplier problem shows that, for h > 0, the function LIH-l(h) has two
circular critical sets:
which is equivalent to
lB.1. Integmble Hamiltonians 527
2 2 h+£ 2 2 h-£
Xl +YI = --,
m
X2 +Y2 = --.
n
Thus, £M-I(h, £) is the product of two circles if h =F £ and h =F -£.
With these bits of information, we now draw a bifurcation diagram for
the level sets of the energy momentum mapping of the harmonic oscillators,
as seen in Figure 18.1. The set of critical values of £M are the darkened
half-lines, the range of the map is the shaded wedge, and the inside of the
wedge away from the boundary contains the regular values. Unlike our
earlier bifurcation diagrams, in the bifurcation diagram of £M we record
not the dynamics of the flow of X H but the changes in the topology of the
level sets of £M. As such, the bifurcation diagram is independent of the
frequencies m and n.
The flow of XH on the level sets of the energy-momentum mapping are
easy to describe. The level set £M-I(O, 0) consists of the origin, which is
an equilibrium point. For h > 0, the circular critical level sets £M-I(h, h)
and £M-I(h, -h) are always periodic orbits regardless of the frequencies.
The flow on each regular torallevel set depends strongly on m and n. If m
and n are rationally related so that m = (alb)n for alb a rational number
528 Chapter 18: Dimension Four
in lowest terms, then the flow consists of parallel (a, b) toral knots, and
every orbit is periodic, as illustrated in Figure 18.2. If m and n are not
rationally related, then every orbit is dense on the torus; see Figure 18.3.
This concludes our answer to the first global question we have posed above.
We now briefly turn to the second global question. A constant energy
surface H-1(h) of the harmonic oscillators is diffeomorphic to the three-
lB.l. Integrable Hamiltonians 529
18.1. Lissajous figures: The projection of a solution of a pair of linear harmonic os-
cillators onto the (Xl, X2 )-plane consisting of the position variables is called
a Lissajous figure. Draw some Lissajous figures for various values of the
530 Chapter 18: Dimension Four
(a) (b)
(e) (d)
(e) (f)
Figure 18.5. Another animated view of 8 3 : The two tori are closely en-
veloping the two critical circles. A third torus is shown as it moves from
one torus to the other. The third torus has been cut into bands to reveal
its linkage with the other tori.
with m > 0 and n > O. This is the energy function of two linear oscillators
"running opposite in time."
(a) Verify that H is completely integrable with the first integral
(18.4)
532 Chapter 18: Dimension Four
Let us begin with a relatively easy local question. Is the origin a sta-
ble equilibrium point of the Hamiltonian vector field XH? The matrix of
the linearized vector field at the origin has purely imaginary eigenvalues in
1 : 1 resonance, and thus its stability type cannot be determined from lin-
earization. However, since the quadratic part of the Hamiltonian function
is positive definite, it is not difficult to establish that the origin is stable.
The important question regarding the global dynamics of the Hamilto-
nian (18.4) is the existence or the lack thereof of an additional first integral.
To gather experimental evidence towards a resolution of this question, we
compute an orbit numerically and manipulate it graphically. For a crude
idea, we first project the solution into three dimensions, for example, the
(Xl, X2, x4)-space. This projection can be rather complicated. To gain
further geometric insight, we then plot the points of intersections of the
computed orbit with a two-dimensional plane, say the (X2, x4)-plane.
Since the Hamiltonian itself is conserved, a computed solution, barring
numerical inaccuracies, usually lies on a three-dimensional constant energy
surface in JR4. If there happens to be another first integral in addition to
the Hamiltonian, then the solution will be confined to a two-dimensional
submanifold such as a torus or a cylinder. If, however, there is no addi-
tional first integral, then the solution will typically wander in this three-
dimensional constant energy surface. These two cases can be distinguished
most easily on the planar section. In the presence of an additional first
integral, the points of intersection on the plane will yield curves; otherwise,
they will be a sprinkling of dots not lying on any discernible curve.
We have plotted in Figure 18.6 a sequence of numerically computed
orbits and their planar sections. In this sequence, we have selected the
initial data with increasing energy values. The results are rather reveal-
ing. For small energy values, there appears to be closed curves on the
planar section. As the value of the Hamiltonian gets larger, the curves
begin to disintegrate, which is a strong indication that the Henon-Heiles
Hamiltonian is not completely integrable.
A cursory explanation of the disintegration of the invariant tori for
increasing energy values is the following. For small energy values, we may
view the Henon-Heiles Hamiltonian (18.4) as a small perturbation of the
completely integrable Hamiltonian (18.3) of a pair of harmonic oscillators
because the quadratic terms dominate the cubic ones. Now the presence of
closed curves on the planar section for small energy values can be inferred
from the Twist Theorem by noting that on the planar section near the ori-
gin we have an area-preserving perturbation of a twist map. Consequently,
for larger energy values the perturbations become stronger leading to dis-
integration of more of the invariant curves.
Despite the strong numerical evidence above, the important question
still remains: Does the Hamiltonian (18.4) of Henon-Heiles have a first in-
tegral that is functionally independent from the Hamiltonian? The answer
18.2. A Nonintegrable Hamiltonian 533
c. _.-. ,.)
CUT PLANE:
-1IiI.5egaa 8.5aelln
Y Min: Y Max:
-1IiII.:seea8 IiI.SaIiMlIiCit
Z Min:
-le.aeaca 18.888aa
!II .&X:
-8.511l1li11 111.511111888
CUT PLANE:
:)
)( Min: :Ie Max:
-1iJ.581iJ1iJ1iJ 11,5889118
Y Min: Y Max:
-1iJ.511'''19 a.5eaaRS
UJ.IiJ"aIiJ8
MAP POIHC:
)( Min:
t·:::.:::~:-::: .._.:. )( Max:
-1iI.51iJ1iJ1iJ1iJ 1iJI.5eaellillg
)( Min: )( MaK:
-8.5I1aH 8.511Mea
Y Min: Y MaK:
-8.5HIiJIiJ GI.51i11iK11aa
Z lIIin:
-iCiJ.Hllla lliJ.iIIlilliHJa
)( Min:
-S.51iJ1iJ1iJ1iJ e.5SIiJIiHlIiJ
Y Min: y ...ax:
-111.5(18"" 8.51iJ1iJ1iJ1iJ8
Figure 18.6. Three numerically computed orbits and their planar sections
of the Henon-Heiles Hamiltonian with increasing energy. The initial data
for the first orbit is Xl = 0.34, X2 = 0.25, YI = 0.1, and Y2 = 0.1. For
the other two orbits, Xl is increased to 0.375 and 0.383. The orbits are pro-
jected into the (X2' Y2, xI}-space and the section plane is Xl = O.
534 Chapter 18: Dimension Four
is not yet entirely satisfactory. It recently has been proved that the ~enon
Heiles Hamiltonian has no analytic first integral; however, the nonexIstence
Q.
of a differentiable first integral has not yet been established. <)
18.4. Consult Henon-Heiles: There are some famous pictures in the original ar-
ticle by Henon and Heiles [1964] which you should try to duplicate on the
computer. How did they pick so many initial conditions on a given constant
energy surface? Do not be fooled by pictures so readily. When inspected in
detail, some of those nice looking invariant curves may not be curves at all.
Substantiate this remark on the machine using extreme enlargements.
18.5. Generalized Henon-Heiles: Consider the Hamiltonian
Bibliographical Notes _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ~
and Rod [1979]; its analytic nonintegrability was established by Ito [1985]
and Churchill and Rod [1988]. General results on the nonexistence of an-
alytic first integrals are in Ziglin [1982 and 1983]; see also the survey by
Kozlov [1987]. The question of C k vs. analytic complete integrability of
certain Hamiltonians is investigated by Oliva and Castilla [1989] and Gorni
and Zampieri [1989].
Celestial mechanics occupies a special place among Hamiltonians; the
classic source is Siegel and Moser [1971]. Moser [1973] describes some of
the more recent developments such as random motions, homoclinic points,
and ar.alytic nonintegrability. For lighter reading, try Pollard [1966] and
Milnor [1983].
A delightful survey of some of the influential numerical experiments in
Hamiltonian mechanics, including the planar restricted three-body prob-
lem, is in Henon [1983]. Many of the examples in this article are stored in
the library of PHASER, if you wish to experiment. All-purpose numerical
algorithms can fail miserably for Hamiltonian systems; see, for example,
Hockett [1990] for spurious chaos in the Eulerian numerics of the central
force problem, which is a completely integrable system. The development
of special purpose symplectic numerical integration algorithms appears to
be promising, as exposed by Channell and Scovel [1990].
Farewell
~
f(x) = L..J
1
Iii! Dd(a) (x - a)1 +
1
(m + I)! L Dd(e) (x - a)l.
lil:::;m lil=m+!
The next two statements are about the derivatives of the composition
of two functions, both in the scalar and vector cases.
Appendix 541
Chain Rule. Let f : lRk _ lRm and g : lRm _ lRn such that f is
differentiable at x and g is differentiable at f(x). Then the composite
function go f : lRk _lRn is differentiable at x, and
The second most important tool in local analysis is the Inverse and
Implicit Function Theorems.
Inverse Function Theorem. Let U be an open set in lRn and let f :
U - lRn be a C k function with k :::: 1. If a point x E U is such that the
n x n matrix Df(x) is invertible, then there is an open neighborhood V of
x in U such that f : V _ f(V) is invertible with a C k inverse.
Implicit Function Theorem. Let U be an open set in lRm x lRn and
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(x, y) E Vm x Vn and y ¥- 'IjJ(x). The derivative of the function 'IjJ is given
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in Patterns and Waves, 221-258. North Holland: Amsterdam.
Poincare map, 376, 976, 978. product system, 185, 187, 192, 192,
derivative of, 130, 136, 377. 214,309.
in four-space, 599. as gradient system, 438.
in three-space, 512. in polar coordinates, 199.
monotonicity of, 376, 977. linear, 187, 189, 191.
of Duffing, 502, 502, 509, 507. simplest, 185.
of one-periodic equation, 116, 119, purely imaginary eigenvalues, 251,
130, 121, 134, 136. 334-364, 995, 99~ 362.
of one-periodic linear system, 258, 258.
quadratic form, 125, 239, 279, 283.
of periodically forced oscillator, 500.
positive definite, 239, 244, 283, 279.
of toral flow, 149, 151.
quadratic system, planar, see Hilbert.
Poincare-Andronov-Hopf bifurcation qualitative structure of the flow, 27.
for vector fields, 211, 344-356,
qualitative theory, 65r.
344, 351, 362r, 410, 515. quasi-hyperbolic, equilibrium, 398.
generic, 359, 959, 961, 401.
periodic orbit, 398.
in FitzHugh neuron model, 360.
quasiperiodic, 132r.
in Lorenz, 521.
in partial differential equations, 363r. Rayleigh Equation, 355.
in predator-prey model, 361. Rayleigh-Bernard convection, 104r.
in Rayleigh, 355. reaction-diffusion equation, 434,
in Van der Pol, 346, m. ill, 439, 440r.
nongeneric, 961. regular value, 525.
m.
subcritical, supercritical, 346, resonance, 255.
Poincare-Andronov-Hopf bifurcation for strong, 481, ill, JJl§., 493r.
maps, 473-483, 474, 451, 493r. restricted three-body problem, 531, 535r.
in delayed logistic, 476, fl!J.. Riccati Equation, 126, 127, 132r.
subcritical, supercritical, 475. root computing, 71, 75, 76.
Poincare-Bendixson Theorem, 366, 367, root of unity, 456, ill, 474, 480, 481.
968, 969, 370, 387r, 415. rotation interval, 166r.
on surfaces, 387r. rotation number, 155, 165r.
Poisson bracket, 524. computation of, 165r.
pole placement, 277. in standard circle map, 159.
position variable, 524. irrational, 157, 164.
positive definite, function, 278, 279. rational, 154-156, 160, 162.
Hamiltonian, 531, 532. S3, see three-sphere.
quadratic form, 239, 244, 283, 279. saddle, 188, 191, 239, Mi, 272, 275,
positive orbit, l!, 10,11· 284,292.
of maps, 444. of map, 415, ill, 457.
positively invariant set, 287, 367, saddle-center bifurcation, 425, ~.
969, 971. saddle connection, 302, !!JM., 392;
potential energy, 414. see also heteroclinic, homoclinic.
potential function, 12, 19,11, 15, and symmetry, 403.
190, 191, 414. breaking, 303, !!JM., 401, JJl!!..
bounded, 420, ill, 492. saddle-focus, see Sil'nikov.
generic, 422. saddle node, elementary, 397.
maximum, minimum of, 423. saddle-node bifurcation, 26, 28, 28, 96,
two-parameter, m. 44, 204, 206, 316, 401, 437.
with degenerate critical point, 425. and even symmetry, 320.
with equal maxima, ~. in Henon, 470.
predator prey, 171, 180, 182, 215r, 961. in standard circle map, 161, 162.
discrete, 483. of fixed point, 86, 468.
first integral for, 195, 196, 202. of periodic orbit, 383, 11M, 401.
principal matrix solution, 219-222. of periodic solution, 141, 149, ill.
Index 567
unfolding, 249, 264r, 411r, 522r. hyperbolicity of its limit cycle, 379.
a zero eigenvalue, 249, 250, 252. in electrical circuits, 172, 172.
double zero eigenvalue, 250, 252, Poincare-Andronov-Hopf bifurcation
253, 406, m,408. in, 346, !M.§., 351.
double zero with symmetry, 408, positively invariant set for, 371.
Ml2.,41O. structural stability of, 392.
gradient vector field, 439. uniqueness of its limit cycle, 380.
pitchfork, 53. variation of the constants formula,
purely imaginary eigenvalues, 250, 251. 111, 254.
universality, in interval maps, 97, vector field, Q, fl, 23r, 178, 179.
100,103r. on circle, 21, 54, 55.
in circle maps, ill, 165r. on torus, 148-155.
unstable equilibrium point, 17, 266, vertical bifurcation, 209, 210.
272-277, 272, 280, 284. vibrating membrane, 435, J!lQ, 440r.
unstable fixed point, 73, 444, 455. Wazewski's Principle, 299, 300, 305r.
unstable manifold, 293, llJM, 295, Whitney topology, 393.
305r, 415.
and generic potentials, 423. zee-zeebar coordinates, 477, 480,
appro;omating, 296. 483,486.
global, 295, 297, 298. zero eigenvalue, 239, ~, 246, 249,
of fixed point, 457, ill. 307-332, 308, 309, 312.
and stability, 310, 311, 313.
Van der Pol's oscillator, 181, 215r, bifurcations from, 314, 316, 317.
276, 291, 370, 387r. unfolding, 250, 252.
forced, 498,~, 508r. see double.
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