Arnold, V.I. - Ordinary Differential Equations - Red
Arnold, V.I. - Ordinary Differential Equations - Red
Arnold, V.I. - Ordinary Differential Equations - Red
V. L Arnold
Page iv
Copyright O 1973 by
The Massachusetts Institute of Technology
All rights reserved. No part of this book may be reproduced in any fonn or by any means, electronic or
,nechanical, including photocopying, recording, or by any ,nfonnauon storage and retrieval syste1n,
\Vithout per111ission in \Vriting fron, the publisher.
Contents
..
Preface VII
.
Frequently Used Notation
Basic Concepts I
5 Nonautono,nous Equations
2
Basic Theore1ns
Linear Systems 95
13 Linear Problems 95
14 The Exponential of an Operator 97
Index 275
Page vii
Preface
In selecting the subject matter of this book, I have attempted to confine myself to the irreducible
minimum of absolutely essential material. The course is dominated by l\vo central ideas and their
ramifications: The theorem on rectifiability of a vector field (equivalent to the usual theorems on
existence, uniqueness, and differentiability of solutions) and the theory of o n e -parameter groups of
linear transfonnations (ie., the theory of linear autonomous systems). Accordingly, I have taken the
liberty of omitting a number of more specialized topics usually included in books on ordinary
differential equations, e.g., elementary methods of integration, equations ,vhich are not solvable with
respect to the derivative, sin&'lllar solutions, Sturm-Liouville theory, first-order partial differential
equations, etc. The last two topics are best considered in a course on partial differential equations or
calculus of variations, while some of the others are more conveniently studied in the guise of exercises.
On the other hand, the applications of ordinary differential equations to mechanics are considered in
more than the customary detail. Thus the pendulum equation appears at the very beginning of the book,
and the efficacy of various concepts and methods introduced throughout the book are subsequently
tested by applying them to this example. 1n this regard, the law of conservation o f energy appears in the
section on first integrals, the "rnethod of small parameters" is deduced from the theorem on
differentiation with respect to a parameter, and the theory of linear equations ,vith periodic coefficients
leads naturally to the study of the swing ("parametric resonance").
Many of the topics dealt with here are treated in a way drastically different from that traditionally
encountered. At every point I have tried to emphasize the geometric and qualitative aspect of the
phenomena under consideration. 1n keeping with this policy, the book is full of figures but contains no
fonnulas of any particular con1plexity. O n the other hand, it presents a ,vhole congeries of fundanlental
concepts (like phase space and phase flows, smooth manifolds and tangent bW1dles, vector fields and
one-para1neter groups of diffeomorphisms) ,vhich remain in the shado,vs in the traditional coordinate
based approach. My book might have been considerably abbreviated if these concepts could have been
regarded as known, but unfortunately they are not presently included m courses either on analysis or
geo1netry. Hence I have been compelled to present them in so1ne detail, without assuming any
background on the part of the reader beyond the scope of the standard elementary courses on analysis
and Iinear algebra.
This book stems from a year's course of lectures given by the author t o students of mathematics at
Moscow University during the acade1nic
Page viii
years 1968 1969 and 1969 1970. In preparing the lectures for pubhcauon I have received great
assistance fro111 R. I. Bogdanov. I wish to thank him and all my colleagues and students ,vho have
com1nented on the preli1ninary mimeograph edition of the book (Mosco,v University, 1969). I am also
grateful to D. V. Anosov and S. G. Krein for their careful reading of the manuscnpt
V. I. ARNOLD
Frequently U1ed Notation
Elements of a linear space arc called v«tors, and arc usually denoted by
boldface kuers (v, {, etc.). Vectors of the space R" arc identified,..;,h sctsof
n numbers, For cxirnptc, wc writev = (v1 , • • • ,v.) = v1 e1 + ··· + c,_.e.,
where the set of n vectors e1, • . . , e. is called a basi.s in R•. The norm
(length) of 1hc vector v in the Euclidean space Jl• is denoted by fvl =d the
scalar product oflwo vcclors v = (u., . .. ,v.), w =- {w., ... , wJ E- R• by
(v, w). Thus
{v, w) = v 1w 1 + · · · + v"w "1
lvl = J(v, v) = Ju: + · · · + •!·
We often deal with functions of a real parameter I called the,;,,,,__ Oilfcr
entiation with respect to I (giving rise to a v,/«ily or rat, of�) is usually
denoted by an ove.rdot, as in X ::;;; dx/dt.
1 Ba1ic Concept•
finite. Thus, for example, the classical (;\1'cwtonjan) motion ol'a system con
si sting of a finite number of patticlcs or rigid bodies comes under this head
ing. In facr, the dimension of the phase space of a system of" par-tides isjus•
611, while that ofa •i·stem ofn rigid bodies is just 12n. Asaamplcsofpro
ccsscs which cannot be described by using a finitc.dimcnsiooa.J phase spatt,
we cit� the motion of fluids (sLUdicd in hydrodynamics\, oscillations of
strings and membranes., and the propagation of waves in optics and
acoustics.
A process is said to be differ,ntiable if il$ phase space has the structure o( a
differentiable manifold and if its change of state with time is described by
differentiable functions. For example, the coordjnatcs and velocities. ofUK"
particles of a mechanical system vary in time in a differentiable manner,
while the motions studied in shock theory do not have the differentiability
property. By the same token, the motion of a system in classical mcclwiics
can be described by using ordinary differential equations, while other tools
arc used in quantum mechanics, the theory of heat conduction, h)·drody·
narnics, the theory of clas,icity, optics, acoustics, and the theoryofshod:
waves.
The process of radioactive decay and the process of reproduction ofbac·
2 Chap. I Buie ConccplJ
ccria in 1hc presence oro suffici ent amount ornucrienc medium afford two
more examples or detcrminiJtic finite-dimensional differentiableprocnsa.
- imenJional, i.e., 1he s1ate orthe proccu
In bo1h casesthe phase space is o n e d
is determined by the quantity or m311cr or the number or bacteria, and in
both cases the proccsJ is described by an ordinary differential equation.
It should be noted chat 1he form o(the differential equation orthe proccu
and 1hc very fact that we arc dealing with a dctcrmini1tic finitc-dime:nsional
differentiable process in the first place, can only be established experimen
tally-and hence only with a certain degree or accuracy. Howcvu, this
state ofaffairs will not be emphasized a1 every turn in what folio""• instcAd,
we will talk about real processes as ;r they actually coinc:idcd with our
idealized mathematical models.
1,2, Phase flow■• An exact formulation or the general principles just
presented requires the rather abstract notions or pl11n, spac, and �fa,,,.
·ro familiarize ourselves with these concepts, we oonsidcr an cnmplc due
to N. N. Konscantinov where the simple act or introducing a� spa«
allows us 10 solve a difficult problem.
Problem I. Two nonintcrsccting roads lead from City A to City B (Fig. I).
Suppose it i s known chat two cars connected by a rope oflength less than 'U
manage to go rrom A 10 B along different roads without breaking the rope.
Can two circular wagons of radius I whose centers move alon.g the roads in
opposite directions pass each other without colliding?
Solutio,r, Consider the square
M = {(x 1,x2):0.;; x,.;; 1,0.;;x,.;; I}
Cars
, Wagons
\
Fig. 2 Phas e space of a pair o( ,·chicles.
Sec. 1 Phase Spaces and Phase Flows 3
(�'lg. 2). The position of two vehicles (one on 1hc first road, thco1hcr on the
se cond road) can be characterized by• point of the square '-', we need only
let x1 denote the fraction of the distance from A to B along the ith road which
lie, be tween A and the vehicle on the given road. Clearly there i, a point of
the square /Ill corresponding to every possible state of the pair olvchicld.
The square A1 is called the phost sp,u,, and its points arc called pJ,,,r, ,-,.,,.
Thus every phase point correspond, to a definit e position of the pafr of
vehicles (apart from their bcing conne ctcd),and every motion olth<evehicles
is represented by a motion of the phase point in the pha,c ,pace. For ex•
ample, the initial position of the cars (in City A) oorrespondi to the lower
left-hand corner of the square (x 1 � x2 • 0), a.nd the motion o( the can
fro m A to /) is represented by a curve going to 1hc opposite (upper right
hand) corner of the square. In juSI the same way, the inilial position of1hc
wagons corresponds 10 the lowrr right-hand oorncr of the square (x, • I,
x, • 0), and the motion of the wagons is represented by a curve leading to
1hc opposite (upper left-hand) corne r of the square. But every pair of cun-cs
in the square joining difTcrcn1 pairs of opposite cornen must interxc1.
Therefore, no matter how the wagons move. there comes a time whc-:n the
pair of wagons occupies a position occupied a.t some time by the-: pair ofca.rs..
A t this time the diStancc between the ccn1crs of1hc wagons will be less 1han
21, and 1hey will not manage to pass each other.
Although differential equations play no role in 1he abo,-c example, the
considerations which are involved closely resemble those which will concern
us subsequently. Description of the states of a process as points ora suitable
phase space often turn, out to be extraordinarily useful.
We now re1utn to the concepts of dcterminacy,finjte-dimensionaJity,and
difl'crcn1iabili1y of a process. The mathematical model of a detcnninl$1ic
process is a plrasejlow, which can be described as follows in intuitive terms:
Let M be the phase space and x e Man initial state of a procc:ss't and lctt'x
denote the state of the process at time,, given that its initial sta.tc is x . For
every real I this defines a mapping
g': M ➔ M
of the phase space into itself. The mapping g', called the l•odDtma IIUJ#i"l,
maps every statex e Minto a new statcg'x e M . forexampl�,t0 is the' iden
tity mapping which leaves every point o f Min its original position. �for�
over
g
r ♦J
= g'i',
since the statey = gJx (Fig. 3) into which x goes after-times, goes after time
1
4 Chap. I Ba.ic Con«pts
t
I
y
R t
t inlo the same state z e g'J a:s the stare z • ,1 • •x into which x goa after
time , + s.
Suppose we fix a phase point x e NI, i.e., an ini1ial s1a1c o(1he proces.. In
the course of time the s1a1c of the process will change, and ,he point x will
dcscribc a phase curve (g'x, t e R) in 1hc phase space i\1. h isjus1 the f,.mily
of I-advance mappings g': M - M that eonstitutcs a phouji,,w, with each
phase point moving along its own phase curve.
We now turn to precise mathematical definitions. In each ca5C ,t\l is an
arbitrary set.
Definition. A family (g') ofmappings ofa set 1\1 into itself, labelled by the set
of all real numbers (1 e R), is called a one-parameter groupoflratlSj•Tllfllholu o(
Mif
g' ♦ s g'ft
• (I)
for all s, I e R and g0 is the identity mapping (which lu,-cs c-·cry poinl
fixed).
PMblem 2 . Prove 1ha1 a onc-paranic1ct group of 1ransforma1ions is a commuu.ln-r group
and th at C\'Cry mapping g': Af - ft1 i..s one-to-one.
t R
P ig . G An integral curve in extended phase s:pa.«.
Definition. The image ofR under the mapping (2) is called apA,,,J,tlm'Wofthc
Aow (M, {g'}). Thus a phase curve is a •uh.ct ofphase space (fig. 5).
Prc blun J , Pr ove that there is one and only one J)Nsc cur,.-c pusing th.tough � p:lllnt
of phatt spac e .
1.4, Vector field,. Let(,11, {g')) bca phase now, given by aone-panmetcr
group of diffcomorphisms of a manifold ,11 i n Euclidan space.
Definition. Uy the pha,, «IMity v(x) of the now,' at a point x e ,If (Fig.7) i1
meant th•• vector rcpr.-,enting the velocity of motion of the phuc point, i.e.,
The left-hand side o f (3) is often denoted by x. Note that the dmvati•e is
defined,since the motion is a differentiable mapping of a domain in Euclid
ean space.
Pro�l,m I.rro\'e tha1
�I
at ,••
g'x - v(1'x),
I.e .. that a1 evcr·y innant of 1imc the vector rq>rntntfog tM \·c:loci1y ol mot.oft ol 1hc'
phue; po i nt equals the \'CCtor rc:.,.-�cntin.g 1hc ph� vc:lociey at 1)1t \ff'Y pouu o( phaK
space c,<cuplttl by 1hc moving poin1 at the gi\'c-n lime-.
Uinl.Sec (I). The i1olution is given in S« .3.2.
Irx1 , • • • , .r,. arc the coordinates in our Euclidean space, so that
x1: M - R, then the velocity vector v(x) is spttifial by • functions
v,: M-.. R,i •I,,., ,n,callcdthccomponmtsofthcvdocityv.:ctor:
v1 ( x ) = !!..I x,(g'x).
dt ,so
Pto611111 2. Pro\·c 1ha1 ,., i, a fu.r)c1iot1 of class �- ;r the on�pan.mc1cr group
1
- .,.---- ·-----
.,.
H
,,,,,,, .r,
\\\ \\\
I I I I I I
I I I I I I I
Fig. 10 1'hc phase plane (or Vt'rtical ran,
i.e., 1hc vcc1or field v on 1hc half line i s dircc1cd 1oward O and lhc: magni•
1udc of the phase vclocily vcc1or is propor1ional 10 x .
Example 2. h i s known from expcrimcnl 1ha1 1/u r,production r•U of• c.'-.,of
bacteria supplied with enoughfood is proportional to tlu 'l"antit., x of6octau pr,,,,.t
at any given time. Again Mis 1he half-linex > 0, bu1 1he vec10, fidd dilfcn in
sign from that of the previous example:
Example 3 . One can imagine a situation whc:rc I.he increase i.1prt>/Jtn'tiDIIJJ.I to tAt
Iota/ 11umbtrofpairs prtsent, i.e.,
Example 4. Vertiealfa/1 ofa particle to the ground(starling from not too great an
ini1ial heigh•) is described experimentally by Galileo's law, which a.ucrts
1ha1 1he accclera1ion is constant. Herc the phase space .I/ is the plan<:
(x,, x2) , where x 1 is ,he height and x2 the velocity, whi le Calilco•s law is
expressed l>y formulas like (3), namely
*, = -g (7)
( -g is 1he aceclcra1ion due to gravi1y). The corresponding vcctoe field of
1he phase velocily has components u, = x1, •, =
-g (Fig. 10).
10 Chap. I basic Con«pu
' ,"-"
�
/
,
,, ,- ,
//
/
.z;
- - ,.,.
Fig, 13 The ty1indriol ph:ue ,,)ace or a pendulum.
Fi3, I◄ Solut ion or tht diffcrtntial tquation l - v(..-) aati,fyinc dw in.ital condition
•(to) - .--o,
regard 1hc phase space of the pendulum as being the surface of the cylinder
(x I mod 2n, x1) ra1hcr than the plane (x1 , x1 ), since changing 1he angle x1
by 2n docs no1 change 1hc s1a1c of the pendulum. The vcc1or ficld o o r
rcsponding 1 0 (9) can also be regarded as defined on 1hcsurfaccof a cylinder
(Fig. 13).
/>rob/em I. Sketch in1cgral cur\'� for Examples 1-3 a.nd �tu�•cs fo,r Examples 4 and>.
VVe now show how 1he operation ofin1cgration (wi1h 1hc hdp ofthe funda
mental theorem of calculus) alJows one to solve diffen:.ntia1 equations d�ltt•
mined by vec1or fields on the line. We begin b y introducing some definitions
1hat will l>e used repeatedly below.
x a v(x), XE U . (I)
t O ffC"rcntial equat ons arc sometimes sa d to be f'qua.honscontaming �--n rlVIC:tioos
i i i
and their dt"ri\'atiVt':S. This is faJst'. For example, the �uatlon
-j; = x(x(t))
d•
r
.. ,. ............ .
I II II
.. .. ,,...JI........
,, JI ,,
tq
(b)
'
fig. 15 A direction fif!ld (a) and intC'gral cun-a (b) in ,,ucndnl pha,c spa<<.
I - 10
J de if v(x ,t- 0,
= O(•> -
•• v(e)
,p(I) = Xo
0)
if v(xo) = 0.
(3)
Remark. Since v(e) is a known function, formula (3) allows us 10 find tbc
function ,t, inverse 10 ,p (t = ,t,(x), q,(1) a x) by quadratures. \Ve can I.hen
use the implicit function theorem ,o find ,p . Thus formula (3) leads to the
solution ofequation (1) subject 10 the condition (2).
14 (;hap. I lb,ic C:onccpu
r
.,
.l'q
I'
'
Fig. 1G A i.olu •i on ,, and i11 i nverK fu nc liou .,.
- v-
I
-
(� . )
Since v(x0) ,;, 0, the function I /v(�) is continuous in a sulficicntly small
neighborhood o f the point {
• = .,
0• and hence
,J,(x) - ,J,(x0) •
J d�
-
•• v(�)
by the fundamental theorem of calculus. This uniqudr dcfin<S t/, in a suffi
ciently small neighborhood of the point x = x0• The function 'l' im.-......- to v
is also uniquely defined in some neighborhood of the point 1 • 10 by the
condition ,p(t0) a x0 (the implicit fimction theorem is applicable .since
1/v(.,0) ,;, 0). 'rhus any solution of equation (I) subject to thccoodition (2
satisfies (3) in a sufficiently small neighborhood of the point I =
10, and the
uniqueness assertion 2) is proved.
c ) We must still verify that the function ,p inverse 10 ,J, is a solution of(I)
and (2). But
dip d,J,- 1
_ =_
di dt x•.,C•>
=
(
-I )- ']
v(x) x•,,ci>
• v(ip(I)), ip(lo) • Xo,
:r
Ill/ /Ill /I/ //
II/I/ /I/Ill II
/Ill/I 1/llb'/II
,,,,.,,,.,,.,,,..,,,..,,,.,, � //;;
,,.,,,..,,, -.,,,,, ,,,,.,,,.,,,,,,,
.,,,,. ,,,,, t
//;}'/I/// '/II/Ill
I/.IIIII// /Ill/I
1/.IIIII/I/ /Ill/
thi, example does not contradict the theorem as stalLd. Ho..,�-cr th<e proof
just given makes no use ofthe differentiability ofvand goes through even in
the ca.�c where the function vis rncrely continuous. Hence Lhc proof cannot
be correct as given. Jn fact. the uniqueness assertion 2) was proved only for
the case v(x0) ,ta 0, and we sec that if the field vis only continuous (and not
differentiable), then uniqueness may well fail for solutions satisfying th<e
condition q,(10) = x0 whc,e x0 is a singular point (v(x0) = 0). It cums out,
however• that differentiability of vguarantees uniqueness C'\-cn in this cue .
2.6. Example. Let v(x) = kx, U = R (Fig. 18). Using (3) to soh-c 1""
differential equation
= kx, k¢0 (4 )
r•<•>
X
of the form {I) subject to the condition (2), we get
( )
I - lo = d� =� In q, t
•o k� k x0 '
'
1:
..
'c.-,
'-- {JOO/
ii
z 10(/
IQ
JJ,0611111 3. Prove 1h,1,1 all ,h� aolu1ion1 or «1ua1wn (♦) a.a1 i,ry 1ng th<t 1,uual cond,UOl'I
•(10) - •o • 0 are Rl,o given by forinu li. (�).
luhould he noted that none o(thc (unctions(�) with x0 ,. Ovanuha for
any value ofI, I lcncc the unique 1olution orequation (◄) such that x0 • 0
is the stationary solutio,1 x • 0. Thu,jormw/o ($) •t<•••l1jor el/ tit, selwt1••11•f
th, 1/ilf,r111lial 1q11atio11 (4).
In particular, the uniqueness assertion o(Thcorcm 2.3 is "�id ror aaua•
tion ( 4). F'rom this one can easily in(er uniquencu for an)' equation (I) with
a difTcrcntial,lc vector field v and for more general equations a.J ��IL
The rea,on for the failure or uniqueness in the case v(x) • x'1 ' is that this
field docs not foll off(ast enough as the pointx • 0 is approached. TMrefore
the solution manages to arrive at the singular poi nt in a finjtc tinx. An
infinite time is required to reach the singular point in the c:ax v{x} = h,
since the integral ·curves approach each other cxponcntiaUy. It i1 char•
actcristic ofany differential equation with a differentiable \.Utor field v
that its integral curve, do not approach each other more rapidly lhan ex
ponentially, thereby accounting for the uniqueness. In particular, lM
uniqueness proof in Theorem 2.3 is easily obtained by comparing tM gen
eral equation (I) with a suitable equation ofthe form (4).
2.7. A comparison theorem. Let v,. v2 be real functions continuous on an
interval U of the real axis such that v1 < v1, and let q,1, q,1 � .solutions of
the differential equation.,
x = v1(x), x = v1(x) ( 6)
respectively, sat isfying the same initial condition ,p1 (10) = q,1(10) = .r0
(Fig. 20), where ,p,. ,p1 arc both defined on the interval a < I < 6 ( - ao �
a < b � + oo).
THEOREM. The inequality
,p, (1) � ,p,(1) (7)
holdsfor all I ;,, 10 in lht interval (a, b).
a t, p t
f ig . 20 The slope or•i is grc.atc-r 1han that or•• at pOints. •ith equal._ but .l'IOt a.tpomu
with equal I ,
18 Chap. I &,ic Concepts
J>roef. The inequali1y (7) is almost obviouJ ("1hc 1lower rider docs no1 go
further").t More e••e1ly, lei T be 1he lc.u1 upper bound of 1hc Jet ofnum
bers t 1uch that (7) holds for all I, 10 < I < t. By hypothc.iJ, 10 < T < •.
l f T < h, then ,p 1 (T) • ,p 1(T) by thc con1inui1yof ,p,,,p 1 and
d,p
-, 1/,p
<-,
di ••T dt ,-r
by hypo1hcsis, so 1hat ,p 1 < ,p1 al all poi nu I > Tsuflicicntly near T. Bui
then Tcannol be the indica1cd least upper bound. Thi, contradiction shows
that T • b, as asserted. I
R,mark. In the same way, it can be shown 1hat ,p 1 (t) ;;. ,,.,(1) for I< 10•
2,8. Complcdon of the proof of Theorem 2.3. Lei .r0 be a stationary
point ofa difl'crcn1iable vec1or field v, so 1ha1 v(x0) • 0. Then, ai we no w
show, the solu1ion of equation (I) sa1isfying the ini1ial c,ondition (2) is
unique,i.e., if ,pis any solu1ion of( I) such 1ha1 ,p(10 ) • .r0, then i,(t) • .r0 •
There is no lo,s of generality in assuming 1ha1 .r0 = 0. Since the field vis
difl'crcn1iablcand v(O) • 0, we have
lv(x)I < klxl (8)
for sufficiently small lx l ,6 0, where k > 0 is a positive constant. The r�
quired uniqueness now follows from 1hc fact that the integral curves of
equation (4) 01hcr than x � 0, which arc steeper near x � 0 than the in1t
gral curves of (I), cannot reach the line .r = 0 in a finite time, as alttady
noted in Sec. 2.6.
This ca.n be prO\'cd mo1c rigorousl y, for �mplc. as follows: Lei • bt a dutioo of
(I) and (2) such that •(10 0 (fig. 21). and suppose•<•,) > o. t, > '� Sintt • is a
) -
con1in uous func1ion, there c.xis'ls an interval ( I , 1.,) with the- follo""ing propatia: I)
i
•Ct,) -r 0, 2) •Ct1) > 0 for 11 <I< 1.,, 3) x. -•(I) satisfies (3) for / < t <: t.,. lo fad,
J
for 11 �·e can choose the grea1cs1 Jower bound oftbc r such that .(r) > 0 for r <, < '•
and for/J any r>oin, t, > l $Uffieiently ne.u ' •
l l
'l'_(t)
t, t t, t, t
Fig. 21 The soluli on • cannot vanish since it approaches zero more dowty thua t.bt
e:xponen tial •i•
Nevertheless we note that the rate of change of • 1 ct• rivm insl.aNCUI bC' largu th.ut
the rate o(change or •l :u the same instant (Fig. 20}.
t
Sec, 3 Phase Flows on the Linc 19
We now com1Ml'c the ll(;lutloo •(I), 1, c. I < I., ...,.hh 1ht ..olu1ior.
.,(1) • .(,,) ,'11•1,t
of equation (it) 11ubjec1 t0 1ht lnl1lal condhton •,(,,) •C•,). 8«,iu,t of (11. 1k c.....
'"'''"on thtortrn lmpllt,
•(I)> i,(t,)t 141 1•1
(o,r all / < t <,,,and hence
J
•Ct,) )i •(f,)tlll1
•1, >Q
)
be divergcrll at -'o•
P,Hlrm 2, Prove the "niqucntNi a.ssenion for chc d1ff'c�ndal c-quati.on J - v(i,, -�
v i1 a d irftrcn1iablc funclio n, iu.,uming the t>d
1 s:tm« ofa .tOlution x •t) utiJying lM
initial co nditi on •(10) • x0.
flint , l.A::t)' - X - .c,). and make a comparlSOR with a suit:abic cqu.atioft ,.., .
We now define a ••t•aduance mapping g': R - R.," carrying tht: initial c:oodj
lion x0 into the solution after time t:
g'xo = l-'xo.
The family of mappings {g'} is called thcpltas,jlow as10<ia1<dt1,itl1t'fl'IUWII (1),
or with the vector field v = k x . ="ote that the- mapping g·' is a linear trans
formation ofthe line, namely an expansion of the line-I' times.. For-a.rbitrary
20
v(x) = !..\
dt ,. 0
g'x, XE U.
·rHEOREM. T/11 motio,i ofihe phnst poi11t cp: R - U, cp(t) = g'x isos.Loti.oftlu
dijferendal equalio11
.i: = v(x). (2)
Proof. \iVc need only shov.• that the velocity of motion of the ph.a.sc point,t'x
at every instant oflime 'o coincides with the: phase velocity at the: pointr x. .
This is obvious. since the transformations g' form a group:
di
dt
g'x
r•t1;1
= !_I
dr r•O
g'•"x = !_
dt r•O
g'(g'•x) = v(.('•x). I
t Note that dijft-re.rttiabilily with respect tO l is implicit in the definition o/a one-pu1mreter
g roup oflinear lra11s(orma 1ion.s g'.
Ser. 3 Plo:1<l' Flm" on tile I.inc 21
li11tnr(y, ,iurc th,· dtoiva tiv,· (d/dt) I,. 0 "it h rt'<p«t to tile p•r.amctcr I o{ tll<'
ru,wtiong(11 x) • g'.:.. wldc- 11i, linear in" i, it�lflinrar in x. t In 1>aruc--ular,
if/, ii tlu. · f<•al linc: R, tlu.·o cvt·ry fu 11<'llon linear in\ i,ofthc fnrm v{.t) A:..t
wllcr'C J: • v( I), Thrrrfi,rc tht' modt111 cp(I) • ,'-.: i,"
�lutton �<'qUahon
(2) wit ii v(,) • 4'·, i.r.. a ,nlu tion of rr1 uotinn (I). Si 11,c the un,q� "'lution
cp or thi, cqtHit ion �" ti,f) •ing tl1t cnndition cp(O) • r i, oft l1r formi' x • l''x,
tl1e proof of1'hcorcm 3.1 is now t·omplc t. c . I
•P,-ohlnn I. Prove lhlll �vNy cOntinuou, Onc-parurK'tt'r group of lino, 1nnJorm.a1ioM
or the linr i� au1oma1irally dilfe�n11ablt-.
/-lint, Kc.·c.all 1he drfiniti(ln of 1he uponrn1ial (1,.11M"1ion fol" intt'gul, nriorw, and .,-ra
tional valuo of the argutnC'nl,
C,,,,.,n,nt, Thu• i,1 dc·fl,1ing a o,1c--•1-...,11mr1C'r group or hn('ar ir;ansfonna,ttoM -.� could
hw: r rr 1,laC'C'd the rt•t:1uirc-mt"nt th.at lht" tran�forma.oom: x' br- d1fTrr<-n t,�bk -.oh�
IO t by the • f"Nl\.lir(•ow,u that 1hcy bt con111luous 1n I
•P,Hlm1 2 . f.',nd all Ollt"•parame1C'r groop, of t,nta.r rra.n-.:rormati QoM o{ tlk follo,.-,ng
linear spat-e s: a) RJ (the t('al planI.'); b) C' (the: comp• cx l line, u·.• tbr ont--d.�t
lin('ar spaC'c: ovt'r the: lie-Id ofC'om1 , lt'x 11umlJ.l•n),
Hint, In Chaf>. 3 we will dc·:1cribc all nn t,.parant<'lt'1" groups o r li n('.-r 1r.1m;b-ma.tions ol
1hc: n-dimcns ion al real a nd complex sp,.--ic:cs R" :1nd C- .
3.4. A nonlinear example. Next we consider the more complicated difT�r•
ential cqualion
X = sin x 1 XE R.
P,obl�m I. 1-�ind thc solution or rhi'I equation sa1i1.()·i ng the ini,�I rondi1ion •'0 x.,
Her't we can again define the /•advance mapping
g':R-R, g'x0 : ip(I),
where ¢(1) is the solution sacisfyi,,g the initial condition 9(0J = x0. 1nc
mappings g' form a o n eparameter
- group of d.iffeomorphisms of th� line,
namely the- phase flow associated with ,he given equation. 1nc phase Row
r
{g') has fixed poinL< x : kn, k : 0, ±I, ... , and the diffcomorphisms
g'(t � 0) are nonlinear transforrnalion.s of the line. TI1c lransformalion
shifls every point:< toward the nearest odd multiple of n i f t > Oand toward
the nearest even multiple of n ifI < 0 (Fig. 22).
/>Tobltm 2 . Provc tha t thc scqucn« offunctions t"•• t, -oo con,•crgcs, bu:1 not uniiorml y .
The above examples give rise to the hope that with every diffe:tt.ntia.J
equation on the line
x = v(x), xe R,
f Note: that the: li near nonhomogcncou.s func1ion/(z} = ax + 6 t'a.ils IO bt linear if• � O.
22 Chap. I Baiic Conttpll
.i · :r
f
rt
t
0
♦
-rr
-
.r :;r:
I 111111
' ,,,,.,,.,,,,,, ,,,,
' ,,,, .,, .,, I' ,,,, '
t iillll
:;r:
I - lo =
J•"l d{
'• r!
{
given by formula (3) ofSec. 2, often written in the form
Sec. 3 Phnsc Flow, on the Linc 23
r • Jrd�'.,
Jdt (S)
I I
t• --+C, X • ---,,,
X I- C
0,1e ,nu,1 not thi nk tluu the last rormula i1 cquivalcn1 to (3) nr 1ha& 1hcfunc-•
1io11 x - -1/(t - C) is a solu1ion. In rac1, 11,c domain of definition oft�
ruuctio,1 x I!!! - I /(t - C) i s oot an interval but ra,hel'" 1w o interval.st < C
and t > C, so 1ha1 1hc rcS1ric1ion orx = - 1/(1 - C) 101hcsc in1crvalsgi,�
two solutions whicharc in no way related to each other (aJ long a.s we c-onfine
oursclvt·s 10 the domain of real t, the only case con.sider«! in thiJ booki.
These considerations show that if the growth ofa population i.1 pn>por-•
tional tothe nurnber orpairs, then the size ofthe population becomes infinite
in a fi11itc time (wl,ercas 11,e usual law ofgrowth is cxponcn1ial,. Physic.ally
thii conclusion corresponds to the explosive nature ofthe proass '.ofcourse,
for I sufficic11tly 11ear C, t lie idcaliz.atiou emailed in describing the process by
the diffcrc11tial equation in qucs1iou become inapplicable, so that the si�c
of,he population does not actually become infinite i n a finite ti� . On the
01hcr hand, we see tha1 /ht formula far tltt t-adcanct mappi� (t'x0 • ,pit
whtrt ,p(/) is the solution satisjj•ing tht initial condition ,p(O) ; x0) dJ,a -Kin a
dijftomorphism g': R - Rfor any t � O.
P1oblt11t I. PrO\C
' 1he i1alic-i:tcd tnst':rliOn,
3. 6. Conditions for the existe.nce ofa phase Bow. The reason why{.(}
in tl1e precedi ng probl�m i.s not a one.parameter group or diffcomor-phis.ms
is 1101 1ha1 diffcren1iabili1y fails or 1ha1 1hc group propertr breaks down, but
simply thal the function g' (t 'i: Oj is not dtfin�d on t.M whole x..
&ru. sintt some
solutions manage to become infinite in a time not exceeding I (fig. 24,.
Howevt>1', if the solutions do 1101 become infinite in a finite time, then the
assertio11 made a1 ,he end orScc. 3. 4 is indeed valid.
Pr ob(e,n I. PrO\-C the: asst':rtion at the: <"nd ofSec . 3.i. an:uming th.tt che (UftC"ltCllft vis difftt-•
entiable and identically uro for sufficiently l:tl'gll" lxl.
.r
I/int, 1'hr 11olu1 l on I• coruaulrd i11 1htt proof or II nH)rf'l'"Mf'ill 1hrortm, .,.h.ll(h Mttth th.a,
('Yrry dirferc.•111IBblr \ '('flOf' 11rld on" ttimf>MI rnan,rold i, LhC" phut \tlor•lY 6rld o/a on,-.
p;, rM11r1rr group or
difrto••�rphi,m, (\tt \rc-. 3'il.
C11mm,nt, Thu• 1hr po\\ibilhy or 1hr rou ,11tttumplt ol \N'. S.� �•rm, (,oan •hr ,••
pnr,uor of ,ht" IInr,
or
1•,H/1"1 1, J'rO\'f 1hr ,.,�rrtion Stt-. 3.!I, lll•'lum11111h.l.1 l"{.dl � Al•I .,. "for .11
when� A :, nd II :trc p-0\ilivc com lAnh,
If."'"
I/int, U,c the rompari1on Thcorcn, 2, 7,
x, = v1 (x,),
=
(I
x2 v2(x2), (2
x = v(x), x e U,
=
(3
{ �• = x, Eu.
x =
v 1 (x 1 ), CR,
(4)
2 v2(x2), x, Eu, C R.
THEOREM. Ifq, is a solution of th, dirt<t prodU£1 (3) ef1k di.ffert1c1u,./ (I).,,.,.ti,.,.,
and (2), then q, is a mapping q,: I - U eftheform q,(t) = (q,1 (1), ,p2 (1,), u:hn-,
q, 1 and q,2 are solutions of equations (I) and (2) d,fin,d on one and th, -i.Jn,;,,lI .
In particblar, if the phase spaces U1 and U1 arc one-dimensional, ""�
Ser, 4 Vc<·tor Field, ,11,d Pha<c flow< in th e Plane 2.S
kuow how 10 ,ol,c carh or1he cqur11ior1, (I) Jnd (2). 1h ercrnrc ... ��n al,o
cx1 llid1ly ,olvc the •y<tem or two differential e'lu•tion, (4).
11 1 f11n, by 1'h<'orem 2.3, tht \Olu1lon • ,ati,ryn13 thf' 1111 11.at rond1d0n•fe) .., ,..• can bf,
found in a 11righborhood or 1hr I"°"" , ,0 from thf' rrb110,u
J• .rn
Au YI J !.fu '• I.l'tt.,.,,�
v, ) ' Xo (•1♦• -' It
irv1(x 1 o) P. 0, Vio(x,o) Ii 0 . lfv1(x1 0 ) • 0, the '1r,1 rclatiot'l l' n:plattd by •• .... ,..
while i( v1 (x10) 0, the: M"<'Ofld rda ti<m ii repl.a«d by"' ,,.. fuu1U y ,/ "• .-,.
v,(x 10) 0 . then x0 is a iingul .ar point of1he \fflOf' fidd v .and an �uil1bnum pouu,o;•
of th«- ll)'lltem f'f), i.e., .,(t) ■ x0. I
4.2. Examples of direct products. C'.c:>nsidcr the following system o( two
differential e<1muion�:
{x, - ·'"11
.\·1 • k:, l·
Problnn I. Sk4"tc-h th'° forrc:�1,ondi113 \'fftOI' fidd,i: ;n 1� pl;1nt-for A-
• ._I. I, 2.
O
V't/e have already solved each or ,hesc equations st"paratdy. Thus the
solution tp satisfying the initial condition �(10) = \'o is ofth<" fonn
,o _ X ,1(, - to)
't'l - 20 (5
Hc.·ncc along evety phase curve., = tp(I) we have chhct x1 :. Oor
lx,1 = C:lx,I', (6)
where C is a con stafll independen t oft.
Prohltnr 2 . r, 1hc (' u n·t: i n 1hC' pha�· 1>lanc (v,, ..-1) gh-m by (6) .i. pha'W:' ('Un.it?
.◄,u. �o.
The family or curves (6) where Ce R 1akcs variou• rorms dependi ng on
the value of 1he parameter k. Irk > 0, w e get a family of •·g eneralized
parabola� of expon ent k,"t where the parabolas arc ta nge nt to the z, .. axis if
k > l and 10 the - <,·axi, irk< l (Figs. 25a and 2:xc). I(k = l. WC get a
family or»raigh1 lines going 1hrough the origin (Fig. 25b). The arrange
mc.·nt of phase curves shown in Fig. 25 is called a node. Fork < 0 the curves
are hyperbolas (Fig. 26),t rorming a saddle poi111 in a neighborhood or ,he
origin. Fot k .- 0 the cutves llttn into straight lines (Fig.27).
I I is cleat from (5) that cvcty phase curve lies entirely in one quadrant (or
or1 one half of a coordinate axis. or possibly coincides with the origi n which
is a pha.se curve for all k). The arrows i n the figures show lhc direction of
I I
d'lt�
�tcr�
Fig.26 A saddle point: Phasccurvoofthesys:trm.i, r1 ,i1 ,b-h.l<O.
-'�
-·-
-·-
-·-
-·-
:=:==
-·-
-,-
.r,·
a:,
f'ig. 28 Pha•c Row or 1he 1yt1em 11 - x1, 1, z...,.
Fig. 29 Phas(' Row or1h( sys1em .i1.,. x1, .t2 - -,r1, Th(' tranJo,-mationst''att nlkd
hyperbolic: rota1ions.
( ,, 0 )
0 ,,
..
in the system of coordinates X It X , The difTcrcntiability of
,2
rx
"';,h respcc•
10 t and xis obvious. Thus the mappings g' form a one•parametcr group of
linear transformations of the plane. The action of g', l = I on a set £ is
shown in Fig. 28 for the case /c = 2 and in Fig. 29 for the case .t = -1.
It should be noted that our one-parameter group of linear t.ransforma
tions g' of the plane decomposes into the direct product of '"''O onc•para•
meter groups of linear transrorrnationsorthc line (namely expansions along
thex, •axis and expansions al"""g thexraxi.s).
P,obltm I . Oocs ('\·�ry on�paramctcr group o(linear translormati,ortS oltbr pbnr dttlOffl�
pos e in the same way?
Hint. Consider rotations ti.u'Ough the angle tor shifts orth� (ocm (x.. xl) -(x1 + X1-�xl) -
28 C:hap. 1 Buie C'.oncepts
5. Nonautonomou1 Equation•
dy
- •
dx f(X,,J),
where 1he righ1,hand side depends on the independent variable x. \Ve begin
our discussion of such cqua1ions with the following example.
.i: • f(x),
{ (1)
j - g(,J),
Herc ., e U c:: R is 1he coordina1e in 1he first phase space and_, e V c:: R is
1he coordina1e in 1he second phase spacc, while/and l arc differentiable
fonc1ions de1ermining vec1or fields i n U and in V . Sup�/{x0) � 0, and
consider the phase curve going through 1hc poin1 (x0,y0). lnen, as -.,e now
show, this curve (Fig. 30) can be given by a curve or the rorm.1 = F(x) in a
neighborhood or 1he poin1 (x0,y0 ).
Parametrically 1he phase curve is given by
X = <p (1),
1 y = 'P,(t),
where 'P = (,p,, ,p 2) is 1he solution or the system (1) satisrying the condition
'P, (10) • x0, <i>,(t0) =
J'o· Sincc/(x0) � 0, we have
d,p,l � 0.
dt t•ro
By the implicit function theotcrn. the function tf,, t = tJ,(x) inversrc lO 91 is
uniquely defined in a ne.ighborhood o f the point x = r0• Let F(x) =
,p,(,f,(x)). Then 1he function Fis defined, continuous, and diffcrcntiablc i n a
!/
:r{t),yft/
Fig. 30 A ph.t.sc curve ofthe system ( I) �nd an integral c:un·c of equation. (2').
Sec. 5 Nonnutonomous Equn1ions 29
2d., - j(x)
g()') (2)
satisfying the initial condition P(x0) - Jo• \Ve call (2) an <fdli• u.il/r
separable variables.
THEOREM. Lt/ the Jun(lions J and g bt dif,tt<d and continuousl.J ,liffna,li_ulc in a
11eighbo,hood rifthepoints x • x0,)' • Yo rtsp«tfocly, uhtref(x0) 9' O,g(J,0) 9' 0.
Tht11 t/1t solutio11 P rif equation (2) subject to the condition P(x0) • .lo uuts
a,/ is
u11iqu1t ;,, a ne,'gl1borhood of/ht poinl x • x0, and 1a1i.sfas the rtlatiM
"" .
d, = kl
t In the s.-:o.sc that any CYl'O solutions coi11cide Yl'hcrc lhcy are dc6.ncd.
30 Chop. I &s,c Concepts
lo
.-ig. 31 Intcgr�I curvn or 1ht tqu;nion t -v(•. t) in the tirntndtd ph;uc-ap.u U.
0
cannot be i;olved by quadralurcs," i.e., ,he solution c.am-.ot be cxprcssc:d as
a finite combination of elementary functions ot algcbtaic functions and
iruegtal� of �uch f1,utc1ion.!l.t Secondly, a complicated formula gi\�ing an
explicit solution often turns out 10 be less useful than a o;impk approximate
=
formula. For c"amplr, the cqua1ion x 3 - 3x 2a can be explicitly solved
by Cardano•s fo 1 m· ula:
.< = Va + Ja2 - I + Va - Ja - I.
Howevc1· ifwc want to solve the equation for o =
0.01, it is useful t o no{C
that it has tlae root x ;::;- -- Jo for small a. a fact which is hardly ob\-ious from
Cardano'sformula. In just the same way, the pendulum equation i + sin x
� 0 can b e solved in explicit form by using (elliptic) integrals, but most
proble,ns involving the behavior of a pendulum arc more easily 10lvcd by
=
startingfro,n theapproximatccquationi + x Oforsmatloscillationsand
from qualitative consideraciorlS which do nor in,•olvc an explicic formula
(see Sec. 12).
t Sec e.g., A . F. Fili pp o\', c.tlution of P,obk11u .,, Difftmrli#l Eqlldieaf (in Russian).
Moscow (1961) and f: . Kamke. Dijf,rc1ttfol £9uatW/ffJ, Al,tltods •f SoltthM aJ s.,:.,_..., I .
Ordimu:, Oij/t,tntiol Equations (in Cerman). Lcipz.ig (19:S6). the Jann- conlaini:ng ,omr
1.6 X IOl cquali ora.
l The r•roof of •his fact rnt"rnblcs the proof or the nonsolv.abi:lity of equations oldeg;itt S
in 1crn1S of radicals (Ruffini•Abc:J.Calois), and is deduced from the not'IIOh-..btlity oi a
cctt:.in gl"()vp. Unlike ordi nary Galois theory, we arc con('Cfflcd here •;th .a. DOftSOlk--a.bk
L iegrou1) ra1hcr than a nonsolvablc finitegroup. The b ranch of mathemaucsdealing:•ith
these prob lem:. is C.'l lled diffiuntial algthra.
Sec.6 The Tangent S }acc 33
1
1 :, ,1 u,,1ion111u�ct' 1>tlblt' 10 f'KMl t0lu1l on 11.r't' o0t'n uwf1.al u nampl", untt lh(-y IIOl'M'
tirne, ohlbh brh,1vlor which orcur, l n mor t' comi)Jl('llf'd ttin u w.dt. FOi' n..1m:pl,t.
1hl-' 1, tror of ,o-olkd "1c-lf-, l mll1u t0lut loni" of a numbrt off'qu..t,on. ol 1N1thmutteal
physlc-1 , Mnm:tvrr , rinding an rnC'lly aol vabl(' problrm ah,.,1y. Of)t't\t lhr po-.abJ1ty of
MJl vl11g ndghbo ri1 1.g r•rohl t'tnJ appro.--imatdy, by ptrlurb.111on th f'Of")', wy '"' �- 9.
1ro........\'f'r It ii d:rngc •rou, 10 r.w1rnd rr,ult1 obtnint'd by ••udr,ng an nacdr ,oh.1bk prot>
ltm to ndghborlng 1>rob1..-m11 of a grn.-rnl form, In (,ut, an f'1tU'tly lnt(lrabk fq"-IIIOI\
I� of1rn ln11•grilblr prtt!M-ly b«aW<' lu solutions•� rnott ,Lmply bth.nnt � ... 1.ho.t o/
11tighborin8 no11i rutgrnbk J)l'Oblt'tt\ll,
Dtfi11ition. Two curves <p1 , ,p,: 1- U (Fig. 32) leaving the same point
.< • ,p, (0) • ,p2 (0) arc said to be tang,nt (to each other) ;r the dis�nce
between the points ,p 1 (1) and ,p,(t) is o(t), t - 0,t
Problem I. Prove that 1wo curve, arc 1angcnt •••point x ifand only iftheir
velocity vector!C at the point x are the same.
The set ofall tangent vectors of curves )caving xis an n--dimcnsional real
linear space (with addition and multiplication b y numbcn being carried
out componcn1 by component), called the tangent spau.
No1e 1hat the coordinate system plays a role in this definition. and the
resulting space seems at fir.a glance 10 depend on the coordinate system.
Thus we would now like to give an invariant definition ofthcvdocityvec«>r
and the tangent space, which docs not depend on the system ofcoordinates�
Definition. A system ofcoordinatesy1: U -R, i = I,... ," in a domain U
ofEuclidean space R• is said to be admissibu ifthe mapping
y:U-R", y(x) • y, (x)e 1 + · · · + y0(x)e.
(with basis vectors e1 in R") is a diffeomorphism.
Problem 2 . Prove that the curvesy • <p1 andy • ,p2 leaving the point7(x) arc:
tangent if and only if the curves q, 1 and q,2 leaving the point x arc tangent
(Fig, 33), so that tangency of curv,s is a gtt!m<ln< co�,pt, ind,pa,dau oftk <>
ordinat, syst,m.
Definition. By 1he velotity vtttor v ofa curve <p: / - U leaving a point x E Uis
meant the class of equivalent Cul"Ves leaving x and tangent to ,p (Fig. 34); in
symbols,
V - ,p(O),
V = ;,l,..-
f Waroing: The '""I" ofthe mapp ings•, and •-i can be: lines pcrpcndiculv .a:lz.. say.
Sec. 6 The ·rangent Sr;•cc ss
-
V
v.
.r,
FiR, 33 Prctetvation o( tangency under' a ditft0m0rphilm.
17t(lrvw
real space R• ofvectors (•1, ••• , •.); the mapping Xassociates the numbcn
v11 • , , , v" with the velocity vector ofthe curve q,.
Definition. Let { e TU,, 'I e TU,. -< e R . Then the linear combination
{ + -<'1 e TU, is defined as
{ + ..,, = x-•<x{ + ix,,,
in terms of the one-to-one mapping X: TU, - R• detennincd by the
admissible system of cootdinatcs xi• In other words, we carry O\."Cr into TU,.
the linear structure ofR•, identifying these sets with the hdp ofthe onc•to
onc mapping X .
Y
. = a, .
iJ,,x. (3)
Equation (3) gives the explicit form ofthe mapping yx-•, and this ""'P
ping is a linear transformation. Thus the operations intn>duccd al,o..-c
indeed e<juip TU:x with the structure of a real n-dimcns.ional linear spa«:
indef)tndenl/y �fthe thoict ofadmissible toordiMlt 17sttm. I
Rtmark. The coordinates x1 and j1 arc fixed i n the domain spacr R• • {x}
and th, range space R• e {y}.According to (3), the matrix of the mapping
yx- • in these coordinate systems isjust ohcjacobian matrix(� il.r).
6.3.The derivative ofa mapping.Let/: U - Vbc a dilferrntiablc map
ping of a domain of n�dimensional Euclidean space with coordinates
x 1: u- R,i = 1, ... ,ninto adomain Vofm..cfimcn.sionalEuclidcanspacc:
with cootdinatesy1 : v- R,j = I, ... ,m. Lctxbca point ofthe domain
U, and lety = f(x) e V be its image (Fig. 36).
Definition. By the derivative of1h, mapping/ at th,point xis meant tht mapping
J.I,: TU, - TV1<••
of the 1angent space to U at the pointx into the tangent space to Vat the
point/(x)which carrit11 the velocity vector�leaving the pointxofthecurve
38 Chop. I 8a1ic C.onttpts
v,.
f
f•
v,
�
,------,
Pig. 36 Oefioilion 0(1hc duiva1ive ofa mappan1/a1 a point K,
,p; I-+ U into the velocity vector leaving the point /(x) of th<: curve
/o t.p: / - V, i.e.,
). (4)
J.1.(��l,.0) • � •·• ( f• ,p
THEOREM. Formula (4) dtjitus a linear mappinif ol, oftlw tangOtl s/>OC< TU, utll>
/ht tangent space Tl'Jt••·
Proof We must verify first that the right-hand side of (4) is independent of
the choice of the n:prcscntativc q, of the class of tangent curves at x, and
secondly that the mapping.f.l,is linear. Let i1 denote the components ofth<:
velocity vector X of the curve ((J at the point x, andJ, the componmts ofthe
velocity vector y oft he curve/• ,pat the point/(x). By the rule lordifl'erenti
ation of a composite Cunction, we have
JJ. -
f.21-x,.
i.J
i• I i)xl
($)
whcrey1(x1 , , • • , x,, ),j: 1, ... , m arc the functions specifying Lhc: map-
pingfin the coordinates x,,:,l' But both assertions o fthe theorem arc con•
tained in ( 5). I
In addition, (5) implies the following
Rtm<Jrk. Suppose that in TU� and TV/(.r) we introduce the componmts i,:,
j1 of the tangent vectors in the coordinate systems x,,:,1 rcspccti\."cly. lne.n
the rnatrix ofthe linear rnapping/.lx: TU$ - TV,.($) is the Jacobian macriJc
(ily/ilx). It should be emphasized that 1/u mappingf.l, indLpmdmJ �1M u
eoo,dinate system, the coordinates being used only to prove the theorem.
Probum J. find the deriV21ivc at x = 0 of the M3pping/: R - R given by the formub
y • xl,
Sec. 6 1'hc Tangen, Space 39
A11s,/.lo is 1hc mappi ng o f the 1inc TR0 into the line TR., carrying the whok: line ,nto0.
P,ol,/m1 2 , Lei/: U - V, 1: V- W be: diffcrcnti•bJc mappinp. Pro\c t.lY1 the aom
posite mllppi,,g Ii -I•/ : U - Wis differcnliablc a,,d that Jli dtti\•.ttiw a, the- pow,#
equa l.s
"·'· -,.1,,�1 .,.1...
Prol,J,m J. Lei/: U - V be a diffcomorphi..sm. Pro\·t tNt the mapping/.L.: TU. - nr,1a,
ia an iwmorphism or lioc.ar spaces. Civc an cumplc showi t\g 1ha1 .., ,_,,,,, u fU
(•« Vig. 37).
ProMm, ii. J..N/: R' - R' be the map 1>ing gh·cn by the ro,mula ( .-, -4 tr,>' ,, + i,,.
i = ✓-:-J, Show thatf.l. (x r;. 0) preserves angles (the Euclidean structutts ffl Tll!, Tll!
are spcc.•ified b y qu adra tic (omu .if + .tj and jf + jf ra.pcc1i,'Cly).
6.4.. The inverse function theorem. Lei/: U - V be a diffcren1iable
mapping from one domain in Euclidean space to another, and let x0 be a
point of U .
TH£0Rt�M. Ifthe dtrivatiu�
J.lxo : TUXo - rv,.,xo•
is an i.somorphism of lintar tran.sJormalions, tlun tltne exists a nngltl»r1-d IV•flk
point x0 .su,h that the rutriction
fl w: W -f(�V)
Jlo W is a dijfeomorphism.
of
P,oof t The dim<;1l.SiOM or the tang�1H spaces ru.._ and rv,t-.,, and hentt t he dimcru;ions
or ,he dom ,1i,u U and V, are thC' same. L ei x1, •••• •• be admW:ible coon.:luu.ccs i n U.
and y1 ,.,., ;, ad mi ssible coordi nates in V . Thie mapping/ is specified by functio,ft,s
y, •/, (x,., .• , x,.), i = 1,• .• , n . Let
,.
1 The inverse fonction th eorem i.s t'a.sily deduced from the i.mplicit function tbcottm,, a.nd
vice vctsa. Hc-tc we deri ve the fovcr'$C func-t.ion theottm from the implicit function d-..c.:o.an..
since the l atter always figures i n coursc-son a nalysis while rhc formtt isusually left umt.1.tcd..
For a proorwhi<'h i.s i ndependent of the ;mpl icit function theorem, sec, e.g...Sec.31.9.
◄0 Chap. I &sic Conccpu
dttcrmln,1nl or (81',/8..,1)1,., ,1.,1 i11 ooorcro, Appl)'lf\l tht 1mpl1C'it fun<hon tM'Ort'M '°
tllt' 1y.1ttm orrunftlom P., i I,.,•, 'fin 1' rK'IJhbomood o(tht' potnl ....... Jr.••,,. •r
flod 1h111
I) In 111uf1lcicntly ,m11II nti3hho1hood F. oftht po1ntJ• ft'-♦) t�� ""'" runc,-om
A'1 .,1 (J".,, ,,Y.-) "uch 1h:.1 P(.-(,1),)) 0:
2) Tht' •>•11ro1 F(A',l) O.)'• R h :n no 01hc-r toluhOfll X nnr ••;
or
!) Thl' vulucw the ru11nlo ,u .-1()) 41,1 thr point.re rqu.al 1hr coord,natn of thit point -.,
and th e 1111 ar c co11tlm1ovily diff(rtoilable the ilfflC' numbt'r ol tl mct ti 1M fuM,,;,,,.,.,./,
111 th e oeighl>o�hond F. o f thf point y0 (F' ig, 38), The (u.nctioru •• dt'tffffllnt a diflnm.
1iablc: mappi ne .,, or 1he ne ighborhood E of thc point,. Ax.) into a nrighborhood of
the-po int x., such 1hM/•• iJ the ide,uity m aJlJ>i"I· 1..(,1• £'),. W. � ah<- mapp.ttg1
/l w: 111 - £ \llld .,: F. -W art mutually in\'crK d1ff�nliable mappnp.. and hmtt
diffc:Ol'no r1>hi.nni, I
Probl,,n J , Prove thnl t1(E) it a ndghborhood of th e point ..-., i.e., ,on&.1.tM .JI poinu of
the do m ain U whi(h a.re aufflcirnlly near the poio1 ••·
y
E
-
r
,,.ablm1 /, Prove 1h11 i( the field v I, dlfttrcnh•blir (u�, i, d«-tcrnmwd by ,.(old oc:,e..
1i 11uou,ly cllfftrtn1l11blc funnion•,,(.-,. , . , , x,.) ,n th,r •y t,ttmo (coord1natn .-,).thim th,
r.rld /.v ,, 11l10 dlfTcrrn1l1bk (�•i1h tht anme ,, if th,r d,fl"tol'n0f'ph1�m/ • ofct .. (;9• '>
/U,,,. Stt rnrmuln (,'\).
Tt1 RORV.,, /.et/: U - Vbtadiflto111orphi1m. ThtnthtdifltrrntialtfWIIIM
I: • v(.v), x 11 U (6)
witii phase spau U dtltrmintd bythe lite/orfirld vis tqufoaltntlo /ht,,,_.,;..,
j - (f.v)(y), 7)
1Litl1pl1ast spate V dt1trmi11td bythtvtctorfieldf.v, i.t., q,: 1- Uu• JM•ti•ef(6
ifando11/yif/•<fJ' t - Visasolutio110J(1).
Proof Obvious. I
In other word-., let.,;/- Ube a wlu1ion of equation (G), .and let ti r) - •,...,... t).
If •(10) • ,,0, thM � lc.1\'t1 x0 aod J.. • le 1wC1-''• /(,c0l. It fol�-. from thit ckfinic..ion
or/. tha1
d1di,.,.f•• 4-1
ar ••0f•• f.1�4-1
ar 1.0
•
d
f.l., ;I
d ,.,.
- (f.v)(,10),
Therefore:/.. ., i.s .t soluti on or equation (7). To compl,ctC' the proor, ""'""C' appl.v- this: ttsuh
10 the n\'Crst diffeomorphis.m/-
i
': V- U.
= -�2,
Example I. Consider 1he sySlcm
�,
{ =
Xz Xi ,
(8)
Hence 1he new vec1or field (f.v)(.r) has componen1s w, = _r,, "'z = -.1,.
and our system is equivalen, to the system
{!• ==
J'z
Y,,
-y,.
42 Chap. I BaJic Con«pu
/
f ig. 41 Th e phase pl ane or 1hc new sy51cm.
0
.- r Zrr
-
0 r
fig. 1 15 Polar "coordinat«,"
0
2rt� - · --_.,-·
r
Fig. ◄6 Phase c:urvcs o( the 1k'ndulum c:quuion1
1 in potar coordina1e1..
{ ;=
(Fig. 46)
() = 0,
-1.
The solution of this system is ofthe form
r(I) = r(O), 0(1) = 0(0) - I,
= ,0
and hence the original system (9) has the solution
x,(1) • ,0 cos (00 - 1), x2 (1) sin (00 - 1).
PT<NJl,m 2. Vcriry Lhal thc:sc forrnul.u gi\'C all the sohalioru of the-SJ$10ll (9) b :all,. •nd
no, just for (x., xJ) « V .
Problon J . Prove th:11 1hc ph:ai1c cul"\'ct att circlies (Fig. 47)� and lh:u thie t�:ach--ancc- m.l �
pings g' form a onc-paramcu:r group of linear transform.at.ions o ( lhc pb..ot. •illltrc6' is a
rOlation 1hrough angl e I with a matrix of th e form
c �/ sin/
( -smt cost )·
Returning to the pendulum equation X = -x, ,,·c find that thependulum
Sec. 6 The Tongcnc Space
:r,
:r, X3
.1; ' I
t
,
I I
I I
executes harmonic oscillations (x = ,0 cos (00 - 1)) whose pttM>d �ual• 2,r
and docs not depend on the initial conditions.
P,oblem -I.What arc the integral cur\'cs o( 1hc system (9)�
or
Ans , Hel ices p itch T = 2,r with common ax.is x1 a x1 = 0, wh,ett the- a.xis is a.l:so an
inttgt: i l CUI'\'(; (F ig. 48).
I
Example 3 .Con s id e r th e syucm
.ti = :c, r x,(I - :cf - Af),
(JO
:f1 - -x 1 + x1( I - xf - xi),
obtained from the sysicm
I '�Jl,),
(II)
6= -1
by g oi n g O\'C-r to tte&aogular coordinates .r1 = , cos (J, x1 =,sin().Actu:aDy, the-q,,usn
( 11) is tquiv aknt (with 1hc usual s1ipula1j&ns in\-olving th e nonuniqucnas of p,obr
eoordin.:i.tcs) to 1hc system
I
.i 1 -x1 /(r),- 1 +xJ>
X, - Jt,/tr),- 1 - Xi,
which r educes to (10) if/tr)• r{I - r1).
Thus we must invtstigate the syst em ( 11) with f(r) r( I - ,1 ). Finl � com-du lM
integr al curves of the <'quation ; - -Jtr) i n the half-pb nc (t, r),, • 0 {F11g. -49r � noting that
o ..
lht v ee tor field on th e li 1, e v =/{,) has three singula r poin ts , - %. I. ,..--b,c,rc th< 6dd
is dirtttcd toward the poi nts , ± I and away from the poi:m , O. The phnc- <.'UJ"-U
in 1he haJf.pl i.lne (,, IJ),,.,.,. 0 a� obtai ned by n-..a.k.i:ng a .-OC.&tion (sintt 8 = 8• - , , .
Rcturniug 10 rectangul ar coordimues. we g e t t he pieru.re shown i n f"'ag. .50 . The cu:n."C
x 1 = x1 = 0 is the on ly si n gular point. Th e phase curves startin g near dais point fflO'lo"'C
46 Chap. I Baiic Conccpu
1:ig. ◄9 Jotegral Ctit\'a or,he e<)u.i1io n f- ,{I - , 1) and ph.ate cun.'ff o/lhcl)'ltan(IO)
in polar coordinrttcs,
, Wildcats
10" '
,' ''• ,', ,
,,
,, ••• '' Hares
/!JOOt90.; 81/l l!llf !!to 1!25 /JJ/J/9JJ
Year
f ig. 51 Oscillatiom of the wildca1 ar,d hare pc,pulations i n C...n.ada.
Sec. 6 The 'l'angc111 Space 47
+
1iwny rroo1 it ru , i nuc;uc, a,ul wi nd around th,c ti rdc .rf xf I from the ,�
iu t - I oo. Thi, drclc i, itJC:lr • ,,hue curv«', called • h••' Ql'l.t, ICo-n�,, 11 the: uwtal
po l n1 lic-1 ouuldc Ille did, xf t xj < I, the n the 1>tuue rur,e wi ncb 41r0Mfld the lm,11 cydt
from the Oub i dt' 111 t , too "nd (Jotl off 10 inflnoy for nq:111h·c ,.
Limit ryda dtt(ribt 1hr 1111b lc 1>crlodie rq:imn of the n101ion of at'I 1u1onomou.
sy,uc ,n. For tic,uuplc', ., 1 11nd A, ml3l11 dr notc the dtH11ion1 of the m,,nbtt of ._,a«kau
11nd 1lie nu,nlwr or harC'1 rrom thdr cquiM>f"ium ,·a.lUC" (the C'orrnf)Oftdn'I «°'°"°'I
cqu�lio,, I, not exactly or the form ( I0>, bu1 has um,lu propuues). "nac:11 the lam11 cyck
corre1pond1 10 the 1�r iodic 011c1ll :uions o( the wild cat and hare popu1atiioim_ ""kich att
field, whh the oscill :11io," in the oumhc-r w,ldnu laggin.g bthind
or
f'"•·
,omtwha1 ahirtcd in phue with ro1>«1 10 Ol'lt' another. This is ac tw.ly obland i.n the:
or
01hcr e,ci,mpl ct the orc"rrrn� or ,ta ble periodic- oteilla11on:1 undct' sta�ry a •
SI).
ternal condhions art" afforded by f'lockJ, iteam rng in es, c-:lec-trk bdls, tht human hon,
v., c uum 1ubt- CMcillators gcner.-uing radio wa\'t:Si and variable- sears of the: Crphod 1ypc::
the opcm1ion ortue'h or thne mecha nisnu i s dCl(ribcd by a limn c,·tlc ,nan app,op,i..k
ph.iuc apace. However, it would be wrong 10 think th�t a. U mc,U�t()f'J prottt10, att d,e.
ndb<"d by liinit cydet, and in fart rnut-h rno� complK.tttd brhu>ior o( phurt cun+o es
pauihlr in :a multidimco1ion11I ph aJC" s1>.1cc. Ji, 1h i:1 rt'g.ard, we ntc tht prtt� of1)...-&
s('opt,, cite mot ion of p lancu and a rtificial satell ites, 1n clud1ng thc-:ir t'OUl.lOnl about WU'
aiu: s (the 0011 :H"r odkity orthCK motion. , r<"!lponsab c-
1 i i ror
l the compln111y ol lM caJ.Nrcbr
: u,d the t.liffi ·
., uhy ir, t>rtdk1ing tides), a, well as lhc motion ofcharged putic.ks in a mac
ne1ic firld (r esponsible for th t' O<'c-urr<'n ce of the Au rora Bornlis). Stt a.ho Stts. 24 and
25.G.
2 Baile Theorems
In 1hltt drnp1t..·1 · WC' formul..uc· 1he haii;k rc-,uh, of tl1r 1l1<"ory of ordm.ary
diffi·r<•n1inl c·,pm1 iou,, df'nli11k with 1lu· r-<i!itrntr and uniquC"n"" of iOlu•
1in11< and nffir.i i11t<'g1·al,, and ..,;,1, 1hr drj><•ndrnrr nf'<llu11on, on 11111,al
data rrnd pnrnmc•tc•r,c, \Vt• l>Mlf>nnr tlw proor, u1111I (:hap.4.confimngour•
!WlvcJ al 1hi� poinl to 3 di,C"u,,ion c1f ho\,1 thr ,·arinu, u·iuh• .art rTlau·d to
one.· anollu·r.
x a v(x), X € (/,
II
:,;, !I,
clirlc:rn Apace rcmtaining 1hc domain ,v, so that 1hc vrrtor e1 h.a1 compo,
ncnl9 I, 0, ... , 0. According 10 Sec. 6, the l:"uic theorem can�formulated
n, rnllows:
T/r, 1/ij/ert11lial ,qua1io11 (I), eonsidtrtd in a s•.ffititnllJsmall nntU..1-#
•,
110,,sr'rt1J11/(1r poi,11 o is tquiraltnl lo lht parlttulo,1.f 1implt tq110IIM
I',f•
ye W, (2
;.e. IO tl,e SJJltnt
t
j, = I, =j. - 0 (31
i,, lltt ,l mai,, l•V.
o
The following is still another equivalent formulation orthcb,uic tMC>rcm:
/11 a suificie11tly small neighborhood V ofa non,ingular point "o, - ca c!w.M ""
admi1sible eoordi11ate system (y,, ... , J'.) such that tquotion (I),,,. w u.riltm i• 14'
sta11dnrdfom1 (3) i11 thtse coordinates.
The basic thcorern is an assertion orthr same character as tM th4!ort'.m of
linear algebra on reduction of quadratic forms or matrices of<>p('n.ton to
normal form. It gives an exhaustive description of the local beha,·ior of a
vector field and orthe differential equation (I) in a neighbo<hood of a non
singular point x0, rcdutingcvcry,hing to the case o f the trivial equation (2
The proof ofthe basic theorem will be given in Sec. 32.
7.2. Examples. The hasic theorem might be called the r«lifaoh• ,,-,,.,,
since the phase curves and integral curves orequation (2) a.-., straight lines.
Fig. 54 shows the level lincsyi = const of the "rectifying coordinates.. for
the pendulum equations.
P,obltm I. Arc 1he rc:c-tifying «iordin :uc:s )', untquc-ly dtfin<d? Pro
..
'C" tlwrit in tht cue:
,, = I tlw mol'di n.at(', is defined 10 within an aflint trandormation.f 9 ..,_ 6.
Prol>lr,n 1. Skccch level lints of rcc:1ifyi ng cool'dinata fOl'" och o( the follo---ing ,--uior- fidds
in the domain U:
a) v =- x,e, + bie,. U.,. (x.,x1 :x, >0}:
b) v = e1 + sill x1e:, U = RJ;
-
c) v - x1e 1 + ( I - ,\'f)e,, LJ (\' ,X :-1< ..,<t}.
-- ----
!11 !It
-- •
r
I'
---
!IJ !Ir
(al CbJ
Fig. 54 Rcctiti.cuj(ul
: or the pendulum cquatiottS.
50 Chap. 2 8.uk Throttm,
•P,obltm 3 , S uppose th,·u in R" we are given a (diffcr-c.ntiable) fie.kl ofu.ncm1 pl;a.no R.1 .
Can th i, fleld a l\.\.• ays be rectified (i.e., mmsformcd int0 a field o( paralld plana) 1n a
n e ghborhood or A po nt with the help <If a suita.blc difficomorphtlm ?
i i
Hinl. If ,he field or planes i, rcclifiable. then it is a 6dd of plana tangu,1 to the family
or 1urfocn.
AnJ , No. Con sider for example the field orplancup«i'Md by the fi.cld o(nof'Mah..2e1 +e .,
in R'. There docs not cxi..st a surface with thi..s d irtttion as lhc normal at each poin.L
•Pro6/1-m f. Suppose a vector field v has no singular points in a domain U . (;aft on,c then
rectify the field in the .,..•holc domain U, i.e., is tht basic th�cm true •·1th V = U?
Hinl. Construe, the field in the plane \.\.'hose phuc curves arc of the form abio--n in Fig. )S.
7..
3. The existence theorem. The basic theorem immcd.iatdy implies
COROLLARY I. T/,er, txiJts a solution oftquation (I) satis#nt ti;, imti,,J C#ditiM
,p(to) • xo .
Pr(}(Jf If v(x0 ) =
0, kt ,p(t) • x0, while if v(Xo) t, 0, then, by the basic
theorem, equation (I) is equivalent t o equation (2) in a �ghborhood of
the point x0 . But (2) has a solution,; (which?) satisfying the initial condition
,j,(10) = y0 = f(x0 ). Hence equation (I), which is equivalent to (2), has a
solutioo satisfyiog the initial condi,ion ',0(10) x0. = I
7.f. The local uniqueness theorem. The basic thcor-cm also immediately
implies
/11m11rk. We will soon sec thnt the restriction v(x0) ,< Ocan be dropped For
11 • I thl, lias alrcndy been proved in Sec. 2.
7,5, Local pha1e flow,. I.ct v be• vector field in the pha-K space U, and
let x0 be a point or U.
D,ji11itio11, Ry n local pha11jlow determined b ythe v ector fidd v in a nnghbor•
hood of the point x0 we mean a triple(/, V0, 1), consisting ofan interv.>I
I • {I e R: Ill < c} of the real I-axis, a 11cighborhood V0 ofthc point x0,and
a mapping g: / x V0 - U, which satisfies the following thrtt conditions:
I) For fixed / e / the mapping g': V0 - U defined by I'• - 1(1, x) ii a
diffcomorphism;
2) For fixed x e V0 the mapping ,p: I - Udcfincd by \1'(10) • ,'x ii a
,olution of equation (I) satisfying the initial condition l'(O) - x;
3) Tire group property g'• 'x - g'(g'x) hold, for all x, s, and I such that
lhc r'ight-hand side is defined, where for every point x e V0 1hctt aisu a
neighborhood V, x e V c V0 and a numbcrcS > Osuch that the right-hand
side is defined for Isl < cS,Ill < cS and all x e V.
Example /. Consider the vector field v = e1 in a domain U of Euclidean
space R", and construct the following local phase flow in a neighborhood of
a point x0 e U . Start with a cube of side 4e centered at x0 (Fig. 56). For
sufficiently small e this cube is entirelycontained in U. Lc1 J.'0 d�tc the
interior orthe smal ler cube of side 2e (one hairthat ofthe original cube) and
the same center,and let I be the interval Ill < •· Then define the mapping
gbytheformulag(t,x) • x + e,1.
P,oblt,n I. Veri(y 1hat condi1ions l), 2), and 3) :11tt satisfied (or thlScx.:unpk.
II
Fig. .i7 The local l)h1!hc flow(/, Y 0, -) lat ob1alo� ftOM 1hc local phaw ft.ow, I. W•• A)
of the rectil'led cc.1vuion
: b)' ap1>lying the diffeomOf'J)hn.rn / '.
=
COROLLARY 4 . UI
{ � ==
tions
v(x, ,x),
IX 0. (4)
Sec, 7 Vcc1or Field Near• Nonsi11gular Poin1
---·-,
F
�-=1
A
C:.:::-- .:�
V
P ig. 58 ·n.e rh :u� spac� o(th� extended Sytlcm j - v(•. •). 4i 0
Defi11ition. I f t here exists a solution cpof equation (I) ,atisfyiog the initial con•
dition q,(10) � t . 0 defined for all I e R,we say that tN,ol•tiMca lNutauhd
i11definit,t.,. If there exists a solution defined for all 1 ;i. 1 0 (or all I.;; 10),we
say I hat th, solution can bt extend,dforward (orbackward) indtjiniut:,.
Let r be a subset o f the domain U. I f there c,c.ists a solution ., of equation
(I) satisfying the initial condition q,(10) • x0 and defined on the interval
10 ,;; I <;. T and if cp( T) belongs to r,
we say that t/r, so/111;.,. ,..,. IN ,xu,u/,,/
forward upto r. Extension backward uptor is defined similarly.
Let F be a compact subset of a domain U containjng a point Xo� and let r
denote 1he boundary of F (i.e., the set of poinlS x e Fsuch that c--cry �igh
borhood of x contains points of the compkmentary set U,F). Suppose, the
vector field v in the domain Uhas no singular points. Then i, i$ no, hard l o
s e e thal 1he basic theotem implies
ef
COROLLAR v 5. The solution <p equation (I) can bt ,xt,mhdforward (..-bad.uard)
ef
eithtt indefinittl., or up to tht boundary F. This extmsion is uttiqw in INmu, IM1
any two .tolutions sati.ifying tlte same initial condition coincidt on tlt.e in.UTs«tit11t .jUtt
inlervals ofdefinition.
P1Hf, First we prO\'C lhe uniqueness. l..(-t T belhe least upper bound of1ht$C.t olttu.m.bcn
t tor wh ch the solutions•• and •., coincide for a.II tin theinter,.·a.l '• !fit t 'i: r (r..g. 59).
i
Sup� T i.s an interior point of both intervals of definition. Then •,CT) - •1(7)
because of the continuity o( •• and • • By the- local uniqueness theorem., •1 <Oincida
J
54 Chap. 2 Bue 1hcorcmt
Fig. !,9 Th e u nic:1u«·ncss or 1h.- txu·ruio n folJows from the local u:niqm tl.«llit.m.
with • .1 io a ncighbothood oftht' poi 1,1 T, so 1ha1 Tcann ot be 1hit leas1 upper bound+
Hence T rnw1 ht the end poi ,H of one o f 1hc in1cn-als ofdiefinition, and 1hr t•--o soh.uiom
coi ncide on the par1 of the intcrsttlion of thc:K i nu:n-als for ••,-;hjch , ;;a, c.,. The" OM
<
t 10 i.s treauxl .sim ilarly.
We now comt r uC'l tht' extension. lf thc twosolutiom coinct<lc on the inlO"tlttlion oltbr
intervals ofdefinition, then 1hcy can be C'.'Offlhin.ed t o form a solution ddinrd on \be u:nioa
of these interva ls {Fig. 60). LA:1 T be tht" lr.:a.st upper bound of 1hr tfor wbirh tbett csisu
a so u1io,, ., of equat oo (I) sat .sfyi ng the initial condition •(t0) = .. a .ad a.ho thr am
< < :.
l i i
dition •(t) E F for all t, t0 t Ry hypothesis,. t0 < T < oo . If T co� lM $Ol!ution
cao he cx1cnd«l indefini1dy fOl"'\"ard. Suppok T < co. Thrn. a� ,,,r t nnw shoil-•, tbttt
exists a solut i0 1l • ddlned f°" all t, lo Ct I (; T . such th.al •Cn � r. In fact,. it rouo..-s
from Corollary 3 tha1 every poini llo f' Uhas a neighborhood V0(-.) ;1..nd ;1. corrcspondi:ng
nunlbcr c:(x0) > 0 such th a1 for all x f V0( x0) 1hc:r c nisu a solution • ytisfying the- in:iti31
condition •(1 0) = x a11d defined for It • lol < c (namely•= ,,-r.s,. SinttFis com
pact, We can choose a finite covc-rirlg of 1hr set F fr0m thctt nc:ighborhonds ol the-points
x0 f F. U1 1: > 0 be 1he ,smallcs1 of the finite numbc:T of NW"rMponcUng n:umbcn -1-. .
Since T i.s a leas t uJ)pcr boun d, there exists a r bc-twttn T - I! and T such t}gt •it) f' F
for all/ in 1hc: i,uerval 10 < t ._; r. In par1icular •Cr)� F, i e. . ? th<- poin.1 • r) isc�·aat
b y one of the Mighborhoocb of the finite 00,"<:ring. He� there cxisu a tolu:ticwt •'
ia1 is-fyi ng the initial condit ion •'(t) = •Ct) and dcfin ied for it - rl < • (F"� 61). By UlC"
uniqueness theorem, •' coincides with • on the whole intersection of the inten-als of
Sec. 7 Vector Field Near a Non,ingular Point
-+---1---,1-'-i-
lo t
:i:,
Fi g. 63 The so lutioo of the pend ulum cqtatiofls C'al\ be01c-ndnl inddinitdy. "'--hilt tt
mainin g in the di .sk P ,
defi nition, Henn· ¾'C t an use• a n d•· to OOflStruet a sol u tio n•" defintd re.- ,. < t < t + L
lil p artic;ular, •"'(r) cxlsu.
Finally, we thow ,hat •"(O) f F if t0 < 8 < T . In (ac� n·ay sol ut.oft • satisfyinc th<
in iti.alcondit ion •(t0) - •o an dddincdfor t0 < t < 8mw:1coincick"'ith•"'(uniqumc.).
rr•"(O) .aa •(O) did not belong 10 F, th en T would not bC' the lcu1 upper bound o/lhC'1< 1
tr: •(t) • F for t0 < t < t}. Morro\•t'f',•"( T) � r. I n fact •"( T) e F, bring I.ht lim.d ofa
«-quencc orpoint s•"(O,) e F, 0, -. T . On the other hand, n-cry intO"\-al with T as its k(t,.
ha nd end poin t oo n ta i ns poin ts I such lh at •"(t) docs not bdon;g 10 F, siott otba--.'"ltit all
the points •"{t) wou ld bel ong to F for all I in $0fflC nci.ghborhood of T.and then T •"Ollld
not be the least u ppe r boun d . This prO\'CS the th(OrC:ffl for cxtmsion fclnq,d, The cut
I < 10 is 1rca1ed .s mil arly. If
i
Remark. �Ve will soon see that the restriction v(s) # 0 for alls e Ucu, be
dropped.
£:cnmplc I. E\'en io the case wher-e U is the whol e Euclida.n spa ee , the solution. ca.tl1'0t
alw a)'S b.: exttridtd indefinitel y, e.g.• when n = I , v(x) = xl + I (Fig.6'2. ) .
£.,ur,npk 2 . <.:oos ider the J)('ndulum cqu:u:ioiu :i-1 = xJ.J:J = -x1• Let U ht tbt pbAie
(x., x2) mi nus th e origi n of coordi nates, and kt Fbe the disk lx111 + lxJ IJ '( 2 ."Incn the
s o ution satisfying the in t a l condition x1,0 - I, XJ.o - 0 ca n be exec-nckd inddinildy
l ii
(f;g, 63).
Ptobl,nt /, For whiu inhl11I C'Oodh lo,u can tht IOlullon or ((IU;ahon, ""ith • htnh c-yck
(Sc:c. O.G, Ex�m,)lc S) be u1c,1drd lndrRnltfly1
Probltm 2 . Su1)po1c: t\'t1 •ylOlu1ion ofequ111lon (I) nn httllltndtd ,ndrfiNldy bod!ro,,..�rd
1rnd '" l.>lu:kwnrd, t,4:11 dc-nott 1hc ,,adWt.n<'C' m1t.ppin1 Currying ('\C'f'Y poan1 oil��
1
11 1>M C' U l,Ho 1hc: valut •(I) of the t10lutlon ..,tlll(ylng 1ht innh,t <'ondahon •0
Prove 1h,t1 f1'J I, fl onc-1 :wrnunt1tr 1roup ordiflf'<lmOrphi,nu o( U.
in W .
Thus the diffeomorph ismf carries ohe point (1, x) into the point (1, y)
while leaving t unchanged. The equivalence means that q,: / - Vis a .solution
ofequation (I) if and only iff• ,p: I - IVis a solution orequation (2).
'The above corollary is eq1,1ivalcnt to the ba."-iC thc-orcm. A direct proofof
the corollary will he given in Sec. 32.
Pro6ltm I , OeduC'c Coro ll ary 6 from the lxuiC' throran.
P,obltm 2 . l)t;ducc· th e ))a_,.i«_, th('Or('m from Corollary 6,
.r
w
t t
f-'ig. 64 R<..-..·tili< ::nion of integral C'U(\"O by a d iffcomorph"'m/of otc-ndnt plQsc"Pt«·
Sec. 8 A1,plicacio11s co che Nonauconomous Ca�
l
Fig. 66 Extension or a solut ion up to the bound.ary or a comp:1ct I/ti F 1n otlNM'kd
llhtue sp,.1.c:e.
sofotio11s satisfyi11g the same initial ,011dition coincide on tht inttrs«tiM -JIM utl.rrw,ls
•f d,fi11itio11.
Proof. Si111ilar 10 that or C'.orollary 5. I
P,46l11t1 I . Prove that Corollary 5 is \1 .:tlid even i(the f�ld v h.au.in.g.,.lu poinlJ.
x::
P,001,m 2 . Suppose e\•ery t0lu1ion ofnt_ua1 ion (1) can be atendtd inddinitdvforward or
b.u:kwal'd, Prove 1h :u ita diffroinorphi"n ofpha� SP-ICt onto i�lt
Probln1t 3 . Sui:,posc, in addition, that the ,,ec,or field vis pcnodic in hrrK\totluit v't + T. •)
.., v(t, x) for all I and x . Prove that the diffcomorphwns ftO'} (" an intqu) form a group..
1,e.1
g;r = A•
where A • t(, Which of the following two rc-lations is true:
t"., • • ,(QA•?
U;z -
= d,p ,.-1
"'
. . . ' u,. = d•-1
di ,. l 'I t-i
2) For every t a I,
d'tp d,p d•- •,p
- - F(t, tp(t), Jl )
dI•-I ,.,
I• • • I
di" ,., u t ,.,
For example, the runctions ,p(t) • sin t and ,p(t) • cos I•� both solu
tions or the equation
d 1x
- - -x' x,; R
di'
for the small oscillations of a pendulum.
The phase space of the pendulum equation is the plane (.r, i), as in Sec.
l.61 Example 5. We now consider the question of the dimer,s;onality of the
phase space cor�sponding to the ,uh-order equation ( I).
THEOReM. Equation ( l) is equivalmt lo the s.,stem
(2)
x. = F(I, X a
,. .. , x,.)
of nfirst-order equations in tlu stnst that if ,pis a solution of ,quaJio• (1), IAnl t/o,
u,etor (II', </J, tj,, ... , ,p1•- •>) made up of tlu dtrioativts of tp is a sol■n- ofIN
systtm (2), while if(tp 1 , • • • , tp,) is a solution oftlu s.,st,m (2), thta tp , isa�
o f{I).
P,0-0f. Obvious. I
Thus the phase space of any process described by a differcnoialequation
of order n is ofdimension n. The whole course: ofthe process 9 is determined
by specifying n numbers at time t0, namely the values at 10of thcdcrivat.ivcs
of ,p oforder less than n .
&a,nplt I . The �ndu lum equation LS equivalen t lO the- system
l jl -xl,
jJ - -Xu
already invcsdgated in Secs. 1.6 and 6.6.
£xampl1 2 . The e q ua tion i - 0 i. s equi valent to the system
*1 - x:,
I 'J - 0 ,
wh ose solution is easily found to be K1(t) - x1(0) - C. x1 (t) • s.{O) +CL.Thus �'tty
solu tio n of the equ ation i • 0 is a polynomial of the fir,t dcgrtt in ,.
Probltm I . Prove that the «jua tion d"x/dt• = 0 is satis6cd by a.JI polync:imia.ls oCdcgTtt kss
1han n and o nl y by f.he54: pol ynomials .
Sec, 9 Applications 10 Equations or Highrr Order 61
exists and is unique (in the ttns, that a'!}' two sol•tions sotis/J·in1 (3) ,.;,,,;1, M t/w
i11ltrSt(lio11 ofthe int,rvals of definition).
We can wrilc the inilial conditions (3) more: condsdy a.t
I - 0, • I, * t, 1\1
I ..., 0, " - I, .t = , I. \')
\Vhat an• the motions of the p,4:nd ulum cormpondm,g lo these folutiom?
Counternample /, Consider the equation 2x = l1X and the initi:a.l conditions
I= 0,, = 0, .t = 0 (Fig. 69). Then many solutions satisfy thcs,, conditions,
for examp le, q,(1) ::: 0 and q,(1) = 12 . The point is tha11hccqua1ion in ques
tion is not oft he foni, (I).
II
I Ill
Fi,g. 68 Five special .1e>l utiom of the cqualion of,� im·a-1«1 �ndulwn.
.2, ·
Fir11 or all, we explain the nature or,he pha,c, space corroponcling 10 the
sy11em (4).
, 11EOll8M. Th, l)'Sltm (4) is ,quil!(J/tnt to• l)'sttm of ••
N • 1: 11 1
I• I
(where Vis the potential energy and the m1 > Oare mJWCS) is equivalent to
the system of2n Ham.iiton equations
. iJH . iJH
q,=-J p,:;;; --, i:;l,... ,n.
op, o q,
where P, ;;;: m;<i,,
" _n2 11 ,,2
T= L5!J.=
2
L �.
l=I 2m,
i•I
and H = T + Uis the total energy. Thus the dimension ohhc phascspaoc
or (5) equals 2n.
Pro/Jinn I. State and prove 1hc theorems on cx.istcncc, uniqucnca;, continuous and diKer-,
cntiab lc dependence on initial conditions, and also t.he a-tension c.hcorcm,, b tbc SJS,tcm.
(4).
C:llap. 2 Buie Th<orcms
T The theol'em on differcntiatfon with rcs�t to tM' initial eonditions can thus t,c, med IO
approximate abundle ofsolut.ionswith initialc ondition,:near 0tt1ainuunpcnurbccr• nNCJ
for whi ch the sol ution is known.
t Since
v0(s) = v0("<,) + •i.'I-.y+ O(e')
for small� .
Sec. 9 Applka1lons 10 l':quaiions or Higher Order
where
A(t) - av.I
bx " ll>
' b{t) • v 1 (a0(t)).
o
Thit is the de!i red equation or variacion.t.Note Ihat yalso sa1Ufics th� in,1ial
c·ondi1ion y(O) • 0, since 1hc ini1ial condhion for a is 1hc sa= lorall ,.
In solving problem, it is easier to derive the equation of varia1ions as
needed, ratherthan attempt ro mcmori�c tl.
P,oblrm I, A body fnlli �r,ir;,ll y in :1 rncdium w11h 1-1'1\all rcsileancc dq,,rndi.o,: on both
�ition :11111 velod1y:
I -A r ,,f'(x, i), '< I.
Calculate the effecl of 1hc rnioancc on 1hc ,nooon,
SoJ11tion. In the :a�cncc of rni11anre (, O). thct;olu1ion is kn�n:
,,(1) - ,(0> ► ,,, - 'I"ll '
A«otding 10 1hc 1hcol'cm on difTcrcn1iu ; ion wi1hropctt 10 a p;ar-amc-ta. lM' tolut,on an
� wri u«•n in the form
K Mo r <)'(I) + 0(,')
( Of" s1n:ill c , where., is che deri"·:uivc- or 1hc soh.i1ion with ra:pu:·t lo th< pua.mctu, for
t ..., 0 . Suo.1-tituting 1his aprcnion into 1he original difl'crcnti.21 <q;U21ion. wc-gct.an <"q'A•
cion for)'. In fact,
c-0,
3nd sin� 1his l'Cl:11ion holds for all ,mall ,. chc codnc:icnt or.any power o/� i, the- t.amt in
both sides of the «(uatio n. In pu1icular1 lhi.sgi,·cs th< rou�;ng c-uilysoh--cd �m.tion o/
1,1ati1uion.s:
j F(x0(1), .i0(1)), .1(0) = j(O) = 0 .
.<ns. ,(1) • ,,(1) + ,J: J: F(, ({). x ({)) d{d t + O(c').
0 0
Warning. Strictly speaking, our argument is valid only for .sufficiently small
I, bu1 in fact i1 i$ easily jus1ificd for anyjinit, time in1crval (II � T,procidd
that t do,s not extttd some quantity dep,nding on T (th, «nslanl impliat in tk tm,,
writlen as 0(1:2) ,'ncrtases with T). 1t i s extremely risky to extm.d the results
ob,ained i n this way to an ,'nfinite time interval: One cannot interchange
the limits as I -+ co and e -+ 0 .
Example I. Coruidcr a bucket of water whooc bouom has a small hole of
radius t (f;g, 70). Given any T, there exislS a value of• so mall that the
bucket remains almost rull during 1hc ti=/ < T. However, forcvtt) ' fixed
t - 0, the bucke1 becomes empty as the time approaches infinity.
Probltm 2 . As is well known. a body of m.a.u m mO\:ing rcb.ti,-c 10 the- earth with ,-dociry •
66 Chap. 2 lbsi<: n.eortrm
--
--
,-o
V
, _.,,
Fig. 70 1'he A,ym1>101ic bch.,vtor or 11\e per1urbed tql,lalio,, a. • -0 and a., t - •·
!I
Fi g. 71 Oellec1ion of a (alling body from the wrtical.
is subject to a Coriolis for<lc F -2mv x O. whttc O i, the angular "flOClty \'tt'IOt olt.hc
earth. Astone is dropped (without initial \/elocity) i ntoa m.iM olckpth250 niia118C btitudc
or Leningrad (l - GO•). How ru from the \/,er1tCal is the ,cone deflccred by tbt- Coriolis
force (Fig. 71)?
Solution. Here we arc dealing with the dilTerential equation
li - g + 2X X 0)
d epending O(I the e arth', angul ar vt'.loc ity O - 7.3 x 10-• xc· • as a puamieta. It ca.a
be predic1cd in ad..,ance that tht-CorioUsforce LS5m3.IJ com�rul: lo tbt •---right. a..nd httlu
0 can be regarded as a llftOII parameter . According to theditft":rcntiability thcoum, •c· U�"'t:
,. = "• + Oy + O(Cl')
for am.all O., where
,.
•• --(0) +•y·
Substituting this exp�ion for a into th e differential cqua.tion, we gn the Uf!W.tioo ot
variations
y = 2,, x 11, Yo(O) = y0(0) = 0,
and hence
,
y - g X Cl I> -l°
21 h X Cl, I>
h-Sy·
l°
= XI. - X-2
•l
X (6)
.r
f. t
Fig. 72 Integral curves ofthe two differcntia.l tqua.tions oompris,ed in the-singleformula
jl = x .
68 Chap. 2 Buie Theorems
' t
,., ,.
.. 13 l1l1tgl'al curves of the tWO«juation1 written t�1Mr as theClaanut ftlu.Ation 6J.
determined by a vectorfaldv in tkphase spau U, and kt Ir': R - RI,, th, sl,;.fi l,ys
carrying the point t e R into tk poinll + J e R . Thm ,p • Ir': R - Uis o,o/,,J;a
of (2) for arbitrDI') ' s. In othn words, ifx = ,p(I) is a solution of (2), lnDI so is
X = <p(I + J).
Sec. 10 Phase Curves of Autonomoui Syst<ms 69
We now consider the set of all p,,riods of 1hc resulting con1inuous func
tion q,,
LEMMA I . The s,t ofall pnio<bof th, contin.,,usf•"'tion ,p: R - Uisac'4ud ,v/,
group of th, group of r,a/ numbers R.
Proof lf,p(t + T,) ii ,p(I) and ,p(t + T,) ii ,p(I), 1hcn,p(I + T 1 ±T,) s
,p(t + T,) =,p(I), where :;= indicale< identical cquali1y in I . :\forco.-er,
if T, -+ T. then
Proof If C ,t, {OJ, then there exjst positive clements in C (if I < 0, then
-t > 0). Let
To = inf{I: IE C, t > 0).
• �lg
• I( •
I (A•I/Tg
• •
f.'lg. 75 A clo.cd •ubgroup or 1he lint.
XI
L
-
,
,%' I
Pig. 7G A clos.cd subg:roup or the plane:.
t The dcfinicjon of a difl"eornorphic mapping of one c.vri.-t onto another is gi,.� r..:. ex•
ample, in Sec. 33.6.
72 Chap. 2 Basic fheon:ms
' ..
lim •(t,) • r, lim 11
• ••
- co.
Many geometric concepts can be described in two ways., cithcT in the b.n
guage of J><>int.s in space or with the help offunctions defined on th<: space, a
duality often found 10 be useful i n various branches of ma1h<:ma1ics. In
particular, vector fields can be described not only by using curves, but also
in terms of dijfertnJiation offunctions . The b.uic theorems can 1hcn befonnu
lated in terms offirst inttgra/s.
d " of
t./1. a - I: - .,, (I)
di ,. 0/•"' -
,., iJx, �
L•.f = -If.
h follows from (I) tha1 if 1he function/and 1he fidd v arc ofclusC, then
the func1ion L,Jis of class c·- 1 •
11.3. Properties of the directional derivative. Le, Fdenote theset ofall
infinitely differen1iable func1ions /: U - R. This sci has 1he na1ural
structure of a real linear space (since addition of functions pttSCnres difTcr-
entiabili1y) and even of a ring (since a produc1 or differentiable Functions is
differcn1iable). Let v be an infinitely differentiable vector field. Then 1he
derivative L.J of 1hc function/eFin the direction of vis again an clement
ofF (1he infinite differen1iabili1y is csscn1ial here!). Thus differentiation in
74 Chap. 2 Buie Th,orc,m,
Comm,nt. The licld c, denoted by fa, b], is calll"d the ,omm"4ti.r or Pow.
braek;t oft he fields a and b.
Probltm 4. Prove 11,e following three properties of1he commuiator:
a) [a, b + Ac] • fa, b] + Afa, c], A e R (linearhy);
b) fa, b] + fb, a] • 0 (antisymmetry);
c) f[a, b], c) + fib, c], a] + ([c, a), b] • 0 (Jacobi's idc,ntity).
Comm,111. A linear space equipped wi1h a binary operation satisfying the:
above three conditions is called a Li, al�bra. Thu, vc:ctor field,, 1aken with
the operation of commutation, form Lie algebras. Other examples of Lie
algebras arc the following:
I) T hree-dimensional space equipped wi1h the operation of vc:ctor
multiplication;
2) The space of all n x n matrices with the operation carrying A, B into
AB - BA.
Problem 5 . Starting from the oomponenu of the fields a and bin tome c,o.
ordinate system, find the components of thc.ir com.mutator.
F ig. 79 A 1>h:ttc cur� lio entirely on one lcwl •uM.acc of I.he fint intqraL
(a) (b) �)
Fig, 81 Which 0(1hetc sys1cms have nonc:onua.n, fin.1 iniqnh?
Example I. Consider the following system whose phase space is the whole
plane (Fig. 80):
This system has no first integrals different from a consrant. In fact, any 6nt
integral is continuous in the whole plane and constant on e�ry raye:rranat
ing from the origin, and hence is a constant.
Problem I. Show that evc:ry first intcg-r-al is com.tant in a neighborhood ol a limit q,dc:
(F;g. 81a) orequai;on (2).
t By the s,t ofl,ttUCofa fu.n cti onf: U - R ismantthefull prcimagcofthc pointC • �i..c:.,
the sc i / - 1C C U .
Sec. 11 Dircc1ional Derivative. l'irs1 ln1cgrals 77
L fl
•
= ,.f, [off (- ?!:!.) + oq,?!:!_?!:f.]
op, oq, op,
= 0. I
11.7. Local first integrals. The absence of nonconstan, firs1 in1cgrals is
rcla1cd 10 1he 1opological struc1ure of 1he collection of phase cw-vu. In
general, the phase cur�.sof a system of differential equa1ions do nor all s1.ay
on 1he family of level surfaces of any func1ion, and hence there is no non
constant first integral. However, the phase: curves do have a simple structu.tt
/O(ally, in a neighborhood of any nonsingular poin1, and nonconstanl fint
in1egrals do exist locally.
Lei Ube a domain in n-dirnensional Euclidean spaoe, let v be a differ
entiable vector field in U, and let x b e a nonsingular point (v(.r) ,;. 0).
THEOREM. There exists a neighborhood V of thepoint Xe u su,:h IMJ ,,pu,lio,t (2)
has n - I functionally indtp,rukntt first integrals f,, ... ,J._, ia V _ A-for-,,
any.first integral ef(2) in Vis afunction off ,, .•.
,J._,.
Proof. The theorem is obvious for the standard equation
j1 • •• • a: j.,. a Q (5)
t It was shown by Hamilton th.at the differential equations ofa great ,--ancty ofp.obkt.ns
cncountcTed in mechanics, opti� c alculus ofvariation� and olhcr brandwsoltc:ic:n«-ca..o
be written in lhc form (4),
t It wtll be recalled from calculus th.at the functioos/1y
•••• f.: U - R. arc fuoclion;al.Jy
indepc1,dcn1 in a n eighborhood of a point x • Uiflhc rankofthcdcri,-a.b,"C/.L.oicbcma;p-
pi ng/: U -R• determined by the functions/11 ••••/. cqulls
: •·
78 Chap. 2 Baiic Thmrcms
v.
in R' (l'ig.82). In fact, the firs, intcgrab arc arbitrary differtntiablc func
tions of the coordina1csy 2, ... ,y,., and the coordinatcsy2, ... ,7. give us
n - I runctionally independent first intcgrab.Thcs.amc i, I.rue roraiuation
(5) in any convex domain �1' of the space R'.t By the basic tMC>rem (Sec.
7.1 ), equation (2) is of the form (5) in some neighborhood ofthe point x in
suitable coordinatcs.1, and this neighborhood can be regarded as• convex
domain in the coordinates)' (otherwise replace it by a smaller convex neigh
borhood). It remains only to note that the property of a function being a
first integral and the property of functional indcpcndcncc arc both i n
dependent of the coordinate system. I
11.8. Tirnc-clcpeodcot first integrals. Lctf: R x U - R be a differ
entiable function in the extended phase space of the equation
x = v(t, x), l ER, XE U, (6)
which is in general nonautonomous (the right-hand side v(t, x) is as,umcd
to be differentiable). Then the functionfis said to be a 1i--,kp,,,Ja,Jfo11
integral if i t is a first integral of the autonomous system obtained from (6) by
adjoining the equation i = I:
X = V(X), XeR x U, X = (t, x), V(t, x) = (I, v).
In other words, eve'.1 integral curve ofequation (6) /us entiret., o,ro,u 1-1scioflM
fun&tionf (Fig. 83).
The vector field V dOC$ not vanish.It follows from the preceding theorem
that equ ation (6) has n functionalf)I imkpmdmt (timL-tkpmdnil) first iaugrols
f,, ... ,f, in some neighborhoodof evny point (t, x) (IJld t!UJI every (tim,-tkpmd,ttl)
first integral of (6) can be expressedin t.trmsoff,, ... J, in this nnglJ#n-1-d.
In particular, the autonomous equation (2) with an n-dimensional phase
t A domain in R" �s 5c-. id to be un«x ifwhen ever it contains t""-o poin ts.. ir USOcon.a.ins the
li ne segmentjoining the two poinu . Giv e.an example ofa first intqral ol(S) .tud:adocsnot
rcduoc to a functio n of,1:• ..., �.. in a nooconvcx domain Wof the spa.er R•.
Sec. 12 Con.1ervu1ive Sys1cms 79
I,
Fig, 83 I nttgral cur\!c, on a level surface ora umc-dq,cndC'nt 6nt intqn.L
(x 1,x2)e/xR. (2)
{I
.r
fig. 8S Potent
i.aIenergyof thependul um neartheJo- cr
· and uppercquilitwiumpoaitiom...
Sec, 12 C',on,ervntive System, 81
12.3. Level curve, of the energy. Turning 10 the pha,e curves o( the
system (2), w e note that each such curve lies entirely on one level set o( the
energy. We now study these level sets.
F(x,) = 0, x, = 0.
Proof We use the implicit function theorem, observing that
ox, =
-
iJE
-F(x1),
Ifone of these derivatives is nonvanishing, then the set oflevd £is thcgnph
of a differentiable function of 1hc form x1 • x 1 (x 2) or x2 = x2(x1) in a
neighborhood of the point i n question. I
Note that the exceptional points (x 1, x2) figuring in the theorem, where
F(x,) = Oandx2 =
O,arcjustthe stationarypoints (equilibrium positions)
ofthe system (2) as well as the singular points ofthe vector field ofthe pha,e
velocity. Moreover, the same points arc the critical points ofthe total energy
=
E(x,,x,), while the points where F(x,) 0 arc the critical pointst o(1hc
potential energy U.
To draw the level curves of the energy, it is useful to think o( a bad
sliding in a " potential well" U (Fig. 86).
Suppose the total energy has a fixed value £ . Since the potential�
cannot exceed the total energy, the projec1ion onto configuration spa«-
t By acritUalpqint of a function ismesnt a point .tt which the 1012) diffcrcntialolthc(UJ'K:tion
vani shes. The YJluc of the (uncti on at such a point is caJJul a aiti4=ol Nt1--.
82 Chap. 2 8as,c Theorems
(,
r,
t,
...,
{I
£6 I
I
I
I
':
II I
II I
I
d o
I I
II I
I
� I I 1I I
I II I
� I ,{t• I I fl
II I
II I
.r,
II o
II I
II 0
II
.T1
"'
11 I
.z;
f'ig. 87 l,t\cl rvr'-''-'' of1ht <'ntrgy for a potMci.al with tvi,· o wtll.t• •
V V
r r
Fig, 88 What is Lh<" appearance- of the lc,·d cun--n ofthe- c-mrgy ror ra<'h ol mc,.t
potential�?
of£ equal 10 the critical valuesof the potential energy U(whe.., U'({) 0) =
and in each case examining the curves with values of£ a little smaller and
a little larger than the critical value.
Example I. Sup1=e the potential energy U ha1 three critical pomu, a
minimum� 11 a local maximum { 2, and a local minimum (3• Then f""wg. 87
shows the level curves corresponding 10 the values £ 1 U({,), U({,) < =
£, < U({,). E, C U({,), U({,) < E. < U(c!,), £, U(c!,), £. > U({,). =
Prob/r m I. Ske tch )C"\'d curvn o(lhc energy for the p(:ndul um <"qUa.tionX -sin xand b
the J)C'nduh.un c:q1.Jatio 1,s ntar the lower and uppc-r equilibrium posJUOM X = -Jt and
i - x}.
P,06/tm 2 . SkctC'h le\'d curvn of the C'nergy for lhC' Kt:J,ln ,ottfllWt
U---+ t (.'
X :,
X
12.4, Level curve• of tho energy near a 1ingular point. In llud),n,t the
behavior orlevel curves ncnr a rri1icnl valut' orth<: tntrgy, ii II uK(ul to kttp
1hc rollowin11 foci, in mind:
R,111ark /, Ift/11 pot,illial ""'II) is a q11ndra11cform (/ • Jk<1, 1/,m IN/,.,,/ n,,,�,
of tlr, llttrgy ,,,, St.f'oud-ord,r """", 2,�· • \� + Ax!,
1 n 1hc a11rnc1iv� ra11;r1 wr havr A > 0 and thrtriuc-al point O i1a m,mmum
or the po1en1ial energy (Fig. 89). The level curves of the energy arc then
homothe1ic ellipses cen1ercd at 0.
In ll1e repulsive case, we havck < 0 and the cri1ical pointOis a maximum
or the po1cntial energy (Fig. 90). The lcvd curves of 1hc energy are then
homo1hetic hyperbolas ccn1ered a1 0, 1ogcthcr with 1hc pair of aspnptotes
x1 • ± JTcx,. These asymptotes arc also called sq,arolrius, sin« they sep;t•
rate hypcrl,olas ordilfen·rn 1ypes rrom one ano1hcr.
:r,
f
Proof We need merely nole 1ha1
= d/(tx)
f =x
J'
I
/(x) .=....,,- ...c. dt = f'(tx)x dt f'(tx) di,
o dt oI o
where
g(x) = J:J'(tx) dt
.r,
t Both lemmas can be exte nded to the cue orru:nctions o(IC"·cral variables..
06 Chap. 2 Buie Thec,rc,m,
LEMMA. Ifa solution existsfor Ill < r, then it satisfi,s tht inequaliti.u
f Naturally, changing the potential c:ncrgy Uby a constant docs noc changecquatioa, ( 1 1 .
Hcrtcc it is onl y essential that Ube bounded from below.
Sec. 12 Conservative Sy11r.rn1 87
Fig. 92 The rectangle which 1he pha.se point cannot leave in time T.
0 b X
F ig. 93 The sc, or poinu K where U(x) < E (Ea noncritical enc.rgy ln-d).
i n the phase plane. Consider the parallelepiped Ill :!;; T, (x,:x,) en in the
extended phase space (x,, x,, 1). By the extension theorem, the solution can
be extended up to the boundary of the parallelepiped. It follow. from the
lemma that the solution can leave the parallelepiped only through those
faces on which Ill =
T . Hence the solution can be extended up toarbiuary
I =+ T, and hence can be extended indefinitcly. I
Prove the ponibtlily of indefinitely alcandi.ng tbt: solutiom ol l.M: .ys-1cm ol
Probltm I .
Newton's cqu.at ions
determines a Jmooth turue, in the plarte (x1, x l). This ain.ie is dijfeomorplw IIJ a circle
a11di. a phase curut of the system (2). Simi/art:,, th, ray a .;; x< a, (o, - co< x
.;;b),where U(x) < £,is the proje<1io11 01110 the x1 -axis ofa phtue cvmem.ff,o_,pi,u
to a straight li11t (Fi?, 95). Finally, i11 the case where U(x) < EM ti,, tt:lw/., lint,
the set oflevel E tousistJ oftwo phase ,urr,es
-'2 = +J2(£ - U(x,)).
Thus the set of level £, where 1he energy£is noncritical, consists of a
finite orcountable numberofsmooth phase curves.
12.7. Proof ofTheorem 12-6. The law or conservation ofmcrgy allows us
to solve !\ewton's equation explicitly. In facl, for a fixed value of the total
Sec. 12 Conscrvaoivc Sy11em1 89
,,
Pig. 90 The ph 1ue point 1raverte1 half the pha)t cunc (fro.n • 10 6) 1n a fwwtc UlM
T/2•t1 -t,.
.. ..
e, to '
f f
Fi g, 97 Uae of reflection 10 extend the 1olu1i0n (J( Newton's cqu.-tioa.
energy£, the magnitude ( but no1 the sign) o(thc velocityii is<ktaminal by
the position x, since
x• ± ✓2(£ - U(x)), (3)
and we already know how to solve this one-dimensional equation�
Lei (x 1,x2) be a point of our level set,whcrex2 > 0 (Fig. 96). 1'1akiog use
of (3), we look for a solution ,p o f equation ( I) satisfying the initial condition
<P(10) = x 1 ,rp(10) =x 2,obtaining
1 - 10 =
J•<•1 d�
,, ✓2(£ - U({)) (4)
•J
forI near 10• \',/e now observe that the integral
T d{
2 = • ✓ 2(£ - U({ll
converges, since U'(a) ,' 0, U'(b) ,' 0. Therefore (4) defines a cootinuoUI
function <Pon some interval 11 .;; I .;; 1, with ,p(1 1 ) = a, ,p(l2) =•·This
function satisfies Newton's equation everywhere (Fig. 97).
The interval (11, 12) is of length T/2. \Ve now extend <Ponto the next
interval of lenglh T/2 by using symmetry considerations: <P(I2 + t) =
,p(12 - t}, 0.;; t .;; T/2, further extending <P by periodicity: q,(t + T) .,
,p(I). The resulting function, defined o n the whole line, sati.sfio Newton's
equation everywhere,and moreover cp(t0) = x1 , 4>(10) = x2• Thus1A--e: ha�
constructed a solution of the system (2) satisfying the initial condition
(x., x2), which turns out to be periodic "'1th period T . The concspondiog
90 Chap. 2 Basic Theorems
V
£
V
£ \. --
b • &
'
(1 Q
�
.r, .r,
' r, r, ._..
{a) (bl (c)
Fi.g, 98 D«-omp01hion ora critical level curve olthe cnffJY into phut cvrva.
closed phase curve is just 1hc part ofthe set oflcvcl £lying ov,:r the int<'.rval
a ,;; x ,;; b. This curve is diffcomorphic 10 a circle, like every cloocd phase
curve (sec Sec. 10).
The case where Lhc interval extends to infinity (in one direction or the
other) is simpler than 1hc case just oonJidered, and is left as an exercise. I
12.8, Critical level curve,. The structure of critical level curves can be
more complicated. Note that such curves contain fixed poinr.s (x1, x2)
(where U'(x,) = 0, x 2 = 0), each of which is itself a ph.lK curve. If
U(x) < E everywhere on the interval• ,;; x ,;; b, except for U(a) • U(i)
• E, and if both end points arc critical points, so that U'(•) • U'(i) = 0,
then both open arcs
x2 = ±J2(E - U(x,)), • < x, < b
(Fig. 98a) are phase curves. The time taken by the phase point to traverse
such an arc is infinite (Theorem 12.5 + uniqueness).
If U'(a) = 0, U'(b) ,t, 0 (Fig.98b), the equation
ixl + U(x,) • E,
determines a noncloscd phase curve.Finally, if U'(a) ,t, 0, U'(i) ,t, 0 (Fig.
98c), then the part of the critical level set lying over the interval a ,;; x ,;; 6
i sa closed phase curvc,just as in the case ofa noncritical lcvd £ .
12.9. Example. The above considerations w;n now be applied to the pen·
dulum equation
X = -sinx,
with potential energy U(x) = -cos x (Fig.99) and critical pointsx1 = ta,
k = 0, ± I, ...The closed phase curves r,scmble ellipses near the point
x1 - 0, x 1 a: 0, and these curves correspond to small oscillations of the
pendulum. The period T of the oscillations depends only slightly on the
amplitude, as long as the amplitude is small.For larger valucsof the cna-gy
Sec. 12 Conserva live Sy 11enu
1 91
.%'1
a:,
constant, we get larger closed curves, until the energy reaches a critica.J
value equal 10 the po1<n1ial energy of the pendulum in the upside-down
position. The period of the oscillations then increases (sin« the time of
motion along the: scparatriccs making up the critical kvd set is infinite).
For larger values of the energy we get nonclosed curves on which x1 docs
not change sign, i.e., the pendulum rotates rather than oscillates, achiNing
the largest value of its ve1ocity at the lower position and the smalltst value
al the upper position. Note that values ofx, differing by 2.b< concspond to
identical positions of the pendulum. Therefore it is natural 10 choose the
cylinder (x I mod 2n, x2) rather than the plane (x,, x1) as the phase space of
the pendulum (Fig. 100).
Taking the picture already drawn in the plane and wrapping it around
the cylinder, we get the phase curves of the pendulum on thcsuriacc of the
cylinder. They arc all closed smooth curves, except for two fixed pojnts
A, B (the lower and upper equilibrium positions) and two ,epantriccs
C,D.
Prohlnn. I . Draw graphs or the funetions x1 (t) a.nd x2(t) f0r tht solutioft wilt-. entJS:t aeu
but iomewhat below the critical energy in the upper po5,itioa\.
92 Chap. 2 lbiic Th�m,
.r,
1,---, '
r, I
t
Fl,g, 101 'n1e angl e or de\'lation o r the pendulum and 1he vf'IM1ty ol 1b fflObOn r«
amplitudes neat n.
A ,u. See Fig. 10I. The functi ons x1 (I) a,,d ,.:,(I) can be cxptaKd i n tcrmt. o/J;ft and Cl'I (lhe
c-ll ipti c ai ne a n d elli 1nfc CO!ine). Ai E approachts the lower c rfocal valw. the OM'ilht,ona
or the pe:1,dulum bt:come apprOXimatdy h ar monic, with s.n and en eoinc in:to 11.n and <'OI.
PrtJhlm, 2 . Al wh at rate docs the period o(the os.ci'.llatioru of apendulum app,oach infinity.
as the ener gy E a1>pr0.tcho the upper crilical val ue £1 !
A1'J , At a logarithmic rate (- CIn (£ - E)).
1
Hi,i l, See formul a (4).
.�i g, 102 Phase cmvo of 1he van dcr Pol equalion and th e i.nercmfflil of mttSY aftitr
one circui1 arou nd the origin.
To calcul at e th e c� rgy increment after one circuit, this (unc'tioo shoukl be in.tqntcd
along a tu r n of th� pha..se 1rajettory, but 1hc l.atte r is unrortu.natdy not known. Bue., n
altt.ady C)l'plaincd, the 1ur11 it close to a circle. and ht"ncc, 10 within an at<'llnl(y o(O,-I),
th e integ ra l can be t ake n along ,h e circl e S or radju1 A:
F(A) -
f /, ,1,, -I, ,1,,
(the integral U tak en alo ng a circle o f radius A lr-a\-cncd in the councadockwile di r «
t ion).
Onc,e havi ng calculat� the func1jon F(A), w e can in"�tigatc lM bma,io, of'thc phatt
curves, Ir th e function F i s positive, the en ergy increment 6£ after one circuit is aho
positive (for small )>Ol'litivc f"), In this CA.st, the phuc cun·e is an un�;ndingspin.I. and lhc
sys1 em executes inrre�ing oscill ations, On the othu h and. f F < 0 , lhcn � < 0 Ji.nd
i
th e phase sp it a l is co ntracting . In the- latter ca.sic. the o,c:ilbtionsdamp ouL
It can happen that th e functi on F(A) chang,cs sign (Fil, 102). Suppo1C F .-t) hu a
simple zero A0• The n for small , the equal.ion
O.£(x.. x1) =0
issa1i sficd by .t dosed curve r in the phase pl a ne, near the circle of radius .... (U'lis l"ollows
from the impl icit function thcottm). Obviously r is a closed plusc c:vn�. i.t, a limit
cycle of o ur system,
The lljgn of th e de ri v a tive
p dFI
• dA ......
d e 1crmines whether n eighboring phase curves wind onto the limit cycle or unwiftd from
it. T he cyrle is unstabl, if�P" > 0 and st.ablt if�P" < 0 . In fact, in the fiffl cue die ,energy
inc rea se afte r one ci r cuit is gr eater than zero if the phase cu.n-c lies outside the- cyde a nd
l es., tha n -zc.ro i f it lies inMde the cycle � h ence th e phase cu.n-c a l wa)"S mG\.-U awayfrom the
tyclc. However. In the i,econd CII\C', 1he phHe wrva approuh the cy'('k both (,om lhe
irHldc 11nd from 1he out\ldc• .u In ►�ig. 102.
Exampl, I. Considrr the equation
/I • -x + ex(I - x'),
culled the va11 dtr Pol ,quation. Evaluating the integral (6) with/, • 0,
/, - x,( I - x/), we get
Linear systems arc alma.I 1hr only large clan o f differential equauo,u ror
which there exists a definitive theory. This theory i , =entiall) a brant'h o(
linear olgcbra, and allows u, 10 solve oll autonomous linear equations.
The theory or linear cquo1ion• is al,o useful as a first approximation 10
the study or nonlinear probl<m,. For example, it allows us 10 invntigatc the
stability or equilibrium 1 >osi1ion, and the 1opoloSic•I rlauifieauon ol•i"8u•
tar points of vector fields in the nondcgcncratc cases.
Ax � di g' x vx cR•.
df ••O
Hint. Sec Sec. 3.3.
Equation (I) is ••id 10 be linear. Thus, to describe all one-parameter
groups oflinear transformations, w e need only investigate the solutions or
the linear equation (1).
Sec. 14, The Exponential of an 01><rator 97
i = I, , .. , n, (I')
where (a,1) is the matrix ofthe operator A in the given coordinate systtm.
This mairix is called the mat,i.< oftht S)'>i<m (I').
For n = I the solution of equation (I) satisf)' ing the initial condition
,p(O) = x0 is given by the exponential
,p(t) = ,''"•·
It turns out that the solution is still given by the same formula in tM"gcncral
case, provided we explain what is meant by thcexponemialofa linear oper
ator. \IVr- now turn our attention to this problem.
,, = ..lim
-oo
(£ + ::!It)' (2)
14.2. The metric space of operators. The ,c, L of all linear opcraton
A: R"' - R" i s itselfa linear space over th� fidd of real numbers (by defini
tion, (A + .lB)x = Ax + .lBx).
P,M>Jtm I. What is 1he di,nensi on or1hc linear spa« L?
Ans. n1 •
Hint. A n opc-rato r is spe cified by i1s matrix.
We now define the distance between two operators as the norm of the
difl'crcncc A - B:
p(A, B) = IA - Bl . (3)
1·HeOREM. TM space oflinear operators witlt the metric pis acompkk -tu sp«,.t
t By a mttrit Jf>or.e is meant a pair consisting of a sc-1 �f and a fUJ'Ktion p: ·" , .\I - R.
called the mtt,it. such 1h a1
>
l ) p(x,y) 0 V x,y • Af, p(x,y) = 0 if and ooJy itx =:,;
2) p(x,y) a: p(;,, x) Vx,y • /&1�
3) p(x,y) < p(x, z) + p(z,J) V x,J, O< M .
Ascqucnec x, or poi nts of a metric spac,c .1'1 t:Sca llrd a c.-',SflfW'IC'irV , > 0:) X: p(z,.z,)
< , V i,j > N . A se quence x, is said 10 tML>n� tO a poin t x if V,: > 0:i N:,<� x,) < •
"Ii > N . The spaec Mis said 10 be compt,u if every Cauchy scqucnec is eonvapL
Sec. 14 The Ex1 >oncnoial oran Operator 99
A1 e A1
is literally the s:1mc ,11 the theory o fnumerical aeries. The 1hcoryofscrio of
functions can also be carried over a t once to the casc offonctions with values
in M.
' -'
r.1, (4)
ft{/j
J• I dl
(5)
is uniformly conuergent, then the series (4) can be dijftrmlialtd i,,.. 67 Ina (1 i., IN
coordi11alt 011 the line R):
!_ f
dt ,. 1J, a f df,.
,• I dt
Hint. The proof for the case M = R is given in advanced calculus and can
be carried over word for word to the general case.
lf.5. Definition oftbe exponential t". Lei A: R• ➔ R• be a linear o pe r
ator.
l+a+ ,,, +
2i
which converges 10 t'. It follows from \Veientr.us' tnt that the series,• i,
uniformly convergent for 1111 4- a. I
Problem I. C:alculatc the matrix,,,. if the ma1rix A is or the form
(0 I 0)
b) (� �); d) 0 0 I .
0 0 0
14.6. Example. (',onsider the set of all polynomials of degree las than a in a
variable x with real coefficients. This set has the natural structure o f a rc;ol
linear space, since polynomials can be added and multiplied by numbers.
Probl,m I , Find 1hc dimtmion o( the 11 ,acc of all polyno,mals of dcgrtt kit th.a.n -.
Anl, n; ro, example, I, x, x1, • , • , x--' i.1 a ba11s.
\Ve will denote che space of all polynomials of dcgree las than a by R".t
The derivative of a polynomial pordegree l as than n is itselfa polynomial of
degree less than n. This gives rise to the mapping
A:R"➔R", IIJ> -.
= dp
dx
(6)
Pro '1ltm 2 . Prove that A i .s a linear operator, and find its kernel a.nd imag,c.
A ns. Ker A= R Im A= R•-1
,
1
•
On the other hand, let N' (t e R) denote the operator of shift by I, cart) ' •
ing the polynomial p(x) into p(x + 1).
Problt,n :J . Prove that II': R• - R• is a linear operator, and find its kttnd. and image.
Ans. Ker/-/'= 0, Im H'= R".
Finally we form Lhe operalort u.
THEOREM. IfA is th,.optrator (6), thtn
e'11 = J/1 ,
G :)
in II b.uis •u e1• Sint-c the cigc,walutt l, 2, 11 0 are rc;al a nd dillti_na. the �n*
A it diago nal with cigcn b.uis 11 = e 1 + e 1, 11 = e 1 - e,. The matrix o/A in this basis
is ju st
(� �)-
Hence the m a tr ix of th e o pera tor,.. i n the dgcnbuis is
i:s nilpotent. Mol'C gcnc:rally, prove that if all the clcmentt of the matru: of an operator
on and below the mai n diagonal :lire �cro, then the operator is nilpocenr.
Ptooltm 2 . Prove th a t the djffcrenti:uion opcr:.tor d/h in the sp3c:e oL au polyftOl'l'liaJs ol
degree lcu than n is nilpote nt.
Sec. 14 The E,poncmial of an Opcr.uor 103
If/111 op,,,,,or A is 11;/pott,11, 1h,n tht s,,i11 r4 1,rminat11, i.t., rtJ.wu t• •fouu
sum.
Probl,m , . (!hlcuh11e r'A (t • R) whc:re A: R• - R• 11 the: optn&or whh Matn.a:
b
(: · .. �)
(I o\•c:r the main diagonal and O c:lsewhc:re),
Jl{,rt, One way or ,olving this problem is co utit Taylor's rormuJa fo r polynombk. l'M
diffcrcntial 01�ra1or d/rlx ha� 11 muriK of the indkatfti typ,t: in tome bun (wllich Ofte?°.l,
For furihcr' dcmils, 1cc $«. 25,
14,9, Q.ua1l-polynornlal1, Let l be a fixed real number."J'Mn b)' a p,u; .
poly11omia/ with expo11t11t l is meant a product of the form e1'p(.s) whcrcp is a
pol,-nomial. The degree ofpis called the tkxru o( the quasi-polynomial.
l'robl,,,, I . Prove ,hat tht" 1c1 of all quasi•1)0lynomial1 with cxpoM:nt A oldc:crtt kll than
n is a liocar spaC'c.What i, the dim ens.ion of this t.patt?'
Ans. n: for example,,"•, ,uh , .... ,t" · ,,..... is a basis.
Remade. There is a certain ambiguity implicit in the con�pt of a quasi•
polynomial,just as in the case of a polynomial. A (quasi-) polynomial can be
regarded as an exp1tssio11 made up of signs and le11en, i n which cas-e the solu•
tion ofthe preceding problem is obvious. O n the other hand,�� can regard
a (quasi-) polynomial as afunllio11, i.e., as a mapping/: R - R. ActuaUy
both conceptS are equivalent (when the coefficients of the polp,omials arc
real or compie• numbcrst).
P,obl,m ? . Prove that C\'C'ry function/: ll - ll which an be writtt".n as a.quui-polynomial
has a uniqne rcprc1cnlation as a quasi-polynomial.
flint. \\'c n<"cd only note that if ,0p(x) = 0, then thie cod6ciients ol the polynomial,�)
all \'anish.
The 11-dimcnsional linear space of quasi-polynomials of degrtt less than"
with exponent ,l will be denoted by R•.
TH�ORtM. The differential op,rator d/dx is a lin,ar opnaiorfrom RA u, R• swJt tllJJJ
e''"" = fl' (7)
for every t e R, where H': R" ➔ R" is the operator ofshift bJ t, i.e., (H'l)(x) =
f{x + t).
Proof. Proving first that the derivative and shift of a quasi-polp101niaJ of
t We will $(l()n c-0nsidcr (quasi-) polynomials with real coefficients.
104 Chap. 3 Linear Sys1em,
0
:) ·;
(,l on lhc: main diagonal, I O\'cr 1hc main diagonaJ. 0 clKWMff). for aampk:, nkubtir
exp ( � : ).
_,
tf IA - A
dt
"'fA.
(2)
(J;;+ IA + 1
; 11 1 + .. ·)(e + sA + s; A' + · · ·)
- E + (1 + s)A + (�+ts + �)A' +
The coellicienl of A' in the product cquab (I + s)1/L!, since formula (l)
holds in the case of numerical series (A e R). The lcgi1imacyofthe tcrm-by
tcrm 11111ltiplica1io11 i.s proved in the same way as the lcgi1imacyor1hc term•
by-term multiplicati01\ orabsolutely convergent numerical series {the series
for e'" and �" arc absolutely convergent, since the series ror eJ•I• and ,l•I•
where a • IA I arc convergent).
To prove (2), we di/Tcrcnciatc the series fore"' with rcspccc 10 I formally,
obtaining a series of derivatives:
., d • ., t'
1: _!...11• •AL -A'.
••o dt k! ••o kt
This series converges absolucely and uniformly in any domain IAI .;; ••
Iii ,;; T,just like the original series. Hence che derivative ofche sum of the
series exists and equals the sum of the series orderivatives. I
\Ve can also pl"(lve (I) b y reducing the proofdirectl y to the numerical c::uc. af1er fi:nt
Pl'O\!ir\g the followin g
U!iMMA, let pc Rl= u ..• • = .... J h, o polJ·nomial in tJw Hriah/1! =., ... , :... IC'itA -ra:
t.oqfi�itnts , ond ltt A 1 , • • • , A,... : R• - R• 6� lintar op,Yo. tor s. 71ttit
IP(A.,.,,, A.)I.;; p(IA,1,,••,IA,1),
Proof. An immediate con sequence ofSee. 14.1, Problem 2 , I
Proofoffarmula ( I). Lott S.{A) denote the partial ,um of the series for rA:
• A'
S.(A) = !: -
,-o k! .
Then S. is a polynornfa.l in A with nonncgati\-C coc.fficien1s.. \\'e mun. J,,o,w that the
di ffctt:oce
a. = S0(tA)S0(sA) - $0((1 + s)A)
c:onverga to O ,u m - oo. Note that A. is a polynomial in sA and rA with�
'-"f.ffititnts. In fact, the terms in the product serieso fdegree: no higher lhan • in .d arc: all
obta.incd by multiply ing the te:rms in the factor series of dqr-cc no higher than• in •◄ .
106 Chap. 3 U...-a, Syst<ms
Moreover, S.((J � 1)A) I, a pullal ,um o rtht produe1 11trtt"l, and hfflft A-• tht- ,vm
or all lernu i11 1hc 1•roduc:t S.(1A)S.(,A) or df'lrtt h•ghu than"" in A lu·t a.SI 1,W. cotfl.
cltnh or 1'1>toduc:1 or1>0lynoml1l1 whh llOIHlC'llh\'C' Cotfficknt:t att �lt'C',
11 (o llow• fro 10 the ltrnma 1h11
141.(tA, 1A)I < 41.(ltAI, l•AI),
l.e1 r and a de110 1e 1he oonotg ,11ive numbt,, IIAI a,\d l1Al1 ,o that
41.(r, <1) - s.(r)S.(o) - s.(r + <1).
Since t't• - ,,•", 1he righ1.ha ,,d 1ide approacht'I O :1.1 • - oo. Thua
...
lim A.{tA, ul) 0,
t ER. (4)
Proof. Lei
d ' g' E
A=-
g
- lim -
dt ,=o ,-o I
Se c. 15 Properiies or 1hc t•:xponen1lol 107
We hove already proved (sec Se c. 3.2 ond Problem I, p. 96) that the, tnj«-
1ory ,p(I) • g'x0 i, • solu1ion orcqua1ion (3) satisfying the initial cond1tion
,p(O) • x0 . Ou1 g'x0 • ,"x0 because or(◄). I
The operator A i, called the i1!Ji11i111imall<•ttato, or1hc group (I)
JJ,al>lnn /, l'rO\ft 1hiu 1he infinht-'•lrnal gener11or 1� unk1ut.Jy d(:tumiMd by th� Jf'OUP
lltmark. Thus there is a onc-to--onc correspondence bctwcm linear differ•
cntial equations orthe form (3) and their phase llOW$ (,'), when: each phase
flow consists oflincar diffco morphisms.
15.f, Another definition of the exponential.
THEOREM, ifA; R" � R" is a lintor operator, lhtn
= ,..-GO ' A)
I'1m (/•. + -
M
eA (5)
fn
•• - (E + A)
-
M
= ., (7I - :=l:-
L CM)A'
m ,.o k.. m
where the series converges since the series fore" converges and
A JtS .
(u -•)
D u ·
108 Chap. 3 Linc:ar s,-.1trm
II
\Ve now find ,•. According to formula (S), we must first form the opu
ator 1;; + (A/m) corresponding to multiplication by l + (•/•), i.e.,
rotation through the angle arg(l + (z/m)) together with cxpaniion by a
factor of I l + (z/m)I (Fig. 103).
Problem 2 . Prove that
arg (1 + .:)-
"' Im;+ o(1),
l+z • I + Re!.+ o(�) (6)
m m m
as m ➔ oo.
The operator· (F. + (A/m))• is a rotation through the angk •
arg(l + (z/m)) together with an expansion by a factor of II + (z/•)i-.
Using (6), we find that the angle of rotation and the �ff,cient of ex
pansion have the limiting values
Remark. 1rwe identiry the comple• numbcr t with the oixration of mul
tiplicotion i>y z, the dcAnition reduce, to a theorem, since the exponential
ofan operator has already been defined.
JJ►ol>lt,H :J . 1:1nd ,o, ,•, ,•, ,••, ,,.,,
p,,,1>111H ./, J>rO\'C th.,t ,,, • •• t' •,•• whcr(l 11, 11 • C.
Remark. Since the exponential is also defined by a series, we halve
z'
•' •l+z+21+ zeC (9)
x ,.,
.J.. lJ Jt t R
N N N
fig. 104 An Euler line.
110 Chap. 3 Linear Sys1cms
,iuing of Nscgme,111 in 1he ex1cndcd phase space R >< R•. 11,i, polygonal
curve i� known ns an F.ultr /i,,,. I c iJ na1Ural co cx�c• tha1 as N - ao thr
sequence of Euler lines will converge to an integral curve, so 1ha1 fo,- large
N the last point x.(1) will be close 10 1he value of 1he solution" al time t
,01i,fying the inhial condition 11'(0) • •o·
'fHP.OREM. For /ht li11tar tquatio11 (3),
-�.,
lim X.(1) • ,p(I).
Proof. II follows from 1he conS1ruc1ion 0(1he Euler line for v(x) • Ax thal
(10)
x. - (E + %)" ••·
Therefore
lim x.(1) • •"•o•
by (5), which implies (JO), by (4). I
Prohltm J . Prove 1hat 001 only docs thc c od poim o( the Euler line �h •fl. bu•
also the whole �c<1ucncc o(pittewise•lincar func1N>ns ••: 1- R•. wl1h 1hc Euler fines u
1hcir graphs, converges uniformly to the solution • on the intcn-al 10. t).
Remark. In 1hc general case (where 1he vcc1or field v deptnds on " .. ,,.
lintady), the Euler line can also be wriucn in the form
where A i s the nonlinear operator carrying the point x into the point v(x).
We shall sec later (Sec. 31.9) tha1 even in this case the sequcntt of Euler
lines converges 10 a solu1ion, at least for sufficiently small 111, Thus the a
pression (4), in which the exponential is defined by formula (5), gives t he
solu1ion of all differential equations qui1e generally.t
The Eulerian 1heory of the exponential (which is csscn1ially the same in
all its variants), from the definition of the number t and ,he Euler and
Taylor formulas for,, up 10 formula (4) for the solution of linear equations
and 1hc me1hod of Euler lines, has many other applications going bc}-ood
1he scope of this course.
t In prac1icc, 1hc wt: of Euler Jina i.s no1 a con"-cnic:n: wayorsolving diffcrcn1i:alequarions
approximately, since 10 ob1ain high -accuracy ""'C mw:1 choote a very ,nu.U ,-aluc of the
"ite p"O. t . More ofl cn one uses various rc-fincmcnts ofth e Eu ler m ethod. in which the inte
gra.1 cun•ci.sapproximatcd not bya line segment. but rat.hcr byanan:Ja puabobolsomc
deg ree or other. The most frcqucndy u.sed m ethods att those: of A� S.mc:1, and
Runge, discussed in books on appr0xima.1c computations.
Sc,,• 16 Oe1crminan1 or the "xponential 111
The determinant of the matrix of the operator A dOCJ not dcp,:nd on the
basis. (n fact, if (A) i s the matrix of the operator A in the buts e 1, • • • , e.n
then the matrix of A in another basis is or1he rorm (B}(A)(B- 1). But dearly
x, x,
I
.,, .,,
is the area or1he parallelogram spanned by the vectors {1 (x,,.11) and =
{, e (x2,y 2), taken with the plus sign ir the ordered pair orvectors ({ 1• {2)
specifies the same orientation ofR2 as the pair of basis vcc1ors {e1, e2) and
.,
with the minus sign otherwise.
The ith column in the matrix of the operator A in the basis e,••..•e. is
r, B1
.�ig. 10.) The dc1ctminan1 of a matrix equals the oricnced aria of the panlldogr.un
sp�nncd by the colurnm of the rnatr-ix.
t The paralltll/>i/ud with edges (,•..• • (. e R• is the subset of R• ronsistingola.I point. of
thc ormx,( 1 + ···+x.(•• O<:x,< l,i= 1.2•...••. For11. =2tbcp.u-a.Oc:kpipcdis
f
ealled a pa,allelog ram. Startin g from a ny definition of ,Uumc:. we ca.o e:uily J>l'°''C tht
italkiud assertion. Otherwise the assertion ca n be: ta.kC11. as the d,fonrioa ol &he ,--olwnt: ofa
paral lelepiped.
I 12 Chap. 3 Lu>ear S)-.1<ms
made up of1hc componenis of1he image Ae1ofthe rth basis ,tttor ••· Hen«
ef ef
th,d1t11mi,umt t/11 optrntor A is th, ori,r1ttd uil•mtoftlw 1ma11 tlil 11111/ n,H (14,
pnralltltpip,d witl1tdgt1 •,. .•. , •.l 1111dtr ti,, mapp,fll A.
Prob/1111 I. Let n be a parallelepiped wi1h linearly independent tdgc-c Prov<
tlrn1 1he '"'io of1hc (oriented) volume ofthe image An ofthe parallclepiptd
under the mappin g A 10 the (oriented) volume of n i, independentof n and
equal, cle1 A.
16.2. The trace of an operator. lly the /rac,ofa matrix A - (•,,·, dcnottd
by Tr A,t is rueant the sum ofiu diagonal clements
Tr A • L
I• I
0 11.
The trace ofthe matrix ofan operator A ; R" - R"' docs not dcptnd on the
bash,, but only on the operator itself.
P,obl,m I. Prove 1ha1 1hc Ir ace ofa matri l'I �u.tls the- sum of .111 11 o(iu �,-,run. •-hilr
lhc dc1crminant equal.s the produ(t or 1he cigcm--alun.
Hin!. App ly 1hc fonnula
(,l - x,) ••• (,l - x.) = ).• - (x, + .. · + ,.),1•-• + ·•· + (-t)••, ... .._
10 the p0lynomial
de, (A - ,le) - ( -).)• I ( -,l)• - I
·-.t .,. +
Since the eigenvalues arc independent of,he bas.is, we ha'\'C the follo"A-ing
D,ji11ition. By ,he tract of an opnator Ai, meant the trace ofits matrix in any
(and hence i n every) basis .
THEOREM. IfA: R" ➔ R" is a linear operator andca real numbu. tlta
det(E + tA) = I + cTr A + O(t2)
asc ➔ 0.
t The t race of A is somctima denoted by Sp A (from 1.hcGcmun �-orct -spur-).
Sec, 16 Determinant or the Exponential 113
S«o"d p,oof, Clt:"atl y •C«) dc1(E + ,A) is a polynomial in , iu("h c..tui, • O I. \\*c
must d1ow 1h11t •'(O) Tr A. �noting 1hc dtmC"r'ltJ o( 1hc maltl'C £ + ,A by x1,. �
'·
have
t I
I a" It-,,
7i •O - ;,J• 1 ax,,
t
<!!Ji '
wher'e A i� 1he dc:tc:tlt'l inant of £ + ,A (x,,). B)' definition. the pa.rtul dffi ntfr�-
8A/ax,Jlc e<1v1tls
�I
11 n �-o
dct (/, ➔ hr, 1),
whctc (,11) i• the: 1na1r-ix whmc only non•cro element is a I in 1hc ith row andjlh column..
Out
lI
dct (£ + ht,1) • 1 + n• if i-j,
if i � j ,
and hence
a1> I
o ;r ; ,;. j,
lxu I,: = I if i =j .
Jc follows tha 1
d,I - I:• -yU
d.Jt
- }:• au =- Tr A. I
-,-
Q t, & •() l•I (1' l•I
lim [1 + � + o(�)]• =
m m ti'
=
m...ao
COROLLARY 2. Tiu operator•• preseru,s tlu onmtalion oJR• (i.e., de1 ,- > 0).
COROl.l.ARY 3 (Liouville's fc,rmula). Tiu t-adlJallet moppi•t t' eftJ,, luwu
equation
X = Ax. (1)
multiplies tlu IJ()/umt ofanyfigure by tlufa<tort"', wlurt a = Tr A.
A•O /1>/J
1- i' g , 107 Oduwior of arc:. undt:r tr:rnsformuiom of lhc pha,c flow of the pmd:ulwn
C(fU.'\tion with cocllkicnt of friction -.t.
{ x,�· + kx,
with matrix
( _ � �)
(Fig. 107). The 1race of 1his ma1rix equals k. Lei {g'} be 1hc ph2Jc Row
defined by 1he above system. Then if k < 0 the transformation l' carries
every domain of1he phase plane into a domain of smaller area. On thcotha
hand, in a syslem wilh negative friction (k > 0), the area of the domain
g'U, I > 0 is larger 1han 1ha1 of U . Finally, ifthere is no friction (.t : OJ, the
phase flow preserves area. This is hardly surprising, since in this last case,
as we know from Sec. 6.6, g' is a rotation th.rough the angle,.
Probltm I. Suppose the real p:aru of all the dgcnvalucs o/ A att neg:aln� Show \b.a.t tbt
tf':\mform,ui ons g' of the p!1.:ue flow or equatiOt\ ( I) thett dec:rcaK ,-01.:tamir (t > 0).
PrQbltm t. Prove that the cigen\•alucs of the opc:ruor r' �ua.l �,....wbett lhc l. M't tht:
e igenvalues or the opcralor A. U5C this 10 pro1.-c Theorem 16.4.
of 1he 01,cratnr ,• in 1he same basis. \Ve bt,gin by solving 1his problem in 1hc
pnr1 i c11larly simple ca,c where A has di11inc1 real eigenvalues
17.1. Diagonal operator■, Consider 1he linear difTeren1ial equahon
xoR", (I)
where A: R' - R' is a diagonal operator. The matrix of 1he opcratO<' A"of
("'··.o)
1hc form
0 "·
in i1s eigcnbasis,t where 1he .I., arc 1hc eigenvalues of A. The matrix of1h<:
(••·• . . o)
operator eA ha.s the form
0 ,..,,
in 1he same basis. Thus the solution ,p of equation (I) satisf);ng th<: initial
condition ,p(O) = (x10, ... , x.0 ) has components
k=l, ... ,n
in this basis.
If then eigenvectors of the operator A arc real and distinct, then A is
diagonal (R' decomposes into a direct sum of one-dimensional subspaces
invariant under A). The procedure for solving (I) in this case goes a s
follows:
I) Form the ehara,teristic (or ,ceu/ar) ,quation
det (A - -lE) = O;
2) Find the roots J.,, .•. , J., of this equation (the J., arc assumed to be real
and distinct);
3) Find the eigenvectors { ,, ... , {, satisfying 1hc linear equations
.:.t #,: 0, k • l, ... , n;
4) Expand the initial condition with respect to the cigenvcaon:
COROLLARY. let A bt a diago11al op,rotor. Tiu• tlu tltmtn/S •f tlv .,.,,uc ,••
(1 e R) ;,, a,!)' basis art li11,a, ,ombi11ations ofth, ,xpon,•tiols , ..
,, re/mt t.lrt � .,,
lhe eig,11valu,s of the mattix A.
{x,-xi,
X2 - -x, - kx2•
The mairix or1he operator A is then
so that
Tr A a -k, det A = I.
The corresponding characteristic cqua1ion
) 1 + k) + I = 0
has distinct real roots if its discriminant is positive, i.e., ift!t > 2. Thus 1hc
operator A is diagonal if the coefficient o f friction k is sufficiently large (in
absolute value).
Now suppose k > 2. Then both roots .1.1, ).2 arc negative, aod � equ.a•
tion takes the form
{ �·
Y2
), < o.
i, < 0
in the cigcnbasis. Therefore, as in Sec. 4, we get the solution
J',(1) = •'•'y,(0),
y2(1) = •'"y2(0),
and the phase curves have a node as in f i g . 108. Ast - +
oo all �solutions
approach 0, and almost all the integral curves become tangent to 1�.1,-axis
iflJ.2 1 > ll,1 (y2 then approaches O faster than,1 1). The picture int� plane
(x,. x2) is obtained from that in the plane (y,,.1,) by making a liocar trans·
formation.
118 Chap. 3 Llnc,ar Sys1em1
Fig. JOA Pha!IC cur� of the pendulum eq uat ion wi1h strong friction lft tht actnbai.t1t,.
I II Ill
r,
Fig. 109 Pha.sc curvn of the pendu lu m c-qu:uion with ,1r0ng friction in the wu:al buiL
17.3. The discrete case. All that has been said about ,he cxponcn1ial ,''
with a con1inuousargumcn1 t applies equally well to thc exJX>llCDtial A•with
the discrete argument n . In particular, ifA is a diagonal operator� A• i.s most
Sc•c. 18 C:0111plrxificr11ion and Oc·compl ...,.ification 119
Commt,tl. The s.:w1C' "rgurnenl rt'duco th e 1tud y of any ,Hw,mt UfWM� ,f•4n l, ddirwd
by II rtl.1tion
logC'du-r with lht first k tanu x0,x1, ..• , x._ 1.f 1 0 lh(' 11udy oflh«= cxpcnm.ual functioft
A• , whr-re .-1: R' - R4 is 3 li nt:ar optt:uor. Tbffd'or" lc,no-;,.•in.g how 1 0 nkvlatc- tM
au
m1ttr·i)( or an C';c.ponC'nl ial cn.:1blts U.i 10 C':tiltulat� rr-tul'rt'fll M"qU('ntts.
Re1utniog lO the general problem of calculating ·we note 1ha1 1hc ,,A,
roots oftloc charactcri<tic cqua1ion dct (A - i.E) = 0 maybecomplu. To
study thi" case, we first consider linear equations with a complex phase
space c·.
t Th e fact 1h at the defi niti on of a ttc-urrcnt scqumcc orord� .t 1tquir-cs know-lNlgc oflhc
finl k h:rm 11 of the.- scquc:ntc is intim a1dy connttkd with the fact that t.ht �_sp,3.tt of2
diffen-ntial c:quatio no fonkrk is o f dimension k. This coooectioo bccoma 2ppattnt ift he
diffen:n1i al cqu :uion i$ writte n as a lim it of diffe�nc:e equations..
120 Chap. 3 Lu,rar Systtnu
""·
and Ice (A) b e I he malr-i x of1hc operator A.F'ind thf' matrix ofth('decomp&tx1.6td optnlOf'
(\
Fig . 110 The opcr-ator of multiplication by i .
iR"
d,pl
di ,=,o
= i.
Sec. 18 Complcxifirntion and Drcomplrxil\ration 123
c'
,..,
I
Examining lhc case n • I in more detail,we note that cuno "'i1h ,alucs
in C can be m11l1iplicd as well as added. since multiplication i, defined i n C:
d 11'1'I d,pI
dt('I', + 'I',) =-+-,
dt dt
d d,p ,
('l','I',) = -'I',
di dt
Comme,it. I n particular, the derivalivc ofa polynomial with complex cor:ffi.
cients is given by 1he same formula a s in the case ofreal c«fficicnts.
Proof. Thi, can be prov<d i11 exactly the same way OJ in the real <AK, but
we can al,o start from the real case. In ract, decomplexiryin,i C", -.e g<1
As oflen happens, 1he complex case is simpler 1han 1he real.-. The com
plex case is imponanl in its own righ1; moreover. investigation ofchc co m
plex case will help us in our study of1he real case.
19.1. Definitiona, Lei A: C" - C' bea C-linearoperator. By a lincarcqua
cion with pha.se space C" is meant an equation
t • Az, z e C". (I)
The full description or (I) is "a system o r homogenrou, lineM diffttcntial
equations oft he first order with constant complex cocfficicnu."
By a solution q, of equation (I) satisrying the initial condition -,(10) = z0,
,0 e R, z0 e C" is meant a mapping q,: I - C" of an interval ofthe real I-axis
into C" such that 10 e I, ,p(t0) = z0 and
for every t e I . In other words, a mapping fP: / - C" is said to� a solution
of (I) if af1er decomplexifying the space C" and the operator A, the mapping
q, i s a solution of the following equation with a 2n..dimcns:ion.al real phaK
space:
19.2. The basic theorem. The following theorem is proved in exactly the
same way as in the real case (sec Theorems 15.2 and 15.3):
THEOREM. Tht solution ,p oftquation (I) satisfying th, initialcoMiJjq,, ,p(O) - z.
is givtn by lht formula q,(t) = t''z0• A1orto1'<r, <ll<TJI o�•P,,,'1Jlldu gTOII/>
{g',1 e R) ofC-lintar transformations oftht spact C" isofth,form
g' a ,..c,
wht,e A : C" ➔ C' i1 a Clinear o/)<ralor.
Our goal is now to investigate and explicitly calculate ,,A.
Sec. 19 Linear Equation, with a C',omplcx Phair S 11acc 125
19.3. The dla1onal ca1e. l,et ,I: c• - C" be aC-linear opc,rator, and con•
)Ider the rharac1cri s1ic- tqu:uion
dr,t (A - AF.) • 0. (2)
TIIKOMtM, Ift/11 n roots A A. 'Iftquation (2) art di11inct, INn C" tl«t,,.,_stS
1, • • • ,
i111011,lirtrl rumC" • c: + ··· + C� nfont-dimtn1ional111b1p,,tt1C:. ,C�
;,warianl untie, A and t'", whtrt in tath one--dimtruional incarioJ trdl/JO<t, J.ll'f
C1, ,,,. rttlutts to mulripHcation by till tompl,x numbt'r t1•'.
Proof. The operator A has n linearly indepc,ndent ci�nlines:t
C" • c: + .. · + C!.
The operator A acts like multiplication b yl, on the lineC), and hcntt the
operator' e'A ac1s like muhiplication bye"''. I
We now consider 1hc one-dimensional case (n • l) i n mon dc1ail.
19.4. Example. Consider the linear equation
:i = Az, z E C, ). e C, I E R, (3)
with the complex line as its phase space. As '-"'c alreadyknow, dx solution of
(3) is just
t t !
t This is ,hr-onl>• plac-e whc:tt th<: ("Ompkx cas,e differs from thf' r<-a.1 CbC. Tht g:ra.ttt
,omplcxilyoflhc r eal case-is due IO the fact lb.at tht-fie-Id R is not .a.JgcbniiaD)·doM'd.
126 Chap. 3 Linear Sys1emt
,,
(4)
The 1ransforma1ion g' of 1he phase How of equation (3) is then an ,''.fold
expansion together with a simuhancous rotation through the angle cot.
\/\ 1e now consider the phase curves in the gc:ncral case. fo,-example, sup
r = �', k = afw
or
I
0• In r .
k
A curve of 1his kind is called a logarithmic ,piral. The phase eun-.s are a lso
logarithmic spirals for other combi nalions of the signs of aand w(rigs. 116,
I I 7). In every case (cxcep1 l = OJ, 1hc point z = 0 is the unique fixed point
of 1he phase Aow (and the unique singular point of the eon-csponding equa-
Sec, 19 Linrar Equation, with a Complr• Pho;, Sparr 127
0
Fig, 118 A c:cutcr,
t t
tion (3) ofthe vector field). This singular point is called af,a,s (we assume
that a 'F 0, w 'F 0). If c, < 0, 1hen ,p(t) - 0 as I - + a:>, and 1he focus is
said 10 be stable, while ifa > 0, 1he focus is said 10 be 11nstal>u. If" = 0,
w #; 0, the phase curves arc circles with the singular point as thdr tmla
(fig. 118).
Choosingthccoordinatez;;; x + iyinC1 ,wcnow invcstigatcthccbange
of 1hc real and imaginary parts x(t) andJ(t) a s the phase point moves. I t
follows from (4) 1ha1
da,npcd o,cilla1io11, if« < O. The changr of x or J wilh 1imr can al,o be,
wrluc:11 in the form
A,'' co, wt + 8<'' sin wt,
whcl'e 1he consrn111s A and Bare determined by 1he initial conditions
R,mark I. By studying equation (3) in 1hls way, we have sunuhan<:oU>ly
invcscigaled all one-parame1er groups of C.lincar transformatiom of the
complex line.
Rtmark 2 . At the same lime, we have investigated the system
{ .t •
a.x - C'JV',
j•wx+ay
orlinear equations in the real plane obtained by dccomplc><ifying cq�tion
(3).
Theorems 19.2 and 19.3, 1oge1her with ,he above calcub,tions, immedi
ately imply an explicit formula for the solutions of equation (I).
19.5. Corollary. Suppost then root, l,, . . . , J.. of1/tl ch4ract,ruliL <q,u,tio• (2)
art distintt. Tl1tn ewr., solution rp oftquation (I) is ofthtform
..
•
(1(1) = :E, ,,.,"�·�., (5)
where the (. are conslanl otclcrs independent oflltt initial conditioru IVIIII 1/w ,.. art
compltx con.rt ants depending on the initial condition.1.Fort.vtrytlt.oiuef
UtlJICM.llaJtU,
formula (5) gives a solution ofequation (I).
Proof. We need only expand the initial condition with rcsixct to the dg,,n
basis:
rp(O) - C' {' + . . . + ,.{.. I
Hz1 , • • • , z,. is a linear sys1cm of coordinates in C:, then 1M real pare x,
and the imaginary par171 of every component of the solution '1'(1) changes
with time like a linear combination of the functions �•cos°''-' and
., Sin
,. . Wi1 J.C.,
•
x, = L ,.,�·�'(cos Wi + o.,)
k=I
'
= I:= I A,1:�111 cos w,,t + B ,e-•• sin w 1,
4'
1c 1 (6)
Sec. 20 C'.om1>lexlfieo1ion of n Real Linear Equation 129
rp(t)
Fig. 120 Corn.plcx oonjugale �utions.
130 Chap, 3 t.i�ar S) 1em1
fig. 121 A M>l ullou with " f(:o1I ioltlal condidon unnot ha.w compla nlua..
Proef. Since
CA(x + iy) = Ax + iAy,
the dccomplexificd equation (2) decomposes in1oa direct product
{ x • Ax, x eR.11,
y = Ay, ye R". I
le is clear from Lemmas I and 2 chat from a knowledge of the complex
soh11ions of equation (2) we car\ fi11d the real .solution.s of equation {I , and
conversely. In particular,jormu/a (6) of S«, /9.5 gi..., tht txplintf.,,,. of t J,,
solution in the cast where the clioracteri"stic equation It.as no multipU r#ls.
20.2. lnvariant subspaces ofa real o�rator. LN A: R• - R• bca real
linear opera1or, and lei ,! be one of the roocs (in general complex) of the
charac1eristic equation dct(A - ,!£) = 0 .
=
LEMMA 3. If { e C" CR• is an e(�envtrtor ofthe op,rator CA uidt �• i�
then ( is an eigenvector with ei_�envalue }., 1\1ore01¥1, ). and 1 hare lite ume ,ulti.
pli,ity.
Sr.c. 20 (<>m1}lcxifico1ion of a Real Linear Equa1ion 131
• •
-t�=ri-:---
• V
Fig. 123 The rul llart of an cigcnvcnor IKI� to an inv.atUo, real p&.a.,w .
Proof Sin,·e lli! • 0A, 1he equa1ion CA{ • l( i1 cquivalen1 10 CAt • J.t
and the characlcri1tic equation has real coefficients. I
Suppose now that 1he eigenvalues ).1, • • • , A. e= C of chc opcr.uor
A: R' - R" arc dis1inc1 (Fig. 122). Among 1hcsc eigenvalues we ha,·e a
certain number v of real eigenvalues and a certain numberµ of complex
conjugate pairs (where\'+ 21, = n,so 1hat thc parityofthcnumberofrieal
eigenvalues equal, 1hc pari1y of n).
THEOREM. The spae, R" dtt-ompos,s into a dirccl sum ofvonc-dimouiMIII sdspaus
x = 2I(�+ {)eR",
Being C-linear combinations of the vectors c! and t the vecton :sandy
belong 10 1he in1e,scc1ion C' I"\ R". The vectors x and y arc C-lincarly
13i Chap.3 Linear Sy>1<n>t
rig.
e• l24 The af11n e imagr o(a Fig. 12.._ An dfipciul rouuoa.
Joga1i 1hmic iipil'.-1.
«<0
be any llncnr equaoion In 1hr 1>lnne, and suppose 1hr roou l1, l, o( ohc
characu:rislic cquntion arr diitinc\, H thr rootJ arr �al and), < AJ, ,h�
«1ua,ion dccomposrs into 1wo one•dirncnsional equiuion1 and \\cgc1 oMor
1hc cn,c, already soudicd in (;hap. I (Fig,. 126, 127, 128).
llerc we 0111i1 ohc boundary cases where l 1 or l, equal, 0. Thew ca.n are
or much lctiJ in1crc:11 1 1incc thty nrc rartly tncountC'rtd and ar'C' no, prc-•
served under an arbitrarily small perturbation; 1hry can be investigated
wlth no difficulty whauocvcr.
Ir ohe roots arc complex, so 1ha1 .l,., • a ± iw, 1hen, depending on 1hc
sign or a, we geo one orohe cases shown in Figs. 129, 130, 131. The ca,e of a
ccncer is exceptional, but is encountered, for rxamplr, in coRKrv.uivr s:ys
ocms (see Sec. 12). The case of rnuhiple rooo, is also exceptional. As an
exercise, ohe readcr should verify 1ha1 Ihe case shown i n Fig. 126 corre,ponds
10 a Jordan block with ,l 1 • .l2 < 0 (a "degenerate node").
20.S. Example: The pendulum with fricdon. We now applyeverything
said 10 the equation
ii= -x - kx
of small oscillaoions of a pendulum with friction (k is the coefficient of fric
tion). The equivalent system
X1 sx 2 ,
{
X1 = -xi - kx 1,
has the matrix
:r
rt! -ft O<l«Z
'
0 �Jf<2
re·lt
1c�2
'
:r
' --1.'-
r
F ig. 134 PhaS(: pJa, ,e of 1hc pendulum with small friction. After a anaiJ'I number ol
re\'olutions, the pendulum begins co swillg near LIM: Jo.,-cr equilibrium position..
Sec. 20 Complcxification or a Keal Linear Equation 137
as 1 -+ oo (Fig. 132). Explicit formula, for the change orx, • xwith time
can i;c ob1nincd fr<'>m Corollary 2 of Sec. 20.3 and the formula, ofS«. 19.4.
Thu,
x(I) • rt'' co,(w1 - 0) • A," en, wl + Bt'' sin wt,
where ohc cocfficicnis r and O (or A and 8) can be determined from the ini
tial conditions.
Hence the oscillations or the pendulum arc damped, with ,ariabk am
plitude re" and period 2,r/w. The larger the coefficient of friction, the
J1 -�
faster the amplioude decreascs.t The frequency
w p
f Nevertheless, the pend ulum still m.t kcs. inlinitdy many swings for a.t1:y va)QC .t < 2.. lf
k > 2, however, the pendul um changa its direction o(morion no more dattoncc.
138 Chap. 3 Ll�ar Syst<rn,
For I his 10 hold. th,co!.l)ici•ntJ of""' r•tcto,s must b, ttal ••dIM1t ,ftt,,.,,tutM·
J11ga1, ,,«tors m1;JI be compt,x conji(g,,t,s, Nott rur1her that cht II compl-t'C con
stant� c 1 , , , • , , nrr uniqurly dtu:rrnincd by the solulion of chc «Jmplc'C
,.
·1•11go•e>i. r:111,y solu1ion of1/11 rcol •quot ion has a ••iv•• rrp,u,•101,.,. ,Jtlt,f,,,,.
' •••
,p(t) - I:I "•····�. +
k•
I:
II••• I
(3)
(for a fix•d cltoict of ,ig•nu«lors), whtr• th, •• are rtal constants Hi tlu ,, lffff/lWC
constants.
Formula (3) i, called 1hc gtntral solution of the equation. \\le can also writ<
..
(3) in 1hc form
.. ..
,p(I) • I; ••••••(, + 2 Re I; ,,,•"(,.
t• I I••+ I
1'o solve linear systems in practice, we can u.sc: the method of undc1t.r•
mined coefficients 10 look forsolutions in the form oflincarcombinatio.nsor
1hc runctions (4).
COROLLARY 2. Lt/ A be a real matrix with real <igerwalU<S ).• ttNI ,.,.,pwc ,iga,
values «t ± fr»t, all of which are simple. Thm tVt.,Y tltmtnl ofIN ,notrix il a ,,A
linear co mbination of th,fun<lions (4).
ProoJ. Every column of ,he matrix e'"' is made up of components of the
image of a basis vector under the action of ,he phase flo\,· of tM: system of
differential equa,ions wi1h matrix A. I
Sec. 21 Classilica1ion of Singular Points 139
As shown above. the general real linear sysu�m (wh� charac1cristic �ua-
1ion has no 111ulti1>le roots) reduces 10 a direct product ofOM-di,,,..mion.1,I
aod 1wo -dimensional systems. Since one-dimensional and t�"'O-dimtnsion.al
sys1cms have already been studied, we arc now in a position to invatigatc
multidimrn.sional systems.
0
f}�
0
2)------0-
0 0
J) J')
0
4) 4?
0 0
S) s?
F;g, 135 £igcovalucs ofa re-al opct1t0r A: R.3 - R.>. Nondcgctttt2tt cues.
7/r u>-4
0
6) $)•••
--t
Fig. 136 Some de generate cases.
140 Chap. S Linear Systems
Fig. 137 Phase apace ofa linear equat ion Fig. 138 The: CUC l, < l.a < 0 < 1.:
in the case A, < A, < A1 < 0 . The phase Contraction jn two dittet.ions and cqM.n
..
flow at a contraction in all three directions.. lion i n the third.
A, 0
l •
,
0
). ). I
F ig. 139 The case Re l i.3 < A., < 0: Fi,g. 140 The caJC )._, < Re 1 1__, < 0:
Contraction i n 1he direc:1fon or(1 a nd rota• Contraction i n the dlr«lionol(1 and rota �
ti on with faster contraction in chc plane or tion with slower contraaioa in tht:pb.oc ol
(1 and(1• (, anc:1(3•
f,
Fig, 141 The c,a,sc Rcl1•2 < 0 < 13: F',g. 142 Equiwlcrtl llc,,,o.
£xpansion in 1hc direct.ion of(1 and rota�
tion with contraction in the plane of (1
and (1 •
Sec, 21 CIMsiflcation or Singular l'ointJ 141
where the real part or one or the roots•• vanishes or equal s the real part of a
root not conjugate to•• (here we do not consider the case or multiple rootJ).
The investigation or the behavior or the phase curves i n each of these cases
presents no difficulty,
Bearing in mind that,,, approaches Oas I - + oo irRc 4 < O (the more
rapidly, the smallerRe 4), we get the ph.lsc curves
,p(t) - Re(,,,'•'{, + ,,r''{, + ,,,'•'C,l
shown in Figs. 1 3 7 1 - 41. Cases l ' ) 5 - ') arc obtained from cases 1 ) -5) by
changing the direction or the 1-a,<is, so that correspondingly we need only
rever1c the direction or all the arrows in Fig,.137-141 .
l"rllhl,m J . Draw phase curve, (or ca.,c, 6)-9) in fig. 136..
21.2. Linear, dlll'erentlable, and topological equlval-«. Each or
lhcsc clM!iAcations is based on some equivalence r�lation. There cxdt at
least three reasonable equivalence relations ror linear systems, corYCSpond
ing to algebraic, differentiable, and topologjcal mapping,.
Definition. Two phase Aows{f'}, (g'}: R" - R" arc said to be 'f'U«Jout ir
there exists a one-to-one mapping h: R" - R" carrying che flow(/'}into the
flow {g'} such chat h•J' • g' • h for every IE R (Fig.142). (\Ve then say
thac "chc flow {f')is transformed inco the Aow{g'} by the change ofcoordi
nates h.") Under these conditions, the Aows arc said to be
I} l.inearly equivalent if the mapping h: R" - R" in qucstK>n ;. a liNd,
automorphism i
2) Dijf,rentiablyequivaunl if the mapping h: R" - R" is a diffto-pms11t;
3) Topologi,a/lyequivaltnl ihhe mapping h: R" - R• is a ,.,,_,,.,p!ris,,,, i.e.,
if his onc•to--one and continuous in both directions.
Problem I. Prove chac linear equivalence implies differentiable cquivalcnoc,
while differentiable equivalence implies copologjcal equivalence.
Remark. Note chat the mapping h carries phase curves or the 8ow (/'} inio
phase curves of the flow (g'),
Problem 2 . Docs every linear aucomorphism h E CL(R") carrying phase
curves or chc flow {f'} into phase curves of the flow (g'} establish a linear
equivalence between the flows?
An.t.No.
Hint. Let n • l,f'x .- e'x, g'x = e2'x.
t The terms"conjugate" amt ..,imilar" arc sometimes used as synonyms b-..icquivdc:nt"
as defined he,re.
142 C:hap. 3 Unc-ar Systrmt
T11£0REM. Lei A, B: R" - R"' be li,tenr opaa1or1 all ofwltou ntt-•colus •rt
simpl,. Th,11 th, syst,rru
i • Ax,
5' - By,
are linearly equivaltr1I ifa11d only ifth, ,ig,ntvzlu,s of th, operators A ad 8 ,,.,.,,k,
Proof A necessary and sufficient condition ror linear t:quiv-akntt of linca.r
systems is that 8 = hAh -1 for some h E GL(ll"), since Ai • hAx = y-
hAh- y (Fig. 143). But the eigenvalues of the oprrators A and AAA- 1 coin•
1
cidc (here simplici ty of the eigenvalues is unimportant),
Converscly 1 suppose the eigenvalues or A arc simple and coincide ";th
the eigenvalues o f 8. Then, according 10 &c. 20, A and 8 dtt0mpo5<" into
direct products of identical (linearly equivalent) one..,dimcnsioml and two-
dimensional systems. Therefore A and 8 arr linearly cquivalcnL I
Probl,m I .Sho w lhat the sy.ucms
.t, = .... + ..,.
I l: =X 1
arc not linearly equival e111, cvel'I 1hough their eigen"'"21ua coincide.
l•
t
Fi g. 143 Linearly equiv alent systems.
Src. 22 Topological C:lauification or Singular Points 143
all ofwhose eigenvalues have nonzero real parts. Let m _ denote the number
ofeigenvalues wi1h a negative real para and m ,. the number 1A;th a positivie
real part, so that m _ + m • = n.
For exa1111>lc, 1his 1hcor<m assens 1hau1able nodes and foci (Fig. I«) are
1opologically cquivalcn1 10 each 01hcr (m_ = 2) but not topologically
equivalenl to a saddle point (m_ = m + = 2).
Jus1 like 1hc index of iner1ia of a nondegeneratc quadratic fonn, the
,,umber m_ (or m •) is the unique topoJogicaJ invaria.nt of a lin�ar S)"St.tm.
Remark. A similar rcsuh holds locally (in a ndghborhood ofa fiMd poin1) lor
nonli,1ear systems whose linc-ar part.s havr no purely imaginary eigenvalues.
In panicular, in a neighborhood ofa fixed poin1 such a system iJ iopologi
cally cquivalcn1 10 ils linear pan (Fig. 145). Herc we will not go into t!K
proof of this proposition, of great importan� in the stud)• of nonlinta.r
systems.
22.2. Reduction to th" cas" m _ = 0. The topological cqui,'lllcncc of
linear systems with identical values o fm_ and m + is a conseqllfflCC of the
following three lemmas:
t-£MMA I. Direet prodru:ts oftopologu:ally equwalou 17sunu are wpohgiull:,• fllU'<l ·
lent. More ,xaetly, ift!IL sysum, sj)«ifad by th, opnawn
A J ' Bl ·• R"'• ➔ R••'
are carried into each ollur by the homtomo,phisms
hi : R"' • - R"'•,
tlun there exists a homeomorphism
S,c. 22 1'01,ological Clauificacion or Singulilr Poincs 14S
* I • 8 1X 11
Proof. Simply Ice
h(x., x2) � (h,(x 1), h 2 (x2)). I
The ntxl lemma i.s familiar from a course i n linea.r algebra.
LEMMA 2. If th, operator A; R• - R' has NJ puuly imagina1,1 ,.,,,.,.1.,_,, 1Jo,,, IN
space R" decomposes inlo 1M dirttl sum R• - R•- + R•· oftw.n,6sf>oUS i a
vnrinnl under /he opera/or A, such that all th, rigmr,altlil ofIM ustri<tio,,efA I• R•·
hav, negnti,� real parts and all th; eigenualun of tit, ,at,i(lion efA u R•· l,,,c,
positive real parts ( Fig. 146).
Proof. This rollows, for example, from the ch,orem on the Jonan normal
form. I
Lemmas I and 2 reduce che proororcopological equivalence to the l'ollow
ir\g special case:
LEMMA 3. LetA: R" - R" be a lintar operator all ofwltostrigmrohta Mttponli«
rtal parts. Then the 1ys1,m
i = Ax, XE R"
Rm.
---Rm.
-1-- -
-
Fig. 146 Invariant subspaces of an operator with no purdy imaginaryciguavalucs..
146 Chap. 3 Llnnr Systtms
"'f/
~ ----•---
J, �i g. 147
/!
All uni1able nodes are lopolQSieally tqu.inl(:nt.
Fig. 150 POOtivc dcf1n11cncss of the form (4) in 1he U.'¢ " • I.
2 Re (Az, z) a 2
.. •
L, Re l,lz,1 2 .
But all the real parts or the eigenvalues _., arc positive, by hypo,hcsis, and
(4)
f, c•
F ig , I.SI Comtruc1ion ofa basis iJl which th('matrnc ol'1he opuuo, K tnangul.a,.
t)
with all ,i,m,r1tsabo,,, th, mai11 diagonal of modulus /us Wn t:
a(s)
-
ollrtr wor,4, ifaform
= L a1,x._x1
A,I• J
S,•c. 22 Topological Clas;ificacion ofSingular Poinu 149
is positi111 d,jinit,, th,n th,r, ,xists ant > 0 Sll(h th,,/ l<'<'Jf••mo(x) + 6(x) u_/vr,
lb.,! < t (for all k, I, I < k, I < rn) is also positiw dtfo,it,.
Proof The form n(x) is t>OSitivr at all poi nu oft hr unit sphrrc-
..' x:
L - I,
on the splwre. Tlwrefore the form a(x) + b(x) is posi<ivr on 1h<, sph<,,... if
c < «/m , a,,d hr,,ce i.s positive- dthnitc. I
2
Remark. Our argument also implif'S that rvery positive dcfi.nit� quadnuic
form a(x) satisfies the inequality
al•! 1 .;; a(x) ,,; Plxl', (5)
cvcryw here.
Problem I. Prove that the set ofnondegcncrate quadratic forms with a gi"·en
sigoature i:( open.
Exampk I. The space of quadratic forms ax'+ 2bx,1 + 91 i n twovariabks
is a three-dimensional space with coordinares a, b, < (fig. 152). Tll<' case
b2 • aesepara1es this space into three o�n parts accordingtot�signatu�.
22.5. ProofofTheorem 22.3. Consider th<' derivative along th<, dir<etion
of 1he vec1or field RAz of the sum of th<' squares of the moduli of 11,,, co
ordinates in the "e•almost proper" baslS chosen in Lemma 4 . Acconiing ro
(3), this derivative is a quadra1ic form in th<' real and imaginary paresofth<,
coordina1cs '• = x, + &,,. Separating th<' terms in (3) corresponding 10 1h<,
+
+
_,
a
rlemenll o( tlu- matrix (A) on the main diagonal from thost 00rrnpondong
10 1h,· ,lt'nwnu o( (A) above the main diagonal, wr grt
L�_..,1 • p + Q.,
wht• rr
Q= 2 Re L o.,z,z
,., 1
•
Since the diagonal tlemtnts o( 1hr triangular matrix (A) arr jun 1hr ngrn•
..
values 1, o(thc operator A, Int quodrolicform
•
P - 2 Re L, 1,(xl + J'l)
in Int oariables x,, y, is p,,sitiv, definite and ind,pmd,nl ofIN ct.«, efksu.t I1
fol lows from Lemma 5 that for sufficiently small t, the (orm P + Q.(which
is close to P ) is also positivr definite. In fact, for sufficiently small c, tht
corfficients of 1hr variables x,,Y, in 1hr form Q. b,,come arbitrarily small
(since l•.,I <c fork</). But this implies (2) and hence (1). I
Rtmatk. Since L,."r 2 is a positive definite quadratic form, it satisfies a n
inequality of the type (5):
Hint. Rcpre5cn l the fom, as the inlqral or its dcriVllltiff along the phut cvn--a.
where,, is the Lyapunov function ofS..C. 22.3, and let/r i,., such that
i) The points ofS arc inval'ianl under h;
ii) If x0 is a point of S, then Ir carries thc- point f'"o of thc- p� tnajc-ctory
of the equation x • Ax into the point g'x0 of the pha,c tnaj<etory of the
equation• • x (Fig. 153):
h(f'x.0) = g'x0 V I E R, x0 E S,
{ (6)
/r(O) = 0.
We must now verify the following facts, whose proofs arc almost obvious:
I) Formula (6) uniquely defines the value of/r at every point x ER";
2) The mapping h: R" - R" is one-to-one and continuous i n both dir«
tions;
3) h •f' = g' • Jr.
22.7, Proof of Lem.ma 3 . fin.t we prove
Ul ,MMA 6. Ltt •= R!' - k" lu on.1 soluti01t ofIN 1ptt.OOII Ai = _. i"isliM.tfi- �. -'J,,ra
th, r,al fundi<»t
p(t) - In,'(•(<))
efth, ru,I ,;oriol,/1 t. Thm th, mltf!Pinzp: R - R is • t6ff� tutd
Moreover
¥
,
111t ldie1 the c,1lru!Hfl (7) bttau'le o((:;'), I
COMOll,Al4Y I, ,.�,.,
.. polNI • 'I O '"" 61 U/JffJtnttl Ut dvfeM
1 (ll
• •/ •0,
clwt
1 • o •$,I• Rand (F) istAt pJ,a.>#Jlow of tJv ,._,- j_ As.
P,oef_ Cotui der the solution .,. wi1h initial condioon .,(O) •· By l.mlm.a6. ,'(•<d) I
forsomct, The point•G •<r) bcio.1g, to lhtsp�S. Se-tting/ -r•.-cCd, ■ -r-.. I
COkOU,MtY 2• .,.,., ,,,,,,j,,
/{lliofT
1 8
( ) is ""i9w.
Proof, The phase Cu I've lc--.-vin,g • ( Fig. 153) is un.iqu<' and intenttb t� sphttc-tn a •nck
poin t ■0, by Lem nm G . The uniquc nCJs oft foUows f,om the- monoc0niclty of,<1). acun
by L.emma G . I
·
nus w c have cons1ruc1ed a onc,to,.one mappin,g of the d 1rce1 product of thrt-lent
the sphere on10 t:uc- lid(".&n apace min us a 1in gle p0in1:
F: R )( s-- I - a�,o. F(t, •ol -/'•o•
Jc follows from the th�cm on the d ep cn dcncir- of tbt so,luuon on tht uutQ..f condHions
tha t F, as ...,,en all th e in verse mapping F- 1, lS c on tinuous (and C\'("n a di.fl"eomorphism).
We now no ie �ha l
t-2
f or the sta nda rd equat ion X • a.. Hence the rN.pping
C( t , "o) - , . ,._
lS 11ls o one--to·or-.e a nd contin uous in bo1h dirtttions.. According 10 lhc definition (6).
the mappingh coi ncides with the mappin g G •F - 1: R•,o-R•,o n-p---y--iattr accpt
a t thc p0 n t O. Thus we have proved that It.: R• - R• is .a ()O('oto-one rm.pping.
i
The continuity oth an d It-' e.. , erywhere C":itCe-pl at the" poi n 1 0 fOilto,,..., from theeonlinuity
or F , F-' an d C, c-' ; actually It i.s a diffcomorph.ism C\'ctyWhcr-c except at the polfn O
( f 'ig. 154). The continuity ofIt and Jr-1 a t O follows trom Lffluna 6 . nu. km.ma nscn
a llows us t o obta in a n expl cit estimat e for r1{ -'(•)) i n tcnns of, 1(• ). <: I:
i
(,'(s))"• <; ,2 (h(s)) <; ,
( 2(s))'"·
In fact, le t•= F(I,ao). t ( 0 . Then /JI< ln, 1(a) < ol and ln,1.
( (\ a)) = 2'� M�"ff,
for x :;, 0 we: hav e: • ,.Pao, and hcn ee
(h•f'J(s) •h(f'(./"'o)) •h(f"'><o) • t""'o
( ' •h)(s),
• g'(t"'o) • t'(h(s)) • g
(Fig. 155). Hence two ,uch systems with identical numb,;rs •-and•• are
topologically equivalent (to each other). Note that 1he subspatts Jt•· and
R "' • are invariant under the phase flow {g"}. Ast inctta.sn-. C\;cry point of
Jt•- approaches 0,
Problem I. Prove 1ha1 g's - 0 as t - + 00 if and only ifs ea•-.
Therefore Jt•· is called the incoming strand ofthe saddle. In just the same
way, a•· is called the outgoing strand, defined by the conditjoo .(s - 0 a s
l � - CO.
We now prove the second part ofThcorem 22.1, namely, i-"1pol.�J
equiualtnl systems have the same numbtr ofeigmva.lu.t.s u,·ithn.tgatiw u.al/JdnS· This
number isjust the dimension m_ ofthe incoming strand.and hen�wt nttd
only show that tht dimensionsof tn, incoming strands oftwo toj!ol•tialiJ,quiMuxt
saddlts art idtnti,al.
First we note that every homeomorphism h carrying the phase flow ofone
saddle into the phase Aow of another must cany the incomjng strand of one
saddle into the incoming strand ofthe other (since approach toOast - + 00
is preserved under a homeomorphism). Hence the homeomorphism /t abo
establishes a homeomorphic mapping of the incoming strand ofone saddle
onto the incoming strand of the other.
151 Chap. 3 Lin�ar Sy,te""
Thr fact that the strands have thr s: .unr dimension now follows from ch�
following key proposition of topology:
Tht dime11sio11 of the spnt,t R• is a topolo1ical inoariant. i,,.1 , 11 •IN1NlliNJllt.is•
h: R.. ➔ R" ca11 exist only btlwe.en spaeesoflhe same dimtn.sion.f
Although this proposition seems "obvious/• its proof is not as)' and will
not be given here.
P,oblun 2 . Pro� that the 4 nddlcs with a thrtt--dimcmi«w phur spa« such da.tt
(m_, m.) � (3, 0), (2, I), (I, 2), (0, 3) >o'C 1opologfllly noncqui..knl <wi1hou1 u,i,-,g<he
unpr-ove<, topological proposition).
Hinl. I\ one-d imensional atra.nd corui..st,5 of three pha.,,e curves, ""+.ilc a IDW.c:icli:rnrmioM.1
strand consists or infinitely many pha.s(" curves (Fig, 156).
Thus for R1 , R2 , Rl we have complei<ly proved 1he topological dassi6-
c ation of linear systcm.s whose eigenvalues ha� nonzrro real parts_ How
ever for R"', n > 3 we are compt:Jled to rdcr to thc aboVC": unpro..-ed proposi
tion on the topological invariance of dim("nsion. I
Prolll-tm 3. Carr y out the copo lo gieal clauifica1i0n oflinear opcr.uon A: R --- -R• with no
cige1w3lucs of modulus I. Show that the unique topological invariant is W number al
eigenvalues of moduJus less than I.
l -- " II
...... \ - '
t -- \ 1-: \
_ ,, I
I
Fig. 157 Oo 1hc phuc : curvesstaninginasuffic.c'n:tlysma.Uneighborhoodofancquiltl:ri
-
uin posilion stay near the cqui1ibrium posi1 ion?
, t
Fig. 158 Difference in behavior or integral tun-u ror i:tab� and unisuhlie �dibriwn
pos.i1ions.
156 Chap. 3 Lin.-ar Sys1rm1
vtO/ t
0
Proldt m 2 , Pro ve 1ha1 the abo..,e definition iscorrect. i-c:., that 1he s1abil1iry o( the: irquiHb
r ium posili on is independent o( the 1ystcrn of cO<>rd,�ta figu ring in tbt ckfiniOOII.
Probl,,,, 3 . Suppose it i .s known that for any N > o. c > 0, thctt Wm a dution • ol
cqualion (I) such that 1•(0)1 <•and 1•(1)1 > Nl•(OJI for som< I> 0 . Do<S llus imply
that the equilibrium pos.i1ion • -0 is unstable?
23.2. Asymptotic stability.
•
••
•
1 IIEOREM. Suppose a/11/i, ,ig,r1va/u,s J., of1/o, op,,alor A /,e ,n 1/t, /ifl Wf-JJJ••
Re,!< 0 (Fig. 161). Thm lht equilibrium po1i11on" • 0 of ,p,,,.•
(I u
aS)•mptulira/{v Slab/,.
Probltm I. Ghc an t'Xomplt' of an unstable- <-qui:iibrium pos.itiou (in thit sn,s,tolLyapu"'°'
oft"quat ion (1) for '-''hich a ll Re l4 .;; 0.
Commtnl. It c.m b\" shown 1ha1 if the rt'al p a r •· of ,1.1 la.st ont c:ig,enoJu,,- ti ,-,sil:itt, Ihm
the equilibri um pOiit ion is u1ts1abk. II\ 1ht' raSt" ol�NO re-al paru,,, thir subllit)' ckpu..ds
o n terms of ,he Taylor SC"rin of ordt'r hi gher ,ha.n 1.
Proof. Clearly
But
so that the second term is much smaller than the first if, iJ small. In fact,
r
l.J• f•• I -
iJJ u1
0X I
for any firld u and any function/, where i n out case u a v ,/ zs ,2, •• as
1
0(, 1) and iJ.fliJx, - 0(,) (why'), which implies (4).
Thus there exist constanu C > 0, a > 0 such that
for all x with l•I < 11 . The right-hand side is no larger than .,,• lor ,uJ!i.
cicntly small !xi, and henc,,
p(I} = In r 1
(,p(t)), I ;;> 0.
By the uniqueness theorem r1 (,p(t)) ,;, 0, so that the function ,pis defined
and differentiable. According to the inequality (3), we ha,·e
• I -rd 2 L.,,2
p = r'ot;pdl • 'P =
r
--,-
� -y .
Ans. We did not prove thal the solution ,p can be extend<-d indefinitely
fon.,,ard.
S,c. 23 Stability or Equilibrium PositionJ IY.l
d
Yo� ,••
f
I
I•
•
Compt,1io11 ef/ht proof. Let a > 0 Ix such that the in,quality (3) holds ror
lxl < a, and consider the compact set
F = (x, I: r2 (x) ,s;; a, ltl ,,; T)
in extended phase space (Fig. 163), L<t 'I' Ix the solution with initial C'.Otldi
tion ,p(O), wl,cn- r 2( 'l'(O)) < a. By the extension theorem, we can extend .,
forward up 10 the boundary or the cylinder F. But the deri,-.ui,..- of the
function r 2,p(1) is negative as long as the point (1, .,(1)) belongs toF. Thcr�
fore the solution cannot leave the lateral surface or the cylinder F (where
,, • a2), and hence it can Ix extended up to the end race or 11M, cylinder
(where I = T). Therefore, since Tis arbitrary (and independent of a), the
solution 'I' can be extended indefinitely forward. Moreover. ,'(9'(1)) < tr1
and the inequality (3) holds ror all I ,i, 0 . I
Remark I. Actually we have proved more than the asymptotic ,rability or
the equilibrium position. In fact, it is clear from the in,quality (5) that the
convergence fP(l) - 0 is uniform (with respect t o initial condicions So
sufficiently close 10 0). tvloreover, (5) shows the rate or convergence (namcl)'
exponential),
In essence, Thcor,m 23.3 asserts that the uniform convergence 100ofthe
solutions or the linear equation (2) is not destroyed by a nonlinear �tur
bation v2 (x) • 0(1>< 2) of the right-hand side or the ,quation. A similar
1
assertion is valid for various perturbations or a more general natutt. for
example, we might consider a nonautonomou.s perturbation v2(x. I) such
that lv2 (x, 1)1 ,,; ip(lxl) where ,J,(lxl) = o(lxl) as x - 0.
l',00/t m 2 . Pro1,•c th a1 under- thit col'Kliti ons o( the lMOJ'cm, equations (I) a.ad fl) a.rit
lopol ogically equivalent in a ncighborhood or the cquilibrium position..
Remark 2 . Theorem 23.3 leads to the following algebraic problem, known
a s the Routh-Hurwilz probltm: Dtttrmint whttlrtr or not all tit, ZLTN ef• p,a
pof;-nomia/ Ii, in th, lift half-plan,. This problem can Ix solved by a finite
number o farithmetic operations on the cocffici�nts oft he pol)'llOmiaL 1nc
appropriate algorithms are described in courses on algebra (HUf°\'itz.'s
criterion, Sturm' s method) and complex analysis {the argument principle,
160 C:hap. 3 Lin,ar S)-.t•nu
24,1. Topological classlflcadon, Supp<>se all th• eig,nvalun l,. ... ,.t.,
of the linear cqualior\
X • Ax. x e R", A: R" - R11 (I)
are purely imaginary. Then under what conditions arc two such equations
topologically equivalent? The answer to this question is not known, and
evidently the problem cannot b e solved by presently availabk mat&mati
cal methods.
P,oblem J , Prove fhat in 1he casc ofthe plane (n • 2, l,.1 • f-irt> � O)• .tgdwaic�,--.
lc.ncc (i.e., cquali1y of cigc.nvalucs} i s a nttcssar')' and su.ffioc.m conditMIO for t.opologia.l
equ ivalcr,«.
24.2. Example. Consider the cqua1ion
*, = w1 x 1
*, =
,
-W 1X1 ,
A.'.2 - +iw 1,
(2)
xJ =
*· - ;.J,4 = +iw2
W2X,o
-WzX3
in R4• The space R4 decomposes into a direct sum R" = R 1•1 + R>.•· of
owo planes (Fig. 164), and correspondingly the system (2) decomposes into
two independent systems
t Sec e.g., A.G . Kurosh, A. UH/rst lit Hig.ht, Alg,lmi (in RussQn). MOIP)W (1968). Chap. 9;
M.A. Lavrcntcv and B . V. Sh aba1, A1ttlr6d1eftN n°'?-.JFrmc:ti4,u ffa CifCn Yaridlt f111
Rus..1ian), Moscow (1958), Chap. 5; N. C . Chcbotattv 2nd N. N. Mctma.n, 71w Jt..dt..
Hu1'U,Jl·t1, Prohltmfo, Poly,.,,,ia/J and &tirt Funtti..s (in Russian). Trvdy M.a.L lnu.. �.
MOS<OW (1949), No. XXVI.
Sec. 24 t:asc or Purdy Imagi nary £i g envalur< 161
A.._+--
= Cl) X ,
{ x,
1 z
*, - -w,x,. (3)
:: WzX 4,
(x3, x4) e R3_4.
{ -�, � -W X ,
.\'4 2 3
:rr
0
Fi g, 166 A map of the torus.
0,
-+--+-+--+
-�,
-+- +--+-+
- _,
0,
Fig. 167 A point winding around 1he torus.
Sec, 24 C:n<t of Purely Imaginary Eigenvalue• 163
0•1« 8-<¥
8.,,,.
8,kf • 8
0,11«
(J,4c& 8•Ja
F ig. 169 Imagcs ofa poi nt of the circle u nder rq,caced a.pplica.1.-0n of a rotation th.rough
che angle CJ.
r A sc.t A is $aid to be #ot.,yw&,# dmk in a space B if thcr-c is at least OtM": poitit o/A. .ift a.a
or
arbitrarily small ne ighborhood every point B . or
164
11ro11p, or 1hr linr (••• See. 10), but wr will prov• ii rrorn ,c�1ch, "an,ng
from 1hr ,implr combinatorial rac1 1h01 i/4 + I ob;«ts art pl««l 1• k ult,,
then 111 ltaJI one ttll tonlain1 ,rrort than one ob;ttl ("Dirirh1ct•i c-t-11 pnncipl,0
).
S11ppo8c wt divide 1hr cirdr into k rqual halr-oixn in1<rvab orlmgth 211 t
Then nrnong the Orn A + I poinc, or 1hr srqurnrt (�). 1htre are 1wo pomc.
in 1hr <nme halr-oprn interval. Lrt 1h,,e poinc, b, 0 + Jt7. and O + ,2
(p > q), and lei s • p - q . Then 1hc angle of rotation si differs rrom a
rnuhiplc of 2n by le,, than 2n/k, and any two consecutive poinis of 1he
sequence
0, 0 + sor, 0 + 2sor, 0 + 3sor, . . . (mod 2Jt) (6)
(Fig. 170) arc the same distance d apart, where d < 2Jt/A:. Hence any ,.
neighborhood of any point of S1 contains points of the sequence (6),
provided only that we choose k large enough to make 2Jt/k < , . I
Rtmork. We did r-.01 use the fact that i i1 incommensurable with 2•, but ii
is obvious that the lemma is false if� i.! commt:n.surablc with 2a.
Problem I. Find and eliminate 1he gap in 1hc proof of the theorem.
Proofofth, th,ortm. The solution of equation (4) is of the form
0,(1) � 01 (0) + w,t, 02 (1) = 02 (0) + w,1. (7)
Suppose w1 arld w 2 arc ta1ionally de�ndcnt, !oO that
are compatible, and their solution gives the period or 1he closed phase cun.'<'.
(7). On the other hand, suppose w, and w, arc rationally indq,mden,.
Then w1 /w2 is an irrational number. C',onsider the consccuti..-e points of
intersection o f the phase curve (7) with the meridian O, � 0 (mod 211). The
latitudes of ,he poin,s arc
O,. = 020 + 2n"' 2k (mod 2n)
w,
(Fig. 171). By the lemma, the set nf points or intersection is csuywhett
dense on the me.-.idian. But ifLis a s,raight line in the plane and if we draw
straight lines through a set of points everywhere dense in /. i n a din:ction
different from 1he direc1ion of L, rhen 1hc lines form a set which is �·Cf)··
where dense in the plane. It follows thar the imagct
t Herc (xJ dcnote"S tht: illkgral pa.rt ofx . i.�·.. chc largest integer <. x .
Sec. 24 C:nsc or Purrly Imaginary �:igenvalurs 16�
' ..
•
':,e,,s.,
•6•SOI
0
e,
0, .
trr•
� .,...
Zrtk 0
1
or the phase curve (7) on the square O ,;; iJ, < 2n, 0 ,;; O, < 2,. is every
where dense. Therefore the phase curve or equation (-l), and hence of
equation (2), is everywhere dense on the roru,. I
Th e foll owi ng probl em s gi ve a nu m ber of s impl e implications o(Theorcm. U outl.idc-
1hc 1h cory of or din ar)' diffcre:n1fal equation�.
Prohltm I . Co nsider 1hc sequence
I, 2. 41 8, I, 3, 6, I, 2, 5, I, 2, 4, 8, ...
of fint digits of c.:onsccuiivc powers of 2. Ooa a 7 C\.'tt appear in thil .seq,,ientt? �fc:i«:
generally, docs 2� begin wilh an arbitrary combination of digitJ?
PYoblrnt 2. Prove 1hat
s up (cos 1 •r sin v'2I) =2 .
0<1<•
Proble m 3. Find all elMcd subgroups of the group s• o f complex num ben of modQlus I•
.'Ins. I , sl, l ;rn.
24.4. The 01ultidimensional case. Suppose the eigenvalue, ofequation
(I) in R2• arc all simple, or the form
i. == +iw 1 , +iw1, ..• , ±iw•.
166 Chap. 3 L,.,..ar S)'lt�rm
I
fig. 172 A ph1uc curve of the 1y11cm d - I, , ✓2, I - ./! is cvff'f""'--hcr-C ck-Mt- on
1hc thrtc-dimrns ional tonu.
Then, arguing as in Sec. 24.2, we can show that the pha.Jc au,·es lie on the
m-dimensiooal torus
r· - S' x ··· x S' • ((0., ... , O.) mod 2n} ; R·,z·
and satisfy the equations
0I • WII {Jl • W 1, , • • ' ()• • OJ,..
P,,./,JH 2. find thl, llrnll 111\d 1how 1h11t h 1-. gttaltt than I.
Co"1111,,.,. Thf' inili11I ,rction of 1hr 1-N)Uf'nff' (�t. 2◄.3, Prob. I) ind ,a� 1M1 t�rt' .,..
rrwel' ,rvrrh. 'l'ht11 1- due 10 •ht r,c, that tht irn1ion.ail numbtr l<>s1 -2 0.,010 • "
\'try cl011.- 10 ,he r.11i on;il r'I\Hn�r 3/10.
The sohuion or
a linear cqua1ion with constant coefficinus f'rou«s to
calculation of the matrix t'A • ,..,,The explicit form ofr4' i.s gi\.--en in Seu.
19.5 and 20.Gfor chc ca,e where che eigenvalues orche macrix are all distincc.
We now use the Jordan normal form t o find t'" in the cast of multiple
eigenvalues.
was indicaced in Sec. 14.9 (ic will be recalled chac chc diffcrcntia1ion
operator in the space of quasi-polynomia1se1'P -c.(t) ha.s the mauix A in the
basis,,= 1',''/k!,O '- k < n). In far1,according10Tayloc'sfonnula,the
matrix 11' • ," i, 1hc matrix or
the ,hifc opcra,or /{1) ~f's + t) in 1hc
indica1ed hasis. Thus
.,,.,, (t + s)'
, = '('
i.., h"'(s)e,,
A.
I.I I
where the clements hu(s) or the matrix.H• arc found by using the binomial
theorem and turn out to be quas.i-po1ynomia1s in s wilh expo�nts .i. or
degree less 1han n.
Another way or calculating eA' is based on the following
LEMMA. If Jlte lint(lr opt1C1lor1 A, 8: R" - R• commut,, JtJ I.Ital .lfB -; BA,
then e" •8 = eAt8.
,•,• = (E + A + �: + · · ·)(E + 8 + :: + · · ·)
= E+ + l(A2 + 2AB + 8 1) + · · ·,
(A + 8)
,••• = E + {A + 8) + l(A + 8) 1 + · · ·
= E(A + 8) + l(A' + AB + BA + 81) +
168 C:hap. 3 Li��• Sy,1erm
The scric• coincide If All • IJA, ,Ince,••' • ,.,, for x,, • R. Bu, 1!,en
,r14111 • e14 1 •, since thr �rit� are absolutely cnnvtrgr:nt. I
Suppose we reprC:4'('111 A in 1hr form
A • it:+ 6,
where
0 I
0
6• I
0
,., = I I
I 1'/2
I
'
I
•''
I"-
, = (I)
,,,.,
,,,
Prmif. Since /l operate'> on the basis e 1, ••• , e,. like a shift O - ,., -
4
• • • _. e , t,. acts like a shiCl hy k places and has the matrix
t2 .-.
l
0 ...
11
, ' • E + lJ.t + ._ +
[\ ,, ' +
e:,:,-•,•-1
{l\" = 0).
2 (11 - I)!
•
,pj(t) = L e''' P11 (I).
I• I
where p1,, q1,, r1, are polynomials with rtal coeffieients of degrtt lt:SS 1han
,,,, ,,,, ,,, respectively. This representation follo,,-s from the faet that
Re u'' = Re ,"(x + (y)(eos wt + i sin wt) = t"'(x cos WI -J sin OJI)
if z = x + f.y,). = a + iw. �loreover, it is clear from these formulas that if
the real parts of all the eigenvalues are ncgativ� then all the solutions
170 Chap. S LiMar S)'Sl<mJ
apr,roach Oas t - + 00 (lls mu,o be 1hr cair, according 10 Sea. 72 and Z3).
25.3. Applica1lon1 co ay,cema ofhlgber-order equaclon._ \Yriting •
sy11ern of higher-order rqua1lo,u a, a l)'Slrm of fir11-ordcr �uation,, "'
reduce 1hr problem 10 the problem con,idcrcd above, which in turn can�
solved by reducing thr matrix to Jordan rorm. In pr.1c1for, ho\o\n<tt, ii is
of1cn more convcnicni 10 proceed dilfcren1ly. Flr11 of all, "' nocc tha1 the
eigenvalues of the cquivalcn1 first•ordcr sy11cm can Ix round without
wfl111"1g down the ma1dx of the system. In fac1, for rvery rigc:nvaluc A. \o\-C
have an eigenvector and hence a solucion fl'(I) - ,1'.,(0) or the equivalent
firs1-order system. Bui 1hcn ,he original sys1cm has a solution of the form
J/,(1) - e1'J/,(O). Thus, substilu1ing ,; - ,1'{ in10 1hc original S)>Stcm, w e
sec 1hai 1he sys1em has a (nonzero) $Olu1ion ofthe given form if and onlyif l
satisfies a certain algebraic equation, from which 1lx eigenvalues 11 can M
dc1crmined. We can 1hcn look for 1hc solutions 1hemsdvo in the fonn o(
sun1s or quasi•J>olynomials \<1i1h cxponcntS l1 and undctttmincd cor:ffi...
cicnts.
Exnmplt I, Lt1
..-4'"• - x . (2)
Sub.,tiru ting x - r''( in to (2), we gel ,t',-"'( - ,-A'(. l' • I. 11 ,2,.,.,.. = I, -I, i. -i..
Thus cvcty sol ucion of (2) is of the for-m
X - Cit'+ CJt-• + c_, co,'+ c.. sin,.
Exampl,- 2 . Let
I�•
:C1
=x1,
= -"•·
(3)
Substitu ting x - t...,{ in to (3), wegd 12{, = {2, A.2(J = (1• This J)'lJc.fflollinar cqua -
1io n• in(,, {J has a non1r-ivial s olution ir an<l only if,t• I. Hcnccnuy solutionol(S)
is of the form
... • Cie' + C2,-· + c_, cos I+ c.. sin ,.
1
P,Hlm, I. t1ind 1he Jord.an norml'I (Ofm or the fourth,.Ofdc:t m.a1r1x <'orrnpond1nc 10
cquMlon (4),
25,f, Tbe ca1e of a 1lngle equatloa of order •· In general, 1hc
m11hiplici1y of 1hc eigenvalues do,:s nol dcltrminc 1hc sizes of 1hc Jordan
blocks. The 1ir11aiion bc�omc1 1implcr if we arc dealing w11h 1hc hncar
opcra1or A corrc1ponding 10 a single di1Tercn1ial cquaiion of order'"
•• e C. (S)
Then Corollary 2 implies
COROLLARY 4, l:.vt'.)' solution oftquotion (5) is of tlteform
•
q.,(1) - !; ,'•'p,(t),
I• I
(6)
Proof. Equalion (5) has a solution of1hc form ,••e if and only if lis a roo1 of
cqua1ion (7). I
Turning to the equivalent system of first-order equations
0 I
0 I
X = Ax, A = (8)
a, • . • Q
I
we get
COROLLARY 5. If t/tt operator A: C•-+ C' /tas a matrix oft/teform (8), tit,,, to
tot')' ti,�erwalut). ofA /}i,r, corr,spontf., precisdyonejordan bt«k of
siu <'i""I I# IA<
multiplicity of,t
Proof. According 10 (6), there i s a single cigcndircction corresponding lO
every ,t In fac1, Jc, { be an cigcnvcc1or of the opcra1or A . Then ,he first
component ,'1e 0 of rhc vcc1or ,''{ is one of the solutioru of (6). Bui then 1hc
remaining components are derivatives: Ct = 11(0• Hence l uniquely
determines the direction of(. To complete the proof, we no,c that each
Jordan block has its own cigcndircction. I
P,obl,m I . Is every line:ir combina1ion ofquasi-polynomials (6) a .solutio,n cleqwition (S)?
172 Chap.3 Linear S)'llems
x, - t• J.'f>,(t1)
I• I
0 I x,.-t x...... t
A{.- I - 0
0
= = {•.
•• .. a, , a, x,. - 1 x.
It is important 10 note that the operator A does not depend on•· Hen« x.
i s one of the components ofthe vector A•(, where� is a constant vector and
the matrix ofA is or the form (5).VVc now apply Corollary 5, ttducing the
matrix or A to Jordan form. I
In making the cakulations, there is no need ci1hcr 10 write do-.,•n tM'
matrix or reduce it to normal form. In fact. any c.igc.nva)uc ofthc' operator
A corresponds 10 a solution or equation (9) orthe form x. = i.". Substituting
x, = )." into (9 ), we find that). satisfies the equation
;. • = a i•- + · · · + a.
1
1
1
••
(I + J2}" - �( 1�..,.✓2�)•
- 2 ✓2 2 ✓�
2
Comm,nt, As n - «>, the fin1 1enn i ncrc.a.ws nponentaally, whil e 1hit littOftd ttnn d,c,
creJuc- expo ncn1iall y. Thcre(ott
( + J2)"
•• .. I 2 ✓2
(or large n, and in p.artkuhar .-".,Ix.;::::::- I + J'l.. -rhft g:iYCS usvery good �tiom
IO J2:
J2 =,x.,., -1t.,,,
••
Choosing-x., - 0, I, 2, .5, 12, 29, , •. , we get
✓2 "' I IO - I' ✓2 "' 5
2
2
- 1.5,
✓._12-5 ✓2., 29 - 12
12 - 1.417 .•.
"- 5 - ,.
."T,
Thes,e arc 1he same appro:xima1 i ons used to c.akub.te ./2 in a.ncimt � a.ad can be
ob1aio«I by expa n di ng ./'1. in :1 con1inuou-1 fnu::1t0n. Mo�·tt (x._ 1 - x.) �. is (ht bat
or all roui onal approximation, 10 ,/'l with denomin aton no1 cxcttd.i ng .-_.
x = -grad U, {II)
1 The \'eCt()t field gtad U is ddined by the condition that dl/(() - (gad U. () b n-uy
v,cc-1or � e TR;, where (, • •) de:notn the Eudidnn st'2la.r product. ln (ortboocnn,,al)
rectangular cootdinat�. the fiel d grad U Nii components (U/2x,. ....iU i-z.,.
174 Chap. 3 Linc�r $)-stems
U(x)
.. ,
= 2 L a:x;,
I •
Fig, 173 U'vcl curve or 1hc pot('nlial c:n"rli(y and d irttlions of th� �ctffHhc OIC'il·
lat ions.
x, .z;
L ,,,•
•-I
/(1)
•=•
G ,,,
rorm (I). By 1he above 1hrorrrn, 1he dimension or X ,qualt •· Bui a linear
rr,apping of the s11acr C" onto a spacr X of thr samr dimrn.sion i1 an i,o..
morphl,111. Therrforr <I> rsrnblishrs an isomorphbm bc1wrrn C" and X I
26.3. lnvarlanco under 1hlru.
,·11&0REM, Tit, S/)11'< X ofso/utio111 of tit, dtjft,tnlial t9uatio• (2) is i11NnMI
u11dtr shifls <or')'ing thtfun<1io11 ,p(t) into ,p(I + s).
Proof. The sl1iCt of a solu1ion i s a solution. as in the c� of anyau1onomous
eq11a1ion (Sec. 10.1). 1
The followi ng arc all a:amplo of 1hi(1°invarian1 tubt5Ncn of,� 1pai« F:
l:.'xar,1p/t I. The onc•dimc:nsiooal ,pace fu"').
Exar,1/Jt 2 . The space orquasi•J>Olynornia.ls (,...,,0(1)) o(di�mion "·
Exar,1p/t 3 . The pl.\nc (r 1 cos wl + ,1 sin wt},
Exar11ft/t ii. The space (/t0(1) cot WI + q0(t) 1m wtl of dimension 2ft .
h can be shown 1hac every finitc-dirncn.sional shirt-invariant subspace or
1he space Fis the space or solu1ions or some differential equa1ion (2). In
other words, such a subspace always decomposes into a direct sum ofspaa:s
or quasi-polynomials. This explain, 1he significance of quasi-polynomials
in the theory of linear differc1uial equations with constant coefficients.
If an equalion i.s invariant under some group of transfonnations, then the
space of functions invariant under the group will play an important role in
solving the equation. This is how various spttial functions ariK: in mathe
ma1ics. For example, there is a connection bc1wee:n the group ofrocations
of the sphere and the fini1e--dimensional spaces of functions on 1he sphere
f1 sphcrical functions") which are invariant under rotations.
•Pro6t,m I. F ind all linitc•dimcnsional sut»paecs of lhc s�cc of$fflOOlda rWICbelnl on the:
drclc which arc invariant under rotations or the circle.
26.4. Historical remark. The theory of linear differential equations
with cons1an1 cocfficienis was creaied by Euler and Lagrange before the
discovery or 1hcJordan normal form of a matrix.They reasoned as follows:
Let .l1 and l.2 be two roo1s of 1he characteristic c.qua1ion. The solutions:
,,
,.1., and Ai corresponding 10 these roots span a two-dimtttSional plane
{c ,,••• + ,,,,�} in the space F (Fig, 175). Suppose the equation changes in
such a way 1ha1 11 approaches 11 • Then e11' approaches r'1' and th� plane
degenerates imo a line for J.1 = J. 1 • The question now arises of whc1hcr •he
plane has a limiting position as l1 - J..1• If J..1 � l.1� we can choose r'•'�
,A,, - e11' ra1hcr than e,1.1', eAJt as the basis. But
e-Ai' - ,,1.,, � (lz - l1 )t,1,r,
Sec. 26 More on Q, ,asi-Polynoonials 179
..,,
Fi&:, 175 Limiting JX>lition o( rhc plane' ,panned by ,A,' and,.¥ in tht-lf>KC'F.
and hence, as A. 2 -+A. 1 , the plane spanned by ,11' and ,1,t - ,,a,,, or tquiva•
lently the plane spanned by ,,,, and (,'•' - ,''')/(A2 - A,), goes in10 tl>c
limlcing plane spanned by e1•' and 1,.' • 11• Therefore it is natural to expect
1ha1 1he solulions of ohe limi1ing equa1ion (with 1he double root l, - 11)
\\'ill lie in the limiting plane {c 1 t111 + ,211'"'1'}, where chc fac1 1ha1 c,,.1., +
c2 tt111 is a solution of the orginal diffcttmial equation can be vcri6ed by
dircc1 subs1itucion. The same tcasoning cxpla.im the appcarantt of the
sohuions t'e" (k < v) in 1he case of a v -fold root.
The above argument can easily be made p,,rfcc1ly rigorom (for example,
with 1he help of the theorem on differcmiable dependence of tl>c solutions
on a parameter),
26.5. Nonhomogeneous equations. Given a linear operatorA: l1 -L2,.
by a salulion of the nonhomogeneous equation
Ax =f
with righ1-hand side/ is meant any prcimage x e L, of the clement/e l2
(Fig. I 76). Every solu1ion of 1he nonhomogencous equation is the sum of a
parlicularsolution x I and the general solution of the: homogmandequation
Ax= 0:
A -'J = x, + KerA.
The nonhomogeneous equation is solvable if and only if/belongs 10 tl>c
linear space Im A = A(L1) c L,.
In particular, consider 1hc differential equation
_.l•> + a,x<•- '' + · · · + a.x = f(t) (3)
(a nonltomogeneous linear equation ef otder n u.;t.Jr. constan.l c«.ffiaDW).
THEOREM. Iftloe riglot-lrand sidef(t) of equation (3) u a sum ofq,uujpolpnti,,b,
then so is every solution of equation (3).
Le,
Q.• = (e''P<.(t) )
180 Chap. 3 Linear Sy,ttrm
Ker A A"'f
Q r
,, 0 •-• IIT1A
1hn1 (/) - ).E)'Q.• • Q.M -•, llut the matrix or the operator D - )E ,nth<"
hMis
,.
e, • -,"'
AI '
i, n 11n ,,01cn1.Jordnn block, i.e. , D - )E acts on the ba,is like a shift.
rlcnce thcoperator (D - ).E)' acts like a shift by v places and map, Q."'onto
Q.•-•. I
COROLLARY 2. u/ ,( bt a r()O/ of mu/liplirilJ v of I� clwr«ttri1tit 19w;.,,
a(,!) • 0, nm/ /,ifE Q.' bt a quaJi-polynomial ofd,g,u ltsJ 1ha11 I:•"" t1tJ»,t<lft l.
The11 equatio11 (3) has a so/11tio11 <fJ e Q.' •• 1Lhith is a ifUOJi•polJ-i•I of«ptt /us
than k + \' tmd ex/xmt111 ).,
Proof. We need only set m • k + v in �mma 2. I
Proof of /1,e theorem. �. I: be the SCI or all p<miblc sums or quasi-poly
nomial�. 1�hen I is an infinitc•dimcnsional subspace of 1hc spa.a: F . By
Corollary 2, the image A(I:) or the operator
A = a(D): I: - I:
contains all quasi-polynomials. rlcnccA(I:) coincides with r., being a linear
space. Therefore equation (3) has a particular solution which is a sum of
quasi-polrnomials. 1, remains only 10 add the general solution or th<"
homogenc.·ous equalion, which, according to Sec. 25.4, is itself a sum of
quasi-polynomials. I
Remark /, If f = ,••p ..(t), then equation (3) haJ a pa,ticu/a, JMati,,,. of IN
form q, = ("i'tt<11(1). In fact, there exists a particular solution in 1he form of
a quasi-polynomial o f degree less than k + v . But the terms ordcgrtt less
than v satisfy the homogeneous equation (sec Sec. 25.4) and hence can be
dropped.
Remark 2 . Suppo,;e equation (3) is reaL Then we can look for a s olution in
the form of a real quasi-polynomial if). is real, and in the form
?'(P(l)cos wt + q(t)sin wt]
if). � a + h o . Here the solution ca1 '1 contain a sine function �·en in 1M
case where the right-hand side or (3) consists only ora cosine.
Prob/mt I. Find 1hc rorm of the particubr-solution o( each of the following <qua.tiom.:
a)X±: -r-t1 b)X;tx-,H; c)X.±x=k-'; d)X;tx-t�sint;
c)X ± x =,,,cost; fJ x J:; 2ix •,:,.,sin to d) ,.cw, + 4x = ,:t' cosL
;
182 Chap. 3 U�ar Systems
- r
1....1
.r
Fig. 177 Ao os(illa1ory 1y1tcm u ,,dcr 1hc act�n ofan c,111c-rnal lorcc/it) - COJ rl.
r ·I
,r
w \I w "
Fig. 178 The arn 1>itud c and ph.a.sc or forcro 0Killa1ions of a frkt onlcn ptndvlum AS
l i
a fun c1ion or thefre quenc yor the external for� .
According to (4), t he amplitude rand th<: phase, 0 hav,: th<: ,,,.Jucs shown i n
Fig. I 78. t The real part of<I> equals r cos(,·t + 0). Henc<: ch<: g,:n<:nl solu
tion of 1 he nonhomogeneous equation is of the form
x • r cos(vl + 0) + C,cas(wt + O,),
where C, and 0 1 are arbitrary conslants.
Thus the oscillations ofa p,11dulum u11d,r the action ,ifon ext,rulf•ra cPSi,t of
''forcedoscil/atio11s" r cos(vt + 0) with thefrcqw"')l,iftlreexternalfarcetn1d"ftu
osci//a1io,u'' with the naturalfrequen<y w. The dependence o(the ampHtudc, of
the forced oscillatior'ls on the frequency of the a.temal fortt has the
characteristic resonance shape: The nearer the frequency of the external
force to the natural frequency w, the more: the external focce "'rocks0 1JM,;
system. This phenomenon ofresonance, observed when the frequency ofthe
external force coincides with the natural frequency ofthe oscillatorysyste.m.,
is very important in the applications. for example, in all kinds ofcalcula
tions involvi,, g engineering structures, care must be taken to� that t�
natural frequencies or the strucLUre are not close to the frequencies oft�
external forces which will be experienced by the structure. O�-isc C'\·cn
a small force, acting over a long time interval, will be able to rock tM
structure and destroy it.
t Thcchoiec8 = -n (rather than +A") forv > wisjustincd by Eurnp&,e3bdow.
184 Chap. 3 Linear System,
,,
l t
,� 13, 170 1"he 1u1l\ or '"-O haHn0t1llt whh ni:ighbor1111 frfqUtt)('1f'I (bratt) and 1h l,m,1
in th� c:m� o( rnonan cc ("roding"),
Jw'
has roots
l 1 ,2= -7+in,
k
Q =
(Fig. 180). Suppose th< coefficient of friction k is posim·<e and SD1'1.IJ
(k' < 4w2), and let the external force be oscillatory:
f(I) = cos vi = Re ,'".
Sec. 26 More on Qu"'i-Polyno111inl• 18.S
IW
,n
. .,
A
-,n
-iw
Fig, 180 f.igcnvaluo of the �ua,ion of the 1>t00 ulum with rnc-1,on.
r -9
"I---:::---
w " w "
Fi8. 181 The• am 1:>litudr and r>h:uc ofromed ote.llario.u of a p«Klutum with (rictior\ a,
a ru,,c1io11 of the frequen<'y or •he excrroal ron-c.
C=��-- - (8)
w - v + ikv
Suppose we wri1e C io 1hc 1rigor1omc1ric form (5). Then, according to (7)�
the graphs of the amplitude r and phase O of the forced oscillations, as
functions of 1he freque,,cy v of 1he external force, have tM: appcarantt
shown in Fig. 181.
Adding the general solution C,,-"cos(Ot + O,) of th<: homogmeous
equation to tl1e particular solution, we get the general solution
A 0, we gtt
• Ax,(1) I A1x1 (1) I A1,,(1) I O(A'),
•o 1hu1
J • Al, I A'J, I A'J, I O(A'),
t At, I A'R, I A1 1, I O(A'),
.to x AM1 ._ A1x1 I Al(N1 i"f> • O(A•),
,. - -..- .
The cqu alion I - -si n x holdt for every A, and hence s1, .,.,, "' satitly thc cqu,ion:t
,, - -x,, (9)
The initial condit ion x(O) - A, 1(0) • 0 alto holds KW f'�Y A, and� tlw- <qm.ho•
(9) talltfy the following in itia l condhioru::
x 1 (0) • I, (10)
Solvi ng the:" first two equal.ion, (9) subject co lM conditiont (10), we Id-
T hus thc dfe<:t of th e n onli neari ty (si n x ,' x) on lhc oscilla tions ol cJac, pendulu m
rc<haca t to 1he presence of an extra term A.,x, + O(A4):
x • A rot t + A [a(cos I - cos 3t) + J1I si
1
n t] + O(A4),
The �riod Tor th e 05Cillati:on, is just the point at wht<:h x(t) ha, its �um. and i,
near 2it fo r small A. To find 1hii po nt, we we lhe eondition .i(T) = 0:
i
A{ -,;n T + A'l(P - a),;n T + 3a ,;n 3T + f/T"" TJ + O(A•)I • 0. (12)
To sol vt" (12) .approxiniately for small A, le t T 2• + •· This gi,n
• th,c, «-qua.Lion
,;n • = A1 [2•// + O(u)J + O(A')
ror 11. By 1hc irn pl icit function cheorem,
• = 2nPA' + O(A'),
1.e.,
T= 2•[ I + 1� + o(A')].
wh t"re o(A') ..,. O(A•) 11ince T(A) is even.
t Here it is meful to recall 1he bucket with 1he hok in its bouom (� tbtwarni"I in Stt..
9S). Frorn the pracncc o f the ".secula r'' tcr-m t s:in tin th< foTmul.a for ..,. we an draw no
co ncl usions whal50<':ver about the bt"ha,,ior o rth e pendulum .u t - oo, Out-�lion
is va lid o nl y for a finite 1imcinter-val, a nd the term O(A•) bccomcslargc(or�LT!wsou:r
,o lution ofthc equat ion for 05Cillationsofa pendulum act·uall ymnains bounded (by A}b
a U t, a s ill apparmt from th e law ofconservation ofcnagy.
188 Chap. 3 Linear S)'llffl\1
AHJ, .,.
P,ob/1,r,1 J, l)tdure 1hc 1ame r("lult from lht" uphdt formula fol' the- P"'"Od (S«. 12.7).
Herc we assume that the cocffidc:nts arc rea l. The complex case is compktdy �
! It is actuall y enough 1oassumc that .'l(t) is com.i nuous (1tt Sec- . 32.6l.
Sec. 27 No11n111011omouj Li11car Equation, 189
tq I t
Fig. 184 ,\ noncx 1cndablc solution orrhc cqua1ion i = x-2 .
f \\'c assume th at some EucJjdean metric has ben1 chosen in R•.
190 Cl,ap. 3 Lln..-ar Sys1<ms
f lL._..,
-11+-I -� 0---;-1,�•--
► • r
Jllg. 18.5 A lldo,i t1timau: 0(1hc grn�·th o f t ht totution on lhc 1ntcrv.ttf-.61,
f'
1I I
, I
,1.,, • <
<EMMA, l,I •(I) b<•,o/o,/i,,,eftf.,.li,,,(I) d(/iotd,..IA,i,,Jnr4/[,., 11, .. .. < t<.
{l•..ig. 18S). Tit.tit •(t) satufits Ult II fm'ori tstima.u
l•C•ll < •'" -•••t•(loll, C'l
J>,oef. The csLi r natc is obvi ous for 1hf" null solution, Ir •(i.J -,. 0, 1hcn • t) #, 0 by 1 hc
u1tiqucncu 1hcQrcm. Let r(r) = l•(r)I- Then the funcrion L(r) = In ,..a is ckfinicd Gw
t 0 < r < r. Bur
l-�<2C
•
because or(2), arid hence
l(I) < l(1 ) + 2C(I - 1 ),
0 0
27.3. The space of solutions of equation (I). Let X be, th<: set of all
solutions ofequation (I), defined on the whole interval/. Sincesolutions a.re
just mappings <p: / ➔ R" with values in the linear phase spaoe R", they can
Sec. 27 Nonau1ono111ous Linear Equaiions 191
Finally, the kernel ofB, equals {O}, since the Solution wi1h initial condition
q,(10) = 0 is identically zero, by the uniqueness theorem. I
Thus the mapping B, is an isomorphism of:<onto R•. TI1is is the basic result of
the theory oflinear equations.
Definition. By a fundamental S.)'Sl1m of solutians of equation (I) is meant any
basis of the linear solution space X .
Probl,m I . find a fondamental system ofsolutions of cquatiOn (I) with
A=(-��)
Theorem 2 ha.� a number ofimmcdia,c coNCq_ucnccs:
COROLLARY I. Ev,ry equation (I) has a fundtm1mtal 1711- of 11 ,#lrdio,u
'1'11 •''I"'"·
"""1i,,,uof
COROLLARY 2. £111,y solution of equation (I)is a /in,a,combinati.of
a fundam,ntal S)'Stem.
COROLLARY 3. Any n + I solutionsoftquatian (!) ar,/ineariJ,d,pacdau.
COROLLARY 4. The (10, I 1)-advan£tmapping
,.,, =
510
Br,e-
ro •,R" - R·
192 (:hap.3 Linear Systetn1
,, I
Fig . 187 1'he linear transformati(m of phase span-produc-rd by ad, •nn.tlC dw tolubom:
o f a linear NtUalion from t0 lo 11,
R + ;l
I + ( I - •')
t' x - 0,
r,111,1,,., I. Ptnd 3 fundamtntal 1ywtt'm or•olullor.1 or<quauon (◄) for dw uw "'"Mn tht
C'Ocflkh�nu ft IHC! com11n1, e.g., for I I u 0.
An.,, tl'•"'I, 0 < , "'- v, where A b a root of muh1phc11y v of th(, c-"3rM1tt-KIIC' eqwihon
{ COIi
111 the t:m: of C1om 1>lu roott A a l i.tv, we mu:11 chanC(' ,,,._. 10 ,•• «» flll, t" ..,. fl/II
1 11 ,,articular, for I -+ u • 0 we have
CIJI, tin wt Ir , ro' 0, _-1,
i.s meant the uurncrical function �V: I - R. whose value at thepoint tequals
q,,(1) q,,(1)
fV(I) = <P,(1) (1),(1)
In other words, Wis just the Wronskian of the system of vector functions
q,,(1): / - R" ob,aincd from ,he q,, in the usual way:
,p,(t) = (q,,(1), \i>,(1), ... , q,(•- "(1)), k=l, ... ,n.
Everything said about the Wronskian of a system of vector solutions of
equation (I) carries over without change to the \Vronskian of.a system of
solutions of equation (4). In particular, ,,·c have
COROLLARY 8 . /flhe fVronslcianofasysumofsolut�nsofequalitm (4) L'OJtisJraal
even onepoint, then i t vanishes ide.nti.cally.
Probltm 2 . Suppose the \\tronskian of two functions vanishes at th e point, ... Docs it Wlow-
1ha1 the Wronskian vanishes iden tically?
COROLLARY 9 . lftlu 1-Vronskianofasysltm of,ollllionsof,quafD• (4) cuuiwal
e1ien one poinl, th.en Jht solutions are linLa.rly dlpauhnJ.
ProbU.m 3 . Suppose th e \Vro ns.kian of tw o funetions vanish� identically. Does it�
that th e functions are linearly depen dent?
COROLLARY I0 . A S)'Sltm of sollllions q,1, • . • , q,, ofequation (4) isfiu,,lama,!,J
if and on9' ifits Wronskian is nonzero al some poim.
Example 4. Consider 1he sySlcm of functions <'•', ... , ,.._,_ Thc:sc func
tions form a fundamental system of solutions of a linear cqua.tion of the
form (4) (which one?). Therefore they arc linearly indepcnde,u, so that
Set·, 27 No11A.utonomou1 Liuc:ar t-:qunuons 19S
., .
,1.., ). '
fV • tJ.1 � '" I ,l,.)f
). ' J..
COS WI sin wt
Wa aw
-w sin wt W cos WI
is constant. This is hardly surprising, since the phase Aow of the prnduJum
equation preserve> area (sec Sec. 16.4).
27.6. Liouville's theorem. \Ne now examine how the volume of figures
in phase space changes in the genetal case under the action of1M' translor
rriation g;0 during the time from /0 to I.
Tll£0REM (Liou ville). Tio, fVro111kian ofa sys/rm ofsol•tio,u oftfrlalUJ• (I)
sati.tfits the dijftrenlial tquatio11
fV(I) = exp { f. a(r) dr} W(t 0) , dct i. = exp { f:<•J dr}. (6)
r••
fl,
Fig. 188 Accio n of che 1>h1.sc flow o n 1he 1>aralldtpiptd II. ,pa.nntd by.a (uncb.m,mt:AI
1ys1cm or solutions.
act i o,1 of 1h e 1ra1t1fonna1ior1 dur-i,1_g t ime t of the ph.atc Row ofthe system
.:t, - 2x, - x2 - x.> ,
,, - x, + Jti + x,.
j.) - x, - JtJ - x,.
A,u. W(t) - t1'W(O) - ,,,, since Tr A 2.
be the (r, r + O).. advan cc mapJ>ing (t-4 ig. 188), where A is s mall. This lanar tramforma.
cior1 of phase 11 >acc canics 1hc value or any solu1ion • of cqu.ation (I) at th,,� r in.In
its value a, the 1i m e r + A. A«o rdi ng 10 (I),
•<• + 6) • •C•l + A(,J•(,)6 + •(�).
i.e.•
g:,. = £ + 6A(t) + o(.6).
Ther c(ore, a ccordi1lg to S ec . 16.J, 1hc coeffic ient of \·.,.umc np.ansion w,dr-r- the tnnc,.
formatio n 1: '"' equals
det g;• 0 - I + 6n + •(6},
wh ere a • Tr A. But W(t') i.s th e volum e or the paralkkpiped n. s:pannittl bylM' ,�ucs
o f our system of s0Ju1ions at the tim e r, and the transformation z: •""
arrics thrsc \---a!UO
into the valuCl or the same system of solution s at the time r..&. 6 . "TIie panlk:kpipcd
n. 4 Sp3 nn cd by the new valu es has vol um e U'(r ..&. .l}. Thiercfott
♦
I !ere 1hc appearance nfthc minu, iig11 strm� from 1hr fact ,ha, in "riting
(4) i11 the form orn •)"Item (I), we 11\1,st iranspmc a,x'• 1 ' ti) the n11ht-hand
side. The inairix oCthc r<'luhing system is
0 I
t
Fig. 189 The phase Ao"" ' ofan asymptotica lly stable linear .system.
198
tr r,(I) • 0 (0 1• all , , h ii lrue 1h.11 ,he· N:fU!llbnum Jl(J'III IOO (0. 0) '" ··••)' t-Otibk ., lk
f{l('llkltotor rrlctloo ,, po,,hlvt?
D,jini1io11.
lly the di11tr,:,nr, or n vcrtor llrld v in the Euclidean spacr R'
with rccrnnKulnr C'oordinn1e1 x1 it meant tht" fonc1ion
❖ Du,
div v • '- , ·
(•I c1X 1
div Ax a Tr II.
The divergence of a vc<"tor field dctrrrnine§ the rate of volume cxpamion
due 10 the corresponding pha..«:: Oow.
Let /J be a domain in the Euclidean ph;uc space of the (not neccssatily
linear) equation x•
v(x), let D(1) denote the image or D under the action
or the phase now, and let V(1) denote the volume of the domain D(I).
(Fig. 190).
COROLLARY I. If div v & 0, lntn Int pJuu,flow prum,,s 11w --- •f OIIJI
domoirr.
Such a phase flow can be thought or as the flow of a n incompressible
uphase fluid" in phase space.
ptestrvu volume.
Fig . 190 The phase Row ofa vcc:tor 6t'ld ofdivagcncc- uro presavaa«:aw
Sec. 21J l.l11enr Equa1ion, wilh Periodic C:O.ffiricnu 199
Vvc will assume that all the solU1ions of equation (I) can � a1end�
indefinitely.Thi.s is certainly true for the linear equations in which wc ar�
par1icularly interested.
The periodicity of the right-hand side of ( I) leads 10 a number ofspecial
properties or the corresponding phase flow.
X
0 r 21 t
Fig. 191 The: extended phuc spa« of an equation with periodic coefficients.
200 Chap. S Llnc,ar S)'S1tms
and moreover
,..,r,s _ g',,,.r
60 - oso ·
Proof. By Lemma I,
_ g•
g11T-ts -
11T o,
Q r l
and h,•,wt•
.,•1'•.i, • .,1tF'•,"1tr • g0...\0
•0
""O "'"T .,. 111T
•
S1•tting I • r, Wt' grc
gh" • 0r • A1"c{,
and hcnct· ,or - A" by induC'tion. I
To ,v,ry prop<>rly of 1he sol u1ior1s of cqua1ion (I)1h,r, corttsponds an
ana logou, prop1•r1y of 1hr p,·riod-advanrc mapping A.
'1'1Ui:OR1-.:M.
I) A poi111 "• is ajix,dpoi11t oj1h,mappi11g A(Ax0 • x0) ,Jantl•ttl:,,ftltLJ.t"1iM
wit/1 illitial eo11ditio11 x(O) • x0 is ptriodit u:11/t P,,1od T.
2) A periodic solution x(I) is stable in l:J•ap.nov's """ (as.,,.ptOlir•II:, st•bl,) if
and 011/y ijth,ji,ud poin t x0 '!/tht mapping A is stab/, in L:,apu110,,'1,nu, (.,.,,,.pt.,,
cally sinbl,).t
f
3) lftlu .<J•Iltm (I)is li11tar, i.t., i v(x, 1) =
v(t )x is a lintor/1'MbMefx, tl,c,
lht mappi11,e A is linear.
4) if, mureou,r, /ht trau ofthe lintar optrator V(t) ranisltu, In IN maJ>PUll A
to,utrvt.t volumt: drt A • I.
Proof. A,s,rtions l)and2)follow from theronditiong!•• 1oAandfrom =
t h e continuous dependence of the solution on th� ini1ial conditions in the
inlerval ro. Tl. Assertion 3) follows from the fac1 1ha1 a sum ofsolutiomofa
linear sys1em is itself a solution, while ass�rtion 4) follows from Liouvill�"s
theorem. I
28.2. Stability conditions. We now appl)• the above theorem 10 the
mapping A of ,he phase plane (x,. x,) on10 iuelfcorresponding 10 the S) -S
tem (2). Since the system (2) is linear and the trace of the matrix of ilS
right-hand side vanishes, we have the following
COROLLARY. The mapping A is lintaranti prum:u ar,a (det A = I). 77r all
solution ofthe sys/em of equations (2) is stable ifandon/:, ifth, maPJ>i•l A is stab/,.
Prohl,m I . Prove thal a rotatjon of the plane is a. stable nupping, •ilik- a. h)pcrbolic:
ro1a 1ion i s unslablc.
W e now make a more detailed study oflinear mappings ofth� plan�onto
itse1f which preserve area.
t A fixed poin1 Ko ofthe ma pping.◄ i ssa.id 1obcstuk ii; L;,o!Jt.m«;'sscnkifV c > 031 > 0
such that Js - s01 < d implies IA*s - A•aol < c: for a.II• - 1, � .•. M>d c,i1,,.,,.,,.'!7
stahl�if A•• - A•.xo, -O asn - co.
202 Chap. 3 Li.,..ar S)'11tms
rrA ,1
t\ I A,
0
TrA<2
",
Fig. 193 Eigcnv afuo of the J.>(r-iod a� dvancc mapping.
with irit,gral and half-integral coordinates, co"uponds to a strongly $/Jlbu spum (3).
e
Thus the set of unstable systems can approach the w-axi• only at the
=
points w k/2.In other words, a swing can be "pumped up" bymaking a
small periodic change in its length only in the case where the period of
change of the length is near an integral numbcrofhalf.pcriodsofthenatural
frequency. a result everybody knows from experiment.
The proof of the theorem is based on the fact tha, fort= 0, equation (3)
has constant coefficients and can easily be solved.
.
Probf.tnt I, Find the matrix of the period.advance nuppi.ng A fc>r the $)'Stan (3, ._ith
i: - 0 in the basis x, i ,
/(1) = I., +- ••
w 1:.
0 .. I < •• • < I,
11 < t < 2,r, (41
/(I+ 2n) -/(1 ).
Solution. It foll ows fro ,n 1hc solution of Pmblcm I 1hat A AJ Aa, when
<o1,1 =Nz« .
Hr.hCC the bounda t)' of the zon e of in.stabi lity has the UJuation
I
,.,
Jl,ub/,m 3. (; 1"' the up 1,rr {U\u1Uy umtablr.-) tqUJl1�tum po1,1t..-m of a ptnd:uh,.m b«om,it
.!ltabl«- 1( thr 1Mli1 11 of 1uiJkmion 04C1ll11r, 111 1ht \t'flkal dirttttOn,
An1, Th e U )fl(r ec1 ulllb, lum J>081t lon br.comft uablr fo r •uff"t<'ttnd,- ,,.pad Ol<iH:at"°"'
or l
1hr 1Xll 111 o f au�J'4"ndon.
Solutl11t, I th(' l ngth of 1hr l'.N'ndu m ind • < / 1M amphtudt oldw OIClllaooM
or ,h J>OI orIUiJ)Ct1don. I.rt thr J)(' iodl of IIW' OKUlation1 of hr point oflutpt""'°" bt
A:'t / l)t" r u
cve,-y tual(-axriod (then c 8«/r1), The �uation of m04k>n an Ix •--nlkn ,n tlw form
A - (,QJ J aJ}.r,
whrrc thr sign c:hango aft er du: hm e r and ,,,, 1/1, a1 ,fl. If th e Ol<ibbOM of tM'
point or tUSJk'mion ar e sufficiently n.pid, then a• > (;,1·1, wM.-e a• &,Jtr•. As •• tht
prette:ll ng 1>robl c:m, wc hl\'C A A1A,, where
A, (
ca,t, i, } , inh kr). *' a ' -+ w'
.t ai n h ,h cm:h 4r
aod
A,•
( c�n, A•;no, ). fl' - a' - w1•
-Cl 5i n O r c os Ch
We now sho w 1hal th.is condition holds ror sufficicndy rapid o,cill.-.tiom o/ tw poioc
of suspension, i.e., for , > I (f < I). Introducing dimcnsion.lo.s nriablcs « .a..nd p sudi
that
! - •' < I •
I '
l-µ'<1,
we have
arc valid ro r small #. and µ with accuracy O(c:' + JI"). Thus the stabaity aoodicioa (9)
bctomes
2011 ( hap. 3 l,11\<'ar Sy,trm,
,, . ✓�s'
or
( 2•
r • !7"
1111\ <'omlition HIii be Y.-riHt'H iu 1he r.-.1 111
whe� A' l/2r ii 1he frc:-qu.:-ncy of fl!M"ill:11 ion oftht- pomt ofsuspc-mion. For �mplir.
if1hc leng1h o( tht 1lC'nduh11n I,/ 20 fm alld •ht- point of,uspcn)ion 0u•n,1a OKilb·
tions ofam1lli1ude II I nn, thf'n N , 0.2 Hl•,'!'iiii]w 10 qlll, 1 n panwubr. tht uppn
equilibrium l '°'il ion i ,1abl (" ir 1hc· f11.<t1 uN11 y of o,,rfllatHm of 1hc- po.nt of wspnuaon
,
C"ffec·ch 3f1, 11ay,
x = f(t), l E /, (I)
corresponding t o the ''simplest'' homogtncous equation
X :;; 0, (2)
J''•
can be solved by quadratures:
C
....
f'
Fig. 197 The c,oordinato of the point c ar� fint integrals of the ��uac.ion.
x � A(t)x. (�)
Suppose we know how 10 solve (5) ands • tp(I) i, its solution. Then in the
extended phase space we use coordinates rectifying the intqral curves of
(5), i.e., the point (,p(t), 1) i• ..
signed the coordinates c • .c,.l
and I (Fig.
197). Equation (5) takes the particularly simple form (2) in the new
coordinates, and we can go over to the rec1ifying coorcUnatcs by malting a
tr-an.sformation linear in x. Hence the nonhomogcncous �uation (4) takes
the particularly simple fonn (I) in the new coordinates, and can easily be
solved.
29,2, Solution oC equation (4). Suppose we look for a solution of the
nonhomogeneous equation (4) of the form
,p(I) • g'c(t), c: / - R", (6)
where g': R• -+ R" is the linear (10, 1 ) -advance mapping for the homo
geneous equation (5). Oiffcrcn1ia1ing (6) with respect 10 1, we get
q, = .fc + g'c = Ag'c + g'c = A,p + g'c,
which gives
g'c = b(I)
after subsii1u1ion into (4). This proves the following
THEOREM. Formula(6)gives the solution of<'f"Olion(4) ifandordJifcS4Sisfastlu
equation
c = C(t), (7)
where C(t) = (g')- b(t).
1
coaot.LAJ<Y. The solution of the nonunear <'f"olion (4) wilh inib4l -,Jjti,,n
210 Chap. S Lin�•• Sy11�ms
,p(t0) • c is giwn by
,p(I) • g'(c + J:. (g')" 1h(t) dt).
I'•'J •- -x,,
X,1.1
wi1h 1hc known system of fundamental solutiom x1 • cos t, X,1. • I .uid .r1 = sin ,.
-si.n
x2 - cos I , In accordance with the general r1>k, �-e look for a tolutH)II ohhc rorm
x, • ,1(/) Cott + t,1. (I) sin I,
To dctcnni nt: t1 and t19 ...,,c have the syst em
l' COi t + la sin t = 0, -l1 sin/+ l cost =f(t).
1.
Therefore
11 = -/(t) s in I, t2 -f<t)cost,
$0 that fin.ally
,(1) = [•(OJ - tj\,) sin r dr] cos 1 + [,(OJ + f.A•J <OS r dr] ,;n r.
4 Proof• of the Baile Theorems
A.r
TIISOREM. lit A: M ➔ Mb, a contraction mappini ofa compkk m,tnt •J>IIC• i\1
into itstlf. Then A has a uniqu,fix,d point, a;,......,
point x E i\1, ,,., Slf'U'N<
of images ofx under application of the operator A (Fig. 199) ,,_,8" i. tl,,fixed
point.
Proof. If p(x, Ax) = d, 1hcn
p(A"x, A""x) ,;; )."d.
The series
exists. The point Xis a fixed point or A. Jn fact, since ('\.-Cry contraction
mapping is continuous (choose a = r.), we have
AX:: A lim A"x a: lim A"• 'x = X.
Moreover every fixed point Y coincides with X, since
p(X, Y) = p(AX, AY) ,;; ).p(X, Y),). < I = p(X, Y) • 0. I
R�mark. The points x, Ax, A 2x, ...are called 111cu1.sii� app,oximatimu 10 X .
Scc. 31 F.xi<icncc, Uniqucnt11, and Con1inui1y 213
(A,p)(t) = x0 + J' ••
v(,p(,}, ,}d, .
Ccomctrically 1hc 1ransi1ion from ,p to A,p (Fig. 202) means using one
curve q, t o construct a new curve A'P whose tangent at C"-ery point t is
parallel 10 1hc dircc1ion field determined by ,p rather than 10 thefield on the
new curve Aq, itself. Note that q, is a solution satisfyingthe initial condition
,p(t0) = "• if and only if ,p = A,p.
Inspircd by the contraction mapping theorem, we now consider ttK
successive Picard approximalionstp, Atp, A 2 , ... , bcginning,say,�ith ,p= :so-
,p
214 Chap, 4 Proof, or th• Ba,it '"-•ms
-:r--
1...... •
.:i.·
'
Fig. 201 An in1.-g,al C\ll'Y( 0(1ht <"qua1 ioo X v(•,I).
(A f!)
t t
:,:
'
;::::
,::::
--
:::::: :::::
::::: -
ro :s
f'
::::, ::::' 119'
::::
t
Fi g. 203 Pic ard approximations for the equation :i • l(l),
t
Fi g . 204 Picard approximations (or the cquarion X = •·
Sec. 31 Exi,1c11cc, Uniquenc:u. and C'.on1inui1y 215
t I, I.el j
6KOffJ,/I f(I),
(A•)(•) .. I J''• r(r) dr
(f'ls, 20.ll), Then the flr1t 11e1> lud, 11 once to an uac:t .olution.
l!:xonipl, 2. IICII i •• I Ip O (F ig, 204). In th,� ta� the con,t-rsmtt of the ap--
1,ro,clmatlom fan written down immediately, In rac:t, •t the p(Mnt I �e h.a,e
A• ., •P + J� • dr • Jlo(I + t),
o
2) t' - J + '
the metric p(x, y) • I• - YI it• complete me1ric spaee. W,: no1e two key
t
lneq1H1 litie<, 11n111cly the trianglt intqun/ilJ
I• J: f(t) dteR'
LEMMA,
t Let us recall 1h e proofor these incq uali1ics. Draw the 1we>-dimcns�l pbnc through the
veclors • and y of the Euclidean space. This plane i.n..hcrits th eEudidea.lld.nli«uTC(rom R...
Bvt in the f: udidean pl an<" both i ncq uali1ics ar\" known from dancn tary g�. This
prO\ 'CS the i n cqual ili\".s in any Euclidean space, forcxampJ'° in R•. In pMticub.r, •·�h.a,· �
pro\ ,cct wirho ul any C'.tlcu Jations a t all tha. t
and similarly
Sec. 31 l•:xi,1cncr, Uniqurncs,, and Con1inui1y 217
:r A:r
P,
__
A
V
I A
Av
rr:r ,
"
(/
,0
_o
�
t;,i- y-z,
r R "'
R"
Fig, 206 The dcrivath·c: ofa mapping f.
R "'
b
r
a
to
J,�ig . 208 The: t.')'li ndcr rand the conc: Ke,
:r
t0 I
Fig , 209 Odlnilion orh(•, I).
lies in the domain U for sufficiently small a and 6 . Let C and L dfflOte the
least upper bounds of the quantities lvl and Iv.I on this cylinder, where here
and subsequently the asterisk denotes the derivative (with rcsixct to s) for
fixC"d t. SirH'C the cylinder is compact, these least uppcrboundsattachit\·cd:
lvl ,.; r., Iv.I ,.; L.
Now let K0 be the cone with vf'rtcx (1 0, x0), 0o·pc-ning"' C. and ahitudc:4',
s o ,hat
Ko = {x, I: 11 - lol .. a'. Ix - •ol .. c11 - 1.1}.
If the number a' is small enough. the cone K0 lies inside the cylinder r.
Moreover, if the numbers a', b' > 0 arc small enough, every cone K_.
obtained from K0 by parallel displaccmcnt of the vertex to the point (lo- x),
where Ix - x01 � b\ also lies inside r. Tht' numbcna"' and•· attassumed
10 be small enough so that K, <= r, and we will look for a solution "of
cqua1ion (2) of1hc form ,p(I) = x + h(s, 1) subject to thc initial condition
,p(1 0) = x (Fig. 209). The corresponding integral curve then lics insidc the
cone Kx.
31.6. The metric space lvl. Consider all possible continuous mappings h
of the cylinder I• - x01 ,;; 6', It - 1 01 ,;; a' into the Euclidean space: a•,
220 Chap.◄ Proof• of the &,,c 1hcorcm,
and let ,W dcnntc tl,r S<'t nf,uch mn 1,pings which -atiify thee<ira roodttoon
lh(•, 1)1 < (.'II - lol (4)
(In par1ic11lt1r, h(•, 10) • 0). WeIntroduce a metri c pin 111, by Klltng
p(h,. h i ) • llh, - hi ll • max lh,(,, I) - hi(x, 1)1.
h •111 <••
11-tol<•'
·rH£0REM. T/111,1 J\1, tquipJNd with the mtlrit p, is II comp/tit 11t1tri, 1/» u .
Ptoof. A uniformly convergent sequence ofcontinuous functionscon"-crgcs
10 a c.ontinuous function. If the functions satiify the inequality (4) bclocc
passing 10 1hc limit, then chc limit function also 1atisfies (4 with the same
constanl C . I
Note 1ha1 the space 1\1 depends on three positive numtxn •'. 6\ and C.
31.7. The contraction mapping A: /11 - /If, Next we introduce a
mapping A: M - NI defined byt
Because ofthe inequality (4), the point (s + b(s, t), t) belongs 10 thecone
Kx, and hence to the domain ofdefinition of the field v.
THEOREM. If a' is suffici1111/y sma/1,formula (5) d,fin,ts a conlr«liM -M>i•t of
1/tt spact /11 inlo irself .
Proof I) First we show that A carriu 1\1 into itst/f, The function Ab is
continuous, since the integral or a continut)US function depending contin
uously on a parameter is continuously dependent both on t� pa.ralllC'tcr
If . If .
and on the upper limit. �1.orcovcr, Ah satisfies the incqualhy (4)� since
l(Ab)(s, 1)1 ,;; v(s + b(s, t), t) d,,,;; C dtl,;; Cit - 101,
Therefore AM c NI .
2) Next we show that A iJ a contraction mllpping* i.c.1 that
QAh , - Ah211 ,;; -lllh, - h i ll, 0<-l<I.
J'
To this end, we estimate Ab, - Ab2 at the point (s, 1), \¥c have {Fig.210)
, hz
,"'
, h ,
tq t
f'ig, 210 (:umpati10n ofv1 and v,.
where
v1(t) • v(x + b1 (x, t), t), i - I, 2.
According 10 Theorem 31.4, for fixed t 1hc func1io1\ v(x, t satisfies a
Lipschitz rondition (in the first argument) with constant L1 and hence
g(x, 1) = x+ J''•
v(g(x, t), r) dt,
vg(,,11
• = v(g(x, I), 1).
,.,
I I fi:>llo,\·� 1ha l g ,.ui,fo...,, rquation ('.l I for fix<-d x and ,fw init.ial condition
g(x,/0) = xlort = t0.i\Jon:ov<•rgi�cnn1inuou,,loimt·he.\l. I
Thui. ,,·t· have· p.-ovc-·d tlu..: <'xi trrn· · tlu..··orcinfot t·qua1ion '3
:-. t and �ib.
i1ed a solution which depends <'on1inuoui-lyon 1lw ini1ial f'onditions..
P, ,;bf,m I. 1'1 , 1 \ t · 1lw uniqu c·m� tht..-Ornu.
222 Chap. 4 Proofs or the B.uic Theorems
Solutlo" I. l.r1 6' 0 In 1hc d.-finl1ion of A f . 1'llrn 1hr uniqur n n, o(dw- fixtd point ofth.t
con1racllon rrn,pplng A: Af • /l1 implir, the uniqUC'nn, of 1hr M>luuon wihJ't•"I t,W.
lnhfal condition .,{,o) •o), I
Sol utio1t 2 . I.rt•• and•, be- two ,olut lmn a.11i,(yln1 tM ,.uuf' initbl c:ond111on •• ,., ..
"1(10) •o 1111d dcfintd for It lt l .c. "· MorfO'n- Irr
m••-c, 1•(1)1,
11•11 • I• •1,I
'
wht1·c O c.: n' < (T. Thcn
Moreover ,1 i� a contrart ion mappin g (i n a su itable" �tric). s:in,e the" clnn�ti,,-� ol tht
fum·tion ¥ is small i n a n(ighborhood o f the poi nt O(b«au 5C' of the-,ondirion •• 0 01 .
Pmhltm 2 . Pro\',c 1ha1 1hc Eulrr linr appm,U'hts a solutlOR as ill s.1cp approat'hr,. XTO.
&lution. Let IA • t' h4 be tht' Eulc·r line wii..h st e p .l. and in.itia1 cottd.ition C:-£ -.. i.,, - •
(Fig . 211). In 01 her wonh, lr-1
ii
�g.(x,1) ~ v(g.Cx. 1(<)), s(I)),
whcr(' J(/) • 10 t- k.) :ind I. is tht' i111('gral part of( I t0) ·.) . Th� di(iucnct �-"ttn fflC'
:c
a'
4 4
t0 t1 sf(} t t
Fig , 211 The Eulc:r line g4{x,I).
See. 32 The Differentiability Theorem
fa
�r,
r,
Eult'r line a nd th e solu1ion I c.tn � es1im at � by wing the formu la m S«. 30.3:
I
111. Ill 'llh . - hll < r=.IIAh. - h.11,
Ou1
I IIjusl lhc- same way, we cau associa1r with th<" difft·rcruial C"qua,tOn
X • v(x, 1)1 xcl'c:R' (1)
a sys1cm of di1Tcr-cn1ial c<.1ua1io11s
{ x = v(x, 1), XE Uc R'\ '2)
y• v.(x, 1), y c TU.,
linear in the tangent vector y (Fig. 213). \Ve call (2) tho �rsuaef,.-•li-ef
variations for equation (1). 1'hc asterisk in (2), and in subsequent fonnulas,
denotes the derivative ,-..•i1h respect to x for fixed t. ThlL, v.{x.1 is a linr-a..r
operator from R" into R".
Together with the system (2). i t is convenient to consider the-S)'Stcm
{ x = v(x, I), xe:UcR",
(3
i. • v.(x, t)z, z:: R" _., R",
obtained from (2) by rcplaring the unknown vector y by an unlmo�'ll lincar
transfonnation :.. We will apply the term equation ofi:artatiOl&I to the system
(3) as well.
Rtmatk. In genetal, given a linear equation
y = A(t)y, (2')
it is useful to considct the associated equation
i = A(t)z, z: R" - R•. (3')
involving the linear operator z . From a kno,,•lcdge of one of the equations
(2') and (3'), we can easily find the solution of the 01her (how?.
32.2. The differentiability theorem,
·rHEOREM. Suppose the right-hand sidev ofequation (I) u tu,ia<MJ�:,diffa
entiab/e in a neighborhood ofthepoint (x0, 10). TirLn tht solution g('JL, t) ef,tp,atit»t
Sec. 32 The Differentiability Theorem
,J,,.,(x, t) • E+ f'
'•
v0(,p,(x. t), r),J,,(x. t) dr. {$)
No1ing that 'l'o• • ,J,0, we deduce from (4) and (5) by induction in• that
q,,. • 1• =1/111+ 1 • Thcrtforc the �cqucncc {\f,,.} is the �ucncc of derivatives
of the sequence {,p.}. Hoth sequences (4) and (5) arc uniformly convergent
for sufficirn1ly small It - t01, bt'ing sequences of Picard approximations of
the sys1em (3). Then Lhe sequence { 'P,.} i� uniformly convergent together
with its dtrivatives with r-�pc:ct to x. Hence the limit function
g(x, t) = lim ,p,(x, I)
is uniformly differentiable in x. I
32.3. Remark. At the same time, we havejust proved the following
TH£OR>:M. Th, derivative g. a(the sol•tion oftq•atian (I) u:ith resp«t i. llv i,ritiol
co11dition x sati.sjies the equalion ofvariatiofl.S(3) u...ith lhe initialcontfition ::(10) = £:
i)
-g(x, 1)
Cl
= v(g(x, I), 1),
i}
,,-g 0(x, 1)
01
= v.(g(x, t), t)g.(x, t),
g(x, 10) = x, g.(x, t ) = 1-:.
0
Jtig. 21◄ Acdo11 or lhc (141, t)-advancc cn.nsform..teon on a curv� in ph.u,ir -'f»tt and on
ii, tangent vcc1.or.
• er•
VE C' => V E c,- J =-- g E • l => g e •-- I•
• er
This proves Theorem T,, since Theorem T1is just Theorem 32.2. I
32.5. Derivatives with respect to x and t. Again let, � 2 be an integer.
THeOReM T;. Under the conditions of Thtorcm T., tlr, solution g(x,1) is a difer
tntiabltfunction ofclass C'- ' with ruJMct to both r,onablts x and t:
VE C' c> g E c;r- •.
Proefq/ tit,''"'"'"· All)' ,ch pariial deriva1ivc of 1M' ful'K'1ion F with ropttt to •hit nnablcs
x1 and, lnvolvlng dlfff'rtntlatlon with mpect 10 t can bt' "p,rOKd in tffffil oil and 1M'
partll'I d•:rivalivt:, or ,h e func1ion r of order INI than ,, and hcnn i• cononuo,ya.
an y ,11! pa r tlAI dc:r iva1ivc with rnpctt 10 chc v1m11bJn x1 b ('ontln u0ut by h� t
"""°"
/>,� eftit, tlt,omn, We have
Ou1 I• C0 b y ScC'. Sl.8 (the &0lu1ion depcodJ con1 inuous.ly on a, 1). Thia compkta lh-t
1 >roororTheotem r;, I
P,01>1,m I. Pr ove lha1 if 1he tigh1,hand ,idc of the d iffamtia l c:.qu.ttJOn I it at1.6n1tdy
diffc"1!nliablc:, then th e: solutil>n is alw a n infinitely djffcrm tlablc fu11ttl0ft o/ 1ht 1nmal
conditions:
v, c• ⇒ 1,c•.
ll�mark, h •·an al so be U1c>wn 1ha t if th e right-h.and side v is ana lyti c (bu a Ta)W taict.
converging to v in a neighborhood of every point), then th e sol ution .I .aho de.pa.ct..
an al)'tically on• and,. It is nat ur al to stud)' differential equations with anal)"lit right�hand
sides: for both complex values: of 1he unknowns and (ol particular i:mportaJlft) (or C'Offl*
plex V3l ucs o f the time t.f
fig. 215 The line L1 is traruvcne 10 the plane L1 i n the space R'.
t Concuniog this thcor)', sec e.g., V . V . Colubao, l.«bdd.,. tlu Aaal..JIO' 1--,-1'Di f e
mlial Equatu>n.s (in Ru ssia n), Moscow (1950).
228
x • v(x, 1) (I)
defined by 1he rormula C(x, 1) - (g(x, I), 1), where g(x, I) is a solu1ion or
(I) satisfying the initial condition g(x, t0) • x. Then, as we nows.how, C is
a rectifying diffcomorphism in a neighborhood or1hc poin1 ("o, 10).
a) Tire mappir\f Cu dijfrrentiable (of classC'- 1 if v EC C'), by Theorem T;.
b) T/1t mapping C lraocs I 1111d1a,,grd: C( <, 1) = (g(x, 1), I).
c ) The mapping C0 carriu the standard v,clor fald e (x =
0, I = I) W• ti,,
giuenjield, i.e., G0c =(v, I), since g(x, 1) is a solu1ion of (I).
d) T/1t mapping Cu a diffeomorphum in a ntig/rborlH>ad of llte fJeilll ("o, 10). In
fact, calculating the r-c.sttiction of the. linear operator C.1,..-. • to the trans-
verse planesR• andR (Fig. 216), we get
1
The plane R" and the line wi1h dir"C-<:tion v + e arc ,rans\-cne. Thcrcfott
C. is a linear mapping ofR"' 1 onto R" • 1 and hence an i.sornorphism (1hc
Jacobian of c. a, 1hc point (x0, 10) is non,ero). h follow, from the in,-us,,
func1ion I heorcm Lha1 C is a local diffeom<>rphism.
Proof of the rtttification thtorem: Autonomous cast (sec Sec. 7.1). Consider the
aulonomous eqt1alion
R" e
R""'
\ :�
r�
(J
�((,ti.
{
'I ,,1
L,•1 cite pl,ast· v,·loci,y v0 a, the point x0 bt-diffrrt'nl from O .-ig.217 . Thm
tlwr<· t•xi"1s an (n - I)•dimensional hnM'rplanc R•-1 c R"' passing
lhrough x0 and 1rarhvcrsr 10 v0 (mor<· exartly. a c-orn1 " 1,ondin� plant in 1hf'
tangl•n I space.· T( '•• 1 rau�vc.·r,1· to I lw liucR wi1h dir«tinn v0 . Let G be the-
I
,napping of chc domain R" 1 x R where R" 1 • {{},R • (I} into ,h,-
do111ai11 R" defined h)' cite formula G({, I) = g({, I) "here, { lin on R•- •
neat x0 and g((.1) i-: 1h<- value of 1hc solution of equation (6 ..atisfyin� thf'
initial c·ondition <?(0) -;;;: � a.1 the time, . Then, as ·wi· no" show, c- • is a
rcc1ifying diffcomorpl1i,1n in a :-.ufficicntly small neighborhood olthc point
({ s x , I e 0).
0
f
·r11EQRE.,, l t/1t ri[1l11-ho11d sidt v(x, 1) oftht d,fftrtnliol tq•oli.,. i • v(x, I u
ro111i1111oust., 1/ijftrt11tiob/1, tht11 t/1t so/111io11 g(x, I) 10111/)11'8 tht 1•1t1•/ tMi111••
g(x, 10 ) • xis a ,0111i1111ou1/y dijftrt11tiab/1/•ntlion ojtltt init10/ <M4111M:
I) veC' o geC'for,;;. I.
2) I v e C', th, ,u1ijjoi11g dijftomorphism conslr•cled in S«. 32.6 i, ,..,11•...i,1
f
Proof. Thr proofof th<' clC.isl("n(<', uniqutll4:Sll, and con1inu.ity th�n\S 5'-t-. 31 Uk:S o,n)y
th e difTcr<'n tiabilit)' with rnp..-c1 to x. for fixed t (ae"1ually only � oisc� of.a Lipac:hiu.
c on diti on in x). Th ercror'" the proor r'C'ITiain s \' alid if the dq,,endcnu on , is :11•1:mrrl t o
bt on ly c on1inuous. I
Note th at the .sol u1i on deptond s linearly on Y• and is a t0ntinuoudy d.i:�tiablr f unc
tion or t . ' f 1letefo r,· the: s o l uti on bit-l ongs the cl.us C' with rcspecl to boc.h Ye and , _
if•�
10
I
X "" v(x, t),
Y = v. (x, t)y
has a solutlon wlri,1, is um'qwl)' detmnined b)' it.1 initi•I tMl!itiMs ad dlfit.'flh ,--·•· _.,.J_, • *•
,onditWns, p,<n.idcd on/,1 tlurt the field v i s �clau C'.
P,oof. By the exi sten ce theorem ofS,c,C, 31.8. the 6nt �ua tion ofthe syst.cm has a tolutioft..
which is uniq1.1d y de iermi n«l by its initial <:onditions Ito· 10 and depends on that a:od:i
ti ons co ntin uo usl y. Su bstit uting 1hjs sol u ti on inco the St:COnd equation.. •� get a linear
equa tion in y whose right-hand side dc:pcnds continuously on t and also on the inilal
Sec. 32 The OifTcrcntiability Theorem 2SI
condition •·u (regarded a1 • puame1cr) or the IOluhoo of the f'ln, equal.oft. Bue.. by
Lemma 2, 1h l .1 Unc:M equation ha1 a tol\lllon wh.Kh ,_. d ctermul<d by ,tt ,n.,bal cbt.1 Y•
And h II con1lrwom fum:IICN1 o r t, y0, a11d 1he parame ter-.. I
'-"''/"''°"' ,,
11,u, lllt tl/Wllion ef UhVfM, tL'"' ;,. IN UJI • • c•, NCHc- that In t.ht atit • • C'
""C pro,·ed that che derlv:.th·e of the talu1ion with rnP«• to •he rnht.AI cbu. Yl•'kt lfw
e,1u11llon or v11riA1iom (S), but 1h11 can no lo1;1t:r
1 be a.1erted, 1-IMt we M.111 do nae know
wlle1her chiJ deriva1lve exi111.
1'o pn>vc.· the <liffetcn 1iability of the so lution v.'t1h l'ftpcct to t.M: inuul �� •-c:
fine c:omidt-r a ,pedal caJC:.
UMMA 4 , Suf,,,.st llit iwto, fit/ti •(•,I) eft.l(ISJ C•--' ill ,l,rrn.,.tt«
' "'• Mil --
• - 0Ju, all I . Tllt1t tltt solution rif1M tqMtJtion 1 ... w(•, t) is iilnmtJM# nil
«.,..,,
,,,,,,a,. a,
im'tinl tondl'l'loJU at t/r, Jtfint • .. 0,
P,O()j, Uy hypothoi.J,
I•(•. 1)1 - •0•1)
io a neighborhood of 1he p oint • - O . U,ing 1M formul a of S« . ,0.3 10 cthrNk tht
error of th� approximati on • - � 10 the 1olution • • •<•)
satidyinc 1he Wt.al concb
,
tio,n •(t0) - •o, we find that
I• - "o .; �
l I -A IJ''••("o, r)••I.; K,.<,i(t
max Iv(-., r)I
for sufficiently small IXol and 1, - t01, wh«-:rc the constant Xis inc:kpcndcn.1 ol-.. Tb.,a
f• - •ol - o(lx0I), which impl es chat • is differen1.i.able '4ith rapo t to -. at ttr"O. I
i
\V� rlOW reduce the g eneral case 10 th e speci al situ.at.ion of Lemma 4. To do� ,,,.T ftttd
a suitabl e coordi na te a,ystem in exk:ndcd phas,e IJM tt . Fi.m � note �1 the
ooly c h()()$("
s olutio n under c onsideration can always � regudftl as tM null solution:
l.SMMA 5. Ut • - •(I)� o so-iutiOII efl/luquotUm X - v(•. t) .,;,A• ritftt.-.1,i,h'!{d..uC1•
d<fint.d in o dornai.n ofutmdtdpliost JfMK-t R" X R 1. Tltttt. tNu ttisu • C'4u ,-S. '!{
txlt.ndtdpltolt. SJN,�� wl,iJI, prt.Stn'tS ti1N. i.t., (•. I) - (•1 (•. I}, t}••
ndCMTW:s &6, _....., •
inJo •• a:. 0 .
P,ooJ. Since • • C•, we nttd only make the shill a:1 = • - •Cl). I
In the system of cOOf"dinates C•u I}, the right-hand side of our equation cqualsO at tb,c,
p oint •• - 0. We now show tha t che deri\>ath•c ol the right-hand side ,,,.;th rcs.pect co••
t.an also be rnade 10 van is h with the help or a suitable change- of eoordinato ,,,.-hich is
linear in •·
l.J!;MMA 6. Under tlu u.mditions ef I.Ammo 5 , tli, coordinatu (•., l) r.an j,, "-ta ia ,_. •
U.1Q)'tllat tlu equoli.on X = v(s, t) is �ic.okn.t Ii) IM eflldt.lm S.1 = v 1 (a1, t). m6a-,IWJ,eUv1
and its dt.riuatic.v: 0v 1/QX1 l>otll 1.vmisA al Ult {NMt x1 = 0 . Mor«tttr. lluf� a1 (X. t) a-. ,lw,
,lrosm "'bt lin,ar (but not n«usa,ily lio mogtl'ltO'W' } in•·
Accordi n g 10 Lemma 5, w e can aNume that v 1 (0.1) • O.
To pl'ovc Lemma 6, we fi rst consider the fol�irlg: special cue:
UWMA 7, TIie 4USt1tion of lA.mma 6 ii 4Jalidfor llv /in,:a.r �liM X - A(t}s.
Proof. Vie need only Ch00$,C a1 to be th«-: value of the solu 1ion s;atilfying the initial cond►
tion •C•) - • a1 a fixed time 10• Ac cordi ng to Lemma I. x1 = B(l}• whc:rc B(t): R• - R•
i s a Linc-ar operator of class C1 in, . But our linea.r equation ta..k.cs the fonn i:1 - 0 in the
coordin ates (• .. t). I
232 Chat>· •I Proor, or the a.,,c Throrc,rm
l',IHlfqJ ltN11111 1 6,
t-'lr111 w1• llnr:ol,r 1hc· rqu:uion A \II( ■, t) a, ,rro. 1"" • """ for m 1h--
c·qu1ulon ol' wuiMio,n
• .1(1)•. .i(I) v0(0, I).
lly hy1>01l1MI\ v ,C•, t1nd IWntl" A ,co, Uy l.rmrn.a 7, '°'"
nn c-hcl,ow, C'�t.tw11n
•• /J(t)x 11m h 1hM •hi" Hnt•,.,i,td <,qu1U1011 I, of1tw- for,n t1 0 1n 1k nn.. C'O<Wdu'\11f't
ll I, t ,• 11y 10 .loe(' 1h i,1 tht' dtcht h111'ld 111idl" o( lht" ot1t11nal non linnr ..-qu,HIOft hat a ttto
0
11,war 1>1111 In 1h11 monhnatt ,�trm. I n race. IN • ,b. f Q.. a Ca, to that
Q o(lxl), C IJ • ), Making tlw..,. 11ub<11ttutiom in thf' rquation X "• •� �• 1M'
d1ffC'r!'ntial n 1uo1tion for••:
AC•, � Q .
Hut, by th e: ddinilion ofC, 1 ht·fi"t l<'f'tn, ffl1 tl1c- ln'1 and nght (th<" tf'nm lliw-ar 1n a1 an
equal, and hrn,c-
i, c- 1 Q(C•., ,, -<l•,11- 1
Combining 1..A""mm:u fi and <f, Wt" dNiuC'f'
1.�M"" 8. 1'Jr, 1elutN111 4th, dijfn11ttml rqu,ntion X •' •· I) tiutA • ,,-,,.,""'-Js-" .Jtt., c•
drP,ttdJ dtjf,u11t,r,h{, M tlt., ,,.,,;,,,, tMdttior,, n, ,,,,,. tiu .J" ,,,, •.,,. inM ,n,,,rt .. 11w,
,, ,:
ir,iti11I eond,litJr, ,11t,..fin IN IJStm, of'lflHlti(Jlf, ,f 1-w,Ml•u
X: = y(x, I). ' :(10) •c F., R• - R•.
Pux,j. Write 1h(" ..-qu:uin-n in 1hr- t"onrd in:uc 'tf'l('m ofLcmm:1 6 and thc-n �pply Lrmma ◄-1
To pron· cl1 (' th("Or"('m, Wf' now nrt-d only , m · fy tM conlinuJl y o(1hr dtti,.tti,"'C" ofthe
sol u tion with rnpt•c'I 1 0 lh( ini1ial c<>nditinn. /\c-C'Ofdmg to Lnnma 8. th.ii dn-i,,111,,r
'
exists and .,.-.,i�fo"\ th< · sysu·m or c-quatioo, <>r, ari�1tOM. ft follows from l...cnum. 1 th.at tbr
soh.11 ion� of 1hi:. s�tem dC'pt'n<I cnnhnuously nn x., a.nd , . and the- tbnwrm K fina.Dy
proved. I
5 Dlff'erentlal Equations on Manifolds
It will be rccalkd that the points of this space arc straigh1 lines passing
through the origin of coordinatc·s in R" • 1• Such a line is specified b y any or
its 1>0ims (other than 0). The coordinates of this point {x0, .r-1, ••• , x.) in
R"• 1 are <·alkd the homogtntnu.s cootdiualu of the corresponding poin1 o f
projectiv<· space.
The la.s1 example i s particularly useful. In comidcring the ddinitions that
follow, it will be useful to think in terms ofaffine <"00rdina1cs in a projective
space (see Sec. 33.3, Exaonplc 3).
R"I
tl:t'@/,@,z
Fig. 218 Exampin ofm:ioi(old s .
234 Chap. 5 Differ<:mial Equa1ions on l\1anirold,
({>,: w, - u.,
(Fig. 220). lf'1hc set• w,
and W1 intersect, 1hcn 1hdr in1crstttioo 11'1 n 11'1
has an image on both maps:
w.•
/ll·.
0
Ko�
c____0___
Fig. 221
cv__)
r'
Ju
St'po1rabili1y.
Fig. 222 Atlns or a sphf'rl!'. Th<' family or drdN on th.- spher e tang:mt al tbc point N
is r,:,prf:'Sc-nt('(i on the ll)Y.·cr map by a family orparallel linc<1 and on 1h.c- upptT" m.ap by a
ramilyorrnn,gcnl drdcs.
7
Fis. 223 Atlas ofa torus.
...,
\Z
z
consider the four maJJ5 obtained when O and tJ, vary i n the intervals
3) An alias for 1he projective plane RP' can be made up of1he following
--
three "affine maps" (Fig. 224):
•o J'1
x,
- X-,
o
.,, Xx, o
if Xo 'F 0,
xo :x1 :Xz �Z 1
X
=-,
o
x, ,, =-xx,, if .< I 'F o.
- Xo
"• - x,
-, ., �-x,x, if x, 'F o .
These maps arc compatible. For example, compatibility of ,p0 and "•
means ,hat 1he mapping cp0_1 of1he domain U0. , = {y1,y2:71 # O}ofthe
plane (y.,y2 ) onto the domain U, ,o = {z1, z,: z1 'F O} of the p�
238 C:hap, 5 Differtntial F,qua1lon1 on 111anif'okb
M
!I O Hf,
Proof. Let {G1) be an open cove ring ofthesc:t F . Then {q,1 {C1 n IV,)} isan
open cove ring ofthe compact set q, 1 (F1) for every i . Lettingj range ovu the
tesulting finite set ofvalues, we get a finite number ofC1 co,·cring F . I
33.5. Connectedoes• and dlmensi.on.
Defi11itio11. A manifold ,W is said to be coM«l<d (Fig. 228) if given any '""
pointsx,y e ,W, there ex ists a finite chainofmapsq,,: �1'1 - U,such that IV 1
contains x, l4'" containsy, W, n W1+ 1 Vi is nonempty, and U1 is connected.
A disconnected manifold .M decomposes intocon11«/td con,/»lfDW .\I�t
Probl.rm I . Arc lhc mani folds defined by
x1 + ,11 - :.2 - C
, C ,' 0
in R.l (in RPl) connccto:I?
Prol>Um 2 . ·me set of aU matrices of order ,a "";th nonzero determinants has ch,c natur11
structure of a diffel'entiablc: manifold (a domairt in R"..
). How m.i.ny coru,ecud com
ponent.s does lhis man ifold h�vc?
q,1 : W; � Ur
or
w,
•
st
.-
r �
-,-----
Fig, 231 A r-1.11'\'(' oo a enani(�d .\1.
bt its mop1. Tl,en all tht li,uar JfMrtS RII l'011.taini11.t th.t domai11.J t ·, lwtv /ill llllW
dimensirm.
dim R" :;; dim S" = dim T" :;; dim RP" ; n.
A dlscooncrtc-d 1nanifold i.s !-.aid 10 Ix 11-dimcnsional if all its conncc-1cd
componcncs have chc !-an1:c dim.en�ion n.
Src. 3 :J l)irTrre111iahl,· Mn11ift1ld, 2H
or
Pttbltm 3. l�,,ulp Il1t irt OfH) II II o, th()fCo.u,I m,t 11(n oford,. , ,, ""'1th thr unw,u.-,r of a
dlffr1t"nlfoltlf' 1111rnlfold. llmd ht ronnf"Htd co1ttJ>011f'nl• and ••• d1rnC'•11MOft.
A•s. O(•l ,fO(•) < 21, dim 01•) ,t(N Il
2
33.6. Differentiable mapping,.
/)efinitiou. Atuappi11M/: ,\I 1 _. .\/1 ofoue(,"-manifold into anotMr dsatd 1n
br d/Der"1tiabl, (ofclass C') if i1 is given by diff,�ntiablr fu1l<'1ions (of cl.us
C') io lfw local coordinales on Al I and .\f 1•
In 01lttr w o rds, ltt "• : 11' 1 - l11 hr a map or ,\I,. arting oo a ot1ghbothood o/ a pcMnt
x•Af1,andt11 : IY1-U1 a mapof.\11,actingOfla nr1ghbothoodof11poim/l'"I• It',
{F ig. 229). Tl 1('11 tlir Oli.lpping ofdoru:aiii. of Euclickan ,part .,,, ·/- •i' dffintd in a
ndghborhood of tht 1 >oin1 •1 (x) mwt be diffcrC'nliablC' of..-la• c�.
lixamplt I. Thr projcc1ion o f a <phcrr 01110 1h, plan, (Fig. 230 is a difl' r r·
c·ntiahlc mapping. Note lhal o clifrcrcnciablc mapping nttd no, cafT) a
clifTcrcntialJlc manifold into a dilferen1iablr rnanirold
C'xample 2 . By a eu,v,t on a manifold .\I k•aving the pnint x e .\fat time t0 8
meant a diff f'te1uiable mapping/: / - .\I or an interval/ of 1)1e: ttal /-axis
containing lhc point /0 iruo a manifold .\I such that/\lo) = x .
t-:.ramplt 3 . By a dijfto111orphis111J: ,\1 1 ... .\11 o f a manirold .\11 on1oa mani
fold �f1 is meant a differen1iablt' mappingf. whose invetw: •:
1- JI1 - .\I 1
exis1s and is cliffcremiable. Two manifolds .\fI and .\I2 arc said co �
dijfoomorphfr if there cxins a diffeomorphism from one on10 the oth(-r. foT
example, the sphere and the ellipsoid arc diffromorphic.
000 ...
f" ig . 232 No ndiffeomorphic ,....
·o,-dime nsional ma,,jfo,lds.
f Synonymous.ly, a pautntdriud tun�, since on�dimcnsional submanifolc:h (ddincd in
Sec 33.8) o(the 1nanifold Mare som,e1j1nd al.so c:1llcd cun·cs on , M . A p:an.mctrittd ai.n-c
csn h: we po,inL'I ofsel (,intersttti01l, cusps, c1c. (Fig. 231).
2•1 2 Chn11, 5 DifT,n:n1ial F..qua1ion• on !'-1anifold,
is an c,carnple ofa subset ofEuclidea 1'I space inheriting the natural structure
of a differentiable manifold from R>, namely 1he s1ruc1Ure o(a nd11tmfald
ofR>, The general dcflni1ion ofa sub manifold goes as follows:
The following fundamcn1al fac, is given wi1hou1 proof and will not be
used subsequently:
t A ma nifoldM i.s sa id to� simpt., �tmn«ILdif�-cry c:lo,cd cu.r,,c i:n 1'1 cat1 bt:cootirt--.sly
$hrun k to a point .
t Sec E. Brieskor-n, B�ispiL.k �"' Diffumtut.ltop,oJo� t'ttlllSUfgJan/4to,., Invcnt. �.f.ath.2( 1966).
1- 14,
Src. 34 'l'anll""' Bundle. V,c1or !'irids on a Manifold 243
Thus 1hc abstract concrpt of• manifold don not actually compri,c, a
larger daH of objects than' k•dimcnsional surfacH in N•di�nsionalspacc. 0
1
Thr aclva11tagc o fthe abs1rac1 approach i s that it includes 1hok cas,:s "h��
no crnbrdding in Euclidean space is s�ifi«:l in advanc,, and whn-c such a
specificaiion would only l<ad to spuriou, complication, (a, in IM ca5"ol'tM
projrctiv<' space- RP"). 'l'hc situation here is the same as for finjte--<li�n•
sional linearspaces (thry arf" all isomorphic to the coordinattspattofpoints
(x1, ••• 1 x,.) 1 but spr-ciCying coordinates ortcn mcrdy complicatesmattcn
33.9. £sample. 1-'inally we coniidcr 1ht' rono�•r\l fivt- in1ffnting ,natuf.okts (F"1g. 23-4,•
I) 1'ht gror;p M1 ,... SO(S) ef o,tho,onol "lotrictJ flH"-r 3 •-' .,_,,,,.JJ,.., -1. Si.nu C""l'
mairix or 1 H1 ha, 9 demenls, J\-11 it a sut�e, of the s:pat'e R•. It t1 uiy to titt tbt thK
subset i1 �iclually a svbrn,.ni(old.
""'°"
2) Tlit Jtl JWz - r,s: ef all vttlors ef fe.11111- I I# 11v sJl!Krt S1 in thr�
Euclidean space. As an cxc.rci�, th e rud er should introcfocc the structure ol a d1ffcr
cn1i ahl e rnanifold in10 . .\1l (t;'(. Sec. 34).
3) TJtt 1Jt,u-dimmsior,n( proj«tir., spnu ,\f, RP.I.
4) TM cmif,g11rnti1>n spact ,H,._ 11/a ri.iid Hd,1r.u1mcd at a fi.xrd Poin1 0.
,',) Th t subsd M, oJ llt t Jf>tJU RO ..- Rc.1ddtrtni.J 6.,1 t4t ftfW.tiMJ
z� + zj + zJ - 0,
1, ,I' + 1,,1• + 1-,1' - 2.
•P,ol>ftm I . Wl,ich of lh c m.anifold1 M 1,,,., Al, uc diffcomo,rphi c?
Wi1h every smooth manifold ,'>f there is associated another manifold (or
1wice 1hc dimension), called the 1a11ge111 bundle or ,'>land denoted by TAf.t
The whole theory o f ordinary differential equations can irnm«lfa.tdy be
carried over to manifolds, with t h e help ofthe tangent bundle.
I
�I�•CA/
,- 9'
9'��
Pig. 235 The rnngco1 ,·l"rtor.
w M
•
TM_..
:r: )-,
r---
M
IP
H
:r:
Fig. 238 t\ ta ngw1 bundl e.
In fa.ct, ('(HI.sider a ny map on •he nl:mifold �I, .and I -cl (x., . .•• i.): U'- Uc R•
(f ig. 237) he local coordin ates i n a n eighborhood 11' ofth e poin t x sptta(yiag this map.
1-:vcry veccor t ta ngent 10 ,\•! at a p0fo1 x., IV i.s determined by its components ( 1 ••••• (.
in the indicated coordin a te syst<"m. In fact, if 1: I- ,\Ill a cu.n,e le3,-in,g • in 1.h,c d ir « •
t ion of ( at t imt" t0, th en
(r *''
= t ,.,. x,(y(I)).
Thus C:\'UY vector-( tangent to ft1 at a point oftht- dom.a o It' is .1pffifio:I by 2a nu.mbcn
f •
x,, ...• x,., (1,•••• (,.. 1he" coordintitcs of the ..pOint ofta ngency .. a.nd the • -com
pone nts" {1• Thi s gi\•n a map ofpart of th e set TAI:
f'{() (z,,, ••IX , ( i,•••I(,.),
,.
Oiffcc: n t map s of T,\•f corresponding to diffttcnt nuP' of thie .atJu of .i.U arc com
patible (ofcla.u c,- • if,\1 i s ofclass C'). In fact, lcty,, ...• J. be anothtt loaJ roord:in.atc
$ys1em on ,W , and let ,,,, ...• ,,,. be the componcnlS ofa v� 1or- in 1his l) "Skffl - Then
)'r•.Y1(x1 ,••·•-",.), llr i:��J (i-l,...,11)
x,
J•li.:A'J
are .smoo1h functio n� of and(,- Thw the S<"I T.lfof all tangenc ,-utors &o .\I a<quira a
smooth ma nifo ld struct ure o f dimension 2n.
Definition. The manifold T,W is called the tang,nt bundk (spa,,) ofthe mani
fold ,11.
There exist natural mappings i: M-+ TAI {the 1t11ll s«tion) andp: T.\f....\f
(projection) such that i(x) is the zero vector of T.11, and p(() is the point x
a, which� is tangent to ,11 (fig. 238).
246 Chap. 5 l)iffcren1ial F,qua1iont on 1',1anifoldt
Pfflblt,n I. Prove that the mappings i and JI arc d1rfcrmt.1ablc. that i '--' a d1ft<Om01phl'lm
or M onto i(M), and that JI .. i: M - J\I is thc identity mapping.
The prtirnages ofthe points x e Munder the mapping p: T.\f - .If arc
callcdjibruofthe bundle TM. Every fibre hasthe structure of a linear space.
The set Mis called the ba,e ofthe bundle TJ\1.
34.3. Remarks on parallelizability. The tangent bundle of th<: affine
space R" or of a domain U c R" has the: structu� of a dittel product:
TU • U x R'. In fact, the 1ange111 vec,or 10 U can be specified by a pair
(x, (), where x e Uand ( is a vector of1he linear space R•, for which 1h<:rc
exists a linear isomorphism with TU, (Fig. 239). This can be expressed
differe111ly by saying that affine space ispara/lL/izablt, i.e., equality isddincd
for tangent vectotS to the dornain Uc R" at different pointsxand,-.
The tangent bundle 10 a manifold M need not be a direct product, and i n
general we cannot give a reasooablc definition or equality for vectors
"attached" 10 different points of M(Fig. 239). The situation h<:rc is the same
as for a Mobius s<rip (Fig. 240), which isa tangent bundle w;th a cin:lc asits
base and straight lines as its fibres, but which is not the direct product ofth<:
circle and a line.
Definition. A manifold Mis said to be paralltli:ud ifits 1ang01t bundle is ex
pressed as a di rec, product, i.e., if a difTcomorphism of TAf• � �,1• x R•
carrying T,l;f_, linearly into x x R' is given. A manifold is said to befJtUol
lelizable ifit can be parallelized.
ExQmplt I . A ny domain in Euclid ean space is naturally pualldizo:i .
Problem I . Pro,·e th :u the tonu T• is parallcli ublc, but not the- Mobius strip.
Src. 34 Tangent 8111,Jle. Vector Fields on a Manifold 247
•nlEOR8M. Only tltrttoftlte splterts S' are paralltluablt, namtlyS', S', wS1• I•
par,;tular, //11 lwo•dimensional sphtre is nonpo,.olltli table:
TS',;, S' X R2.
This implies, for example, that a hedgehog cannot be combed: At lea.st
one quill will b e perpendicular to the surface (Fig. 241 ).
The reader who has solved the problem at the end of Sec. 33.9 will find it
easy to prove the nonparallelizability of S1 (Hinl: RP3 1: S' x S1). -n,.,
parallelization of Sl is obvious, while that of S1 is an instructive cxcrclS('
(Hint: S' is a group, namely the group of quaternions of modulus I). A
complete proof of the above theorem requires a rather deep penetration
into 1he iul�ject o f topology; in fact, the theorem was proved only relati\·ely
recently.
Analysts arc inclined to regard all bundles as direct products and all
manifolds as parallelized. This miStake should be avoided.
34.4. The tangent mapping. Lctf: .\1 - N be a smooth mapping of a
manifold Minto a manifold N (Fig. 242), and lctf., denote th,, induttd
mapping of the tangent spaces. The mappingf., ( =f.l,) is defined a.sin
Sec. 6.3, and is a linear mapping of one linear- space into another-:
J•• : T1\1• - TAf<x >· (I)
Let x vary over ,11. "then (I) defines a mapping
f.: TM - TN, f.1,.... =f.,
of the tangent bundle of ,11 into the tangent bundle of N. This mapping is
246 Chap . 5 OilTerential Equation• on �tanirolch
TN
·-
r.
I'
Fig. 243 The tangent mapping.
ur�
x7-
\ '''
--
\
/of
---o ---
- M
$
A ,·r('1nr field.
differentiable (why?) and map< the fil>rn of T.lf linearly into the fibtts of
TN (Fig. 243).
The mapping f., is called the tan.enll mappin.e off (the notation
TJ: T iW -, TN is also u.-d).
Pm/)/tm I. 1�1 /: M - N ancl 1: N - /,,; he �uWWlth nt.apping,.. •;th <'OmpolitlOn
g ..f; :\1 - K . Prove that (t •f). t• .f., i.r•• that
N TN
s'
s
Fig. 245 A vdochy ritld,
·/ \·,w
/11-�4
t ·
is commutative, i.c.,p(v(x)) = x .
Remark. If 1\f is a domain of the space R• with coordinates x1, • • • , x.» thi,
definition coincides wi th the old one (Sec. 1.4). However, the prcw:nt de:fi.ni.
tion involves no special system of coordinates.
Example. C onsider 1he family of rolations g' or 1hc sphere S' through the
angle I abno1 1he axis SN (Fig. 245). Every poin1 or 1he sphere xe S' de
scribes a curve (a parallel oflatitucle) under 1hc ro1alion, with velocity
v(x)
d
& -
di ,.og'x e TS;.
This gives Ihe mapping v: S1 -+ TS1, where obviouslyp v = £, i.e., vis a
vec1or field on S'.
J n general , a one•paramc1cr group of diffcornorphisms g': Jf -- .\I ofthe
manifold /1/ gives rise 10 a vec1or field or1hc phale veloci1yon .If, in precisely
the same way a s in Sec. 1,4. The whole local theoryor (nonlinear ordinary
differential equations can immediately be carried o,rer to manifolds> since
we were careful a t the time (in Sec. 6) to keep our basic concc:pts iJ'li<k...
pendent o f th e coordinate system. In parti<'-ular, the basic local tlxott.m on
rcctifiabili1y o f a vcc1or field and 1he local 1hcoremo on existence, unique-
1'less,continuity,and differentiabilitywith respect to initial conditions carry
over to manifolds. ·rhc specific character of the manifold comes to thc- fott
only in considering nonlocal problems. The simples, or� problffllS
250 Chap. 5 OifTcrcntiol Equations on !11anirolcb
35.1. Theorem. lei A1 br a smooth manifold (of c/a,s C•, , ;i, 2), 9' ut
v: 1\,/- T,\,f b r a vr<tor fold (Fig. 246). Afortour, /rt tlw vrclor v(x) k '1furt1t
from th, ,rro V<Clor of T,W, onl., in a compacl subs,t K ofth, manif,ld ,\f. n,,,, 11tu,
exists a one-parameter group of dijfromorphisms g': ,\1 - A1 for u:/uclt v is dupl,as,
wlo<il.)!fold:
d
-g'x
dt
= v(g'x). (I)
. . ..
M
= =
--'= = ::L M
-
:;,:
g'•� • .ll'.lf7
i/111 < c, Isl < •• II + sl < •·
In fact. the poiot x has an image onsome map, and our asse,rtion has btt.n
proved for equations in a domain of affine space (sc:c Chaps. 2 and 4).t
Thus the compact set K is covered by neighborhoods U from ,-hich ""'
can selc<-1 a finite cov«"ring { Vi}. Let tj be the corresponding numbers e:, and
choose to a min c1 > 0. Then for ltl < c0 we can define diffcomorphisms
g': M -, /11 in 1/,e large such that g'x = x for x outside K and i'., = l'I' if
ltl, Isl, II+ sl < c0• In fact, although the solutions of equation (2) with the
initial rondi1ion x (fort • 0) defined by wing different maps arc different
a priori, ,hey coincide for Ill < to because of the choice of�o and the local
uniqueness theorem. l\forcovcr 1 by the local theorem on diffc�tiability,
the point g'x depends differcntiably on I and x, and s.incc ,,�-• = £, the
1
!!.
di ,=o
g'x = v(x). (3)
exisc. ,111<1 i� unique). The difrcomn,phi,m, �• ond x' haw alttady l>ttn
dcnnctl. Wriring II' •
(i'G/1)".(, w e get• difrcomoqihisrn or.If onto !of. f'or
ltl < c0/2 1hc new dcr.11i1lo11 .,grec, wirh 1h31 orS«. 35.2. and hence (3)
holtk Moreover, i1 i " rai)' 10 ,cc 1ha1
,t i •- 1.•1.' (I)
ror arbil rn.,·y .t and/.
In foc:1, lc1
'.
cu1r.1 ar(' pooible:
l)mln *• p I q •
2)11,ltt k - I, pIq
Dul the difft:0111011,hi�H" .(••"• K"• and t' roo1rnule". �n,t" f;I ,.2. lf1 Thn"•"2..
in,plit" (4) in boil, thr firs• and SCf'Ond caf<C" (1'1'1# ,•z• sin«- J;J. lf'1• :,1 ' -l•
p "9 . � � ,).
We must s1ill vcriry that lhc point .f:'x dcJXnds diffcrcntia.bly on t and x .
This follovv", for example. from the fart tha1 i
= (g' 1�'\'V, while f' 'x dc--
pencls c.Jiffere 'l1 1iably on t and x rnr �unicie:,uly large N, by S«. 35.2.
Thu1' {i} is a one-parameter group or diffcorttorphisms o(the manifold
1\1, and V is rhc corn,,ponding field or 1hc phase "clocil)'- The pmol' of
Theorem 3:·,. I i� nr,w complc1e. I
Fig, 2i7 Exu:n.sion of:. lint'�r \'tttor fidd onto proj«IJ\t' t.»i«,
tRP""'
'-' I I I
''\ I /
I/ .½
k = 2, ..., n .
i=l, ... ,n
254 Chai), 5 OifTcrcntial Equations on �ian,folch
di - -)', (•11 +
V', •>'EI ...,.),
�, a,.,.,, -Y4 a + 'E 11.1,),
dy ,
dt • 011 + ,;- 11 II k >I
( I> I
i11 the new coordinates (Fig. 248). From thnc formulas, valid for.11 ,t, 0, it is
clear how to complete the definition of the field atJ, - 0. For.,, • 0 we
get dy,/dt • 0, thereby proving the lemma.
Solution 2 . An affi11e transformationcan berrgardcd as a ptojcctive tran.d"or•
mat ion, leaving the plane at infinity (but not its pointJ) fixed. In panicuia.r.
the linear transformations t141 can be extended to diff'cornorphismsof proj«
tive space leaving the plane at infinity fixed. Thnc difTcomorphisrns form a
one•parnmeter group, with v' as its phase velocity field.
through any singular points of the field (Fig. 249), and suppose, a point
makes one circuit around the curve in the J>05:ltive dirtttion. Then t.Jx fidd
vector at the point in question will rotate continuously as the point move,;
around the curve.t When the 1>0int returns to its original position, having
I
8
6
.5
fig. 249 A eurvc ofindex I.
gone around the curve, the vector also return.s 10 its original position, but in
doing so, it may make several revolutions in one di�ction OI' the other. Tht
number of revolutions made by the field vcc1or in travening thecunc once
is called the index of the curve. Herc the number of revolutions is taken with
the plus sign ifthe vector rotates in the direction specified by the orientation
of the plane (from the first basis vector to the second), and with the minu,
sign otherwise.
Example /, The indices ofthe curves"'• P, 7, and cS in Fig. 250 arc I,0, 2. and
- I respectively.
Example 2 . Let O be a nonsingular point of the fidd. Then the index ofc--ery
curve lying in a sufficicntly small neighborhood ofO equals zero. In fact, the
direction of the field at O is continuous and hence changes by less than 11/2,
say, in a sufficiently small neighborhood of0 .
Problem I. Suppose Wt specify a vector fidd in the plane R' - ac ... ;thout
the point Oby the formula v(z) • r', where n is an intcgtt "•hich is not
necessarily positive. Calculate the index of the circle z = ,,..oriental in the
direction ofincreasing IP (the plane is oriented by the frame I, i).
Am. n.
36.2. Properties of the index and their implications.
PROPERTY I. Tk index of a closed curue do,s not change ,md,r ,� tkf-o
tion, as long as thecurve dou,u,t go through an)' singular poinl.S.
In fact, the direction of the field vector changes continuously away from
the singular points. Therefore the number of rcvolu1ions also dcpcnch
continuously on the curve, and hence must be constant, being an intcgc.... I
PROPERTY 2. The index'!/a curve does not change U1UUTcon1inuou.stkfomuzti.,.oftk
vtctorfold, provided onry that there au no singular points oftkfold OIit1teaoT<dt,ri,,g
lk wholecourse 'Iftk d<formatio11.
256 Chop, 5 l)ifTcrcn1ial Equa1ions on 1'1anirold•
T11ROR6M I. Ch�r, a W<IOrfitld i11 th, plant, It/ D bf• tir<•I•• dulc •-'S111 ...,,,,.
••)'• t/J lht indtx of tht '"""Sis nonuro, tlt,n ,,.,,, is 01 ltoJI .,., 11,w,t,,, ,,...,
i,uidt D.
LftMMA. l,t v b, the vet/orfold in th, plaM of the compux variabk z �,,.,. b:, IN
formula
v(z) = z" + a I z"- r + · · · + n,.,
so that lht singularpoinls ofv arejusl lh, root.of1q,,a1t011 (I). Tit,,, the iadex in th,
fitld v o f a circle ofsujfidtnl/:, lar.�t radius equals n . §
same in the original field as in the fie ld t". But the index equals• i n the field
•"· I
/>roofof T/1tortm 2 . Let r be the same as in the proof of the lemma. Then, by
Theorem I 31'1d 1he lcr1una1 thcr-c is at lca11 one singular point ofthe v«tor
fie ld, i.e., at least one root of equation (I), inside the disk of radius,. I
l'UEOREM 3 (Fixed point theorem). Etvry s,,.,.11,t mappint f: D - D •f•
singular point O do,s not d,p,nd OIi tlu radius •Itlu ,fr,u, provid,d MCIJ lMl""' ,.,r,.,
is sujft<ient!� small.
Proof Any two such circles can be continuously deformed into each other
without going through singular points. I
Note alJo that im1tead or a circle, we can choose any other curv� going
around O once in the positive dircclion.
Definition. The index of any (and hence every) sufficiently small poouh·ely
oriented circle centered at an isolated singular point ofa vector6e:ld iscalled
the index efthe sin.�ularpoint.
Examplts. Suppose the singular point i s a node, saddle point, or focus (or
center). Then the index of the singular point i s +I, - I, or + I rcpecti,dy
(Fig. 252).
A singular point of a vector field is said to Ix simple ifthe operator ofth<:
linear part of the field at the point is nondcgcneratc. The class ofsingular
point1 in the plane consists of nodes, saddle points, foci, and cxntcn. Thus
the index ofsuch a singular point is always +I.
P,�l,m I . <..:onstruct a \. CC
' IOt field wi1h a singular point o(indcx "- ·
or
Proble,n 2 . Pro11 e that lfle index of a si.ngvl. a r point isindq,cndcnt of thtcboitt:oforicn t 1 -
tion the p lane.
Hint. Changi ng lhe orientation simultan('Ol,l,S(y changes both the p<IUlRT dirtt.tion of
travcning the c ircle and the positive direction o r counting the number of rC"-dvtions.
36.4. Index ofa curve in terms ofindices ofsingular points.. Ltt D be
a compact domain bounded b y a simple curve S in the oriented pb.nc.
Suppose S has the standard orientation ofthe boundary ofD, i.-,., sup�
D lies to the left of an observer traversing Sin the J>O$ilive direction. This
means that the positive orientation of the plane i s given by th<: dihedral
made up of the velocity vector along S and the normal vC'Clor dircacd
inside D.
Sec. 36 Index ora Singular Poi111 ofa Vector f'i<ld 2.S9
Now suppose we arc given a veclor field in 1hc plane, wath no t1n"ul1r
poi1H1 011 the curve: Sand ouly a finite number or singular pornts mqdc the
domainD.
1 IIV.ORP.M, Th, ir1d,x rif the'""' S ,qua/, th, sum of tll, mdtCtJ ofllw Jl1'l"I••J»t•IJ
of th,j,,ld IJ•i1111 i11,itf, D.
First we prove that the index of a curve hu the followi"'! additivity
property:
u:MMA, Cit.1tn two oriented cun·t.s y I and y2 goin1, tltrou1.lt 1l11 somt1»i1t1, Ul 71 + 7 1
bt the ntw oritnltd turt't obtained by trat1trsin1fi,st y I ond afl«u:c,ts 71• T1wa tltt
indexrify, + y1 ,qunl,1h,s11mof1/re indicuofy,andy1•
Proof The field vector makes n 1 turns in going around y1 and •1 more ,urns
in going around y2, and hence 11 1 + n 2 turru in all. I
P,oofof//,e th,orem. We 11artition D into parts D1 such that thett i• no mott
1han one singular poin1 of the field imidc each part (Fig. 2.S3), and no
singular point.s at all on the boundaries of Lhc parts. �1orco\·cr, we assign
each of the cutve5 y, boundiog 1he parts Di the orientation appropriate to
1hc boundary (Fig. 253). Then, by the lemma,
ind LY, indS + Eind 61,
&
I ;
where the closed curve 61 is made Uf) of a part of the boundary ofD1 lying
inside D and i s traversed twice in opposite dircclioos. The index or each
curve 01 equals 0, since 01 can be contracted into a po int ";thout passing
through singular points (sec Sec. 36.2). The index of the cun.·e 71 cqua.ls the
index of the singular point surrounded by y1 (or O if the domain D1 s u r
rounded hy y, contains no singular points). I
P,oh/,m I. Lee f>(:) be a pol ynomial of degr« • in a compla: ",aria.hie z. and kt D be a
domain in the z-pla ne bound«! by a curve $ . Suppost there are no un:,,. of the poly.
nomial on S . Prove lhat the number orzeros of the �ynomial inside D (wich multipli
cities la.ken into accounc) equals the index of the cunre Sin lhe field ,.. = J(z). i...e:, lht
II
fig . 2�'.3 The index of che curve S equal s t.he wm of the indtCCS ofth e cu.f''CS 7, a.od 7
2 •
260 Chap, 5 Differential F,quatiom on t.lan,foldt
numbt-r 6f' revol ut ioru (wiodlng numl>tr) or 11\(c-urvt Jll(S) around tlw on11n.
Com ,.,, ,.,, Thi, 81 vo II way or .olvlng the Rov1h,Hurw111. pmbltffl ol Stt. tS4: FvJ 14,
n1m6 n. q/ u,us ,ifIIiir:,,11 po(J,, ,m'lfl {,. tA, l(ft ..
1/-;IMt. To thlt- md. cOftMdn-a ha lf •
di,k or 1uffirlt 1ly largt nidlu1 In the lr-0 hatr-plane- with 111 «ntrr at th«- point , 0
1 u o 'At
n
and lu dl,une1.:r 111011g the irm13in1ry uii. � numbtr of u·roa in th<- l,rt\ haN'•plllM
�1u11I, the indu or 1he boundary So( the hlll(-d11k ( i r 1he radius II largt" <nouch and ,/
chc: polynorn li.l h1 u no purtly Imaginary terot), To nkulatt the lndn o/ 1hir � S.
we nt"t"t'I only find lht' number o( revolu1ioo1 " around 1hc On1in of th(, """I"' ol tk
i ma gi nary axi l (oriented fro,n i 10 +I), J r, rate. it "tas.,ly "'f:rifiNj Wt
•Problem I. Prove 1hat 1hc index ora singular point ora vector field in IM
plane is invariant under a diffcomorphism.
Thus the index i s a gcomciric concep1 which i s indcpcndmt oftM coo r
dinate system. This fact allows us to define the index ora singular point not
only in ,he plane but a l so on any two-dimensional manifold. In fac'l, we
nee. cl only consider the index ofthe singular point on anymap, and the: index
will then be the same on lhe other maps.
s
Fig. 2S4 A vector field on th� sphere withtwo singular points ofinda 1.
Sec. 36 Index oro Singulor Poin1 ora Vec1or Field 261
Suppos,· we ore giv,·n • vec1or field on 1he sphere wilh only otolatcd
singular pflinu. Thrn 1hrrc arc only• Oni1r number of such pomu, .,.,.,� the
sphere- is c·ompncl.
•·ro180•11... Th, s11m efthe indic,s ef•iitht 11•,t•l•rpo,nll ef•jirltl • 11,, sp/tnt 11
i11dep,11d,111 of//11 thoit1 ofth,fald.
his rlcar from 1he above example thatthrss•m tq••b 2.
/d,n efthe proof Consider a map of the sphere covering 1he whole sphere
cxccp1 for one poin1, which we call 1hc pole. Then consider the fitld ol'the
basis vector e 1 in the Euclidean plane of1hi.s map, and carry the field over to
1hc sphere. This gives a field on 1he sphcrc (defined except at the pole) which
we continue 1o dcno1c by e 1 .
Now con,idcr the map of a neighborhood of the pole. In the plane o{this
map wr can also draw 1hr vcc1or field e 1 on thesphen; defined cxttpt atoM
poin1 0. The appearance of this field is shown i n Fig. 255.
LIOUt
, A. Tht i11d1x ofQ cloud turvt RDim: onu aroundtht point O u, t4t Jll•NTfold
juJI eo,utructtd equals 2.
Proof \¥c need only carry out explicitly the operations described abcn�,
choosing for the two maps, for example, maps of the spherc under stereo
graphic projection (fig. 222). Parallel lines on one map then go into the
circles shown in Fig. 255 on the second map, from which ll is dear that the
index equals 2. I
Completion o f1h, proof Consider a vector field v on 'the sphtte, choosing a
nonsingular point of the field as the pole. Then all the singular pointsofthe
field have images on the map or the complement of the pole.The sum ofthe
indices of all the singular points of the field equals the index of a circle of
sufficiently large radius in the plane or this map (by Theorem 36.4). \\le now
carry this circle over to the sphere, and then from the sphere to the map ofa
neighborhood ofthe pole. The resulting circle on the lauer map has index 0
in the field under consideration, since the pole is a nonsingular point ofthe
------------·
·--------
J-'lg , 2-'6 On every Mand 1he a.urn o( the nurnbtt ol' peaks and 1hr nu.mbf,r of valkys b
I grca1e r 1h :rn 1he 11umbtr o(paU("I..
field. S1aying on 1he new map, we can interpret the index or a circle on the
finl map as the 11umbcr or revolutions o f the field v rtlotitw t• tlttfi,tli e1" in
11
goi ng once a round 1he circle. This num bcr cquals + 2, sinceas �c go around
the circle surrounding the point O on the new map in the positivcdittc-tion
for 1hc first map, the image field e1 o n the n<"w map makrs -2 N:'\'olutions
while 1hc field v rnakcsO revolution,. I
• Prob/,-,1t 2 . l,c:1/: S' , R1 \)(' a 11-noo1 h runc tion on tM aphttc-, all oJ..._.tao., cntinl points
a. re simple (i.e., whoac 1e<ond diffC'rcnti al is nondcgcncn.tc a l n·ny ("'nua.J poantJ. Prou·
that
m0 -m1 + m,: • 2,
where m1 is 1he m.imbcr of' critical poi ,Hs whose Hessian matrix (i;Zf!ZzP,) bu i ncptn-c
eige nval ues. I11 01 hcr words, tlit numlNr tfrtrini.. ..;- , IN __,. .J.M.r ,,.._, Jll,a di,
numbt, ofmaxi,no alwo)'S tqtMJls 2.
For example, th e 1otal number of mou 1nai11 pctb oo euth plus tbr total nW'l'lbtt of
v al ley s i s 2 greater th an the numbc·r of pa.ssn . If we restrict ou.neh·o to an nh.od or a;
c:on1ioent, i.e., itwe coos.ider rv11c1io ns on " diik with no sil'\gular points on i1:1 boundary�
1h en ni0 - m1 + ,nl ,, 1 (fig. 2:.6).
Hint. Coosider the gradil'flt of the fonc1iooj.
•P,oJ,lmt .1. Prove Eultt's llrlonm on pol;Jrtdra, wh.ich asscru that
a0 -a1 +a,-2
for every bou•�ded (Onvbc p<>lyh«tron with a0 \·cntCes, tr 1 edges., and ol &as..
Hint. ih is problem c an be reduced 10 1he pre«ding problem.
•P,obltm I. P,or.v: that lht sum X of tht indius ef IN sut,pl.dr jHlutlS .j • ttd# JtU • olfll!1
' msiOMI niani/old f! itul,pmdmt of11\tfol.d .
compael two,.dim
The numbei- x in question is call(() 1heEutl r dtoroftnistiic of thf: manifold.. For cumple.
we have ju se s«n th .it 1he Evler ch ar.acreristic z(Sz) of the sphere equals 2.
ProJ,U,n 5 . fi nd th e £uler characteristic: ol 1he torus, o/the pretzel, and oftbr � •-ith
n ha ndles (Fig.232).
An.,. 0, -2, 2 - 2n.
•f>ro61mt 6. Extend the results of Problems 2 and 3 from the sp.hctt to aAY cunpact 1w1>
dime,uion al mao ifold, i.e.• prove 1ha1
m0 - m, + ml • a0 - tr1 + a, = x(M).
Sec. 36 l11dcx of a Singular Poin 1 of a Vec1or Field 26S
where S, is a circle going around the ith singular poinr in the. positi,-c: direc-
1io n (Fig. 257). Applyi ng Green 's formula to the domain lY and the
264 Chap. 5 OlfTcrcn1ial Equa1ions on �lanifolch
The lefi-hand side vanishes, since 1he form (2) is locally a total difl'cnntial.
Bu11henindS - EindS1,becauscof1hedefinition(3). I
•1>,01)/111, J , Pl'ovc thal the indu ora dOltd cun·� i1 an in1qtt.
•Probkrn 2. Give cornplc1e proofs or the aMCt1iOn1 in Stts. 36.1·36.3.
= sgn dct A - (- I ) • -,
deg-.., A
where rn_ is th t" number oteigenvalues of the opt"nuor A with a negatn--c real pat'L
Sec. 36 Index ora Singular Point oro Vector Field 26.S
con,idcr the embeddingor the ncgotive real axis in the run real axi1.)
Jlunark 3. The number or poin11 In the prc:imagc (without rqp,rd for sign)
con be dilfcrc:nt for difTcrc:nt regularvalues (for example, in fig. 258 thc
valucy ha! of ur such points, whiley" has prc:ciscly two).
Defi11itiori. Dy the dt,trt• efthe mnppi11J:fis mcant the sum of the dcgrttS ol'/at
all the pointsof the preimageora rcgularvalucor/:
deg/• L
x•f • 11
deg,/
"'
P,obt,m S . J, i n d th e d eg r« of th e mappi ng of th e circle l.zl - I onto itd si'ttft by 1ht
r0tmula/(z:)- r.'\ n • O, ±1, t21 • • ,
Ans . n,
P,ob/,,,, 6. Fi n d 1hc d<"grce or 1he mappi r\g of 1ht' unit ,p� i n Eudido.n � R• cmto
itsdf g i ven by th e formul.- /(,.) Ax/lAxl, whtt'c A: R• - R• is .A nomir.,du Ii.nor
Ol>fflllOf',
Ans. dt:. g/- sgn dN JI.
Pto6l,m 1 , Fi n d the d e gree o f 1hc mapping of 1hr complex projective liaw CPt onso itsdf
given by 1hc formula
•)/(:) = :•; b)/(:) = ,·•.
Ans. a) 1•1; b) -1•1•
Probttm 8 . Fin d the degrtt ofthe mappi n g ofthe- comple x li ne- CP1 011\o itsdt&""m by ai
polyn omial of d egree n .
•Problnn 9. 1.A:1 f: U' - S1 be th e mappi n g constructed in Stt. 36.6 with lhot hdp of a
\'(ClC)r field v in a domain u·, Ice y: s• - U' bc ack,scdcurve.and let A-/• 7:S' -s•.
Definition. By the index ofan isolated singular point O or a vector field v defined
in a domain of Euclidean s p a c e R"' containing O i s meant the: degree orthe
mapping h corresponding to the field, i.e.,of the mapping
h(x) = rv(x) .
lv(x)I
P,obltm 10. Provi e that if th ie operato r vu or the linear part o/ the 6dd vat a sangubr
poi nt O has an inverse, th en th e indc,c or O equ.ils the dcgr« of•••·
Probl�m II. Find th e in dex of th e singular point O o( th ie field in R.• corrcspoadit1g IO thie
«ruati on i- -Jt.
Ans. (-1)".
Sec·. 36 Indcx or a Singular Point or a Ve, tor field 267
Fig. 2$9 Li nea riiat ion of a differe nti a l equatio n on a sphere near its singub..r paints-
268 Chap. 5 OilTen,n1iol Equations on 1',fanifolcb
Hr.nee the: north l)Olt hat indt'x (-1)• tlld 1hc: .ou1h po� hat ;ndu (-t 1)-♦ t0 tM,
x(S-) • I I (-1)".
In p11rlituh1t'. h foll ow, 1h11 mr, f-rctfl /lfld ,_ • rm,,.Ji,,.,,.,..,.,/
"'1,,r, ..... ., ,,_.,, _,
,,,.,,.,,,, polNI,
11,obl,m J!J , Conururt • vc:tlor fidd without Mn1ular poinu on th� odd..4f.VMftMOn.ll
1phert SJ... ',
/lint. Con.sider the scrond-ordc:r diffcrcnth1I «tU�tion R • x , ,r • R.•.
Sample Examination Problem•
I.To ,1op a boa1 a1 a dock, a rope is 1ht0wn rrom 1hc boat which is then
wound around a po,1 auachcd 10 1hc dock. Wha1 is 1hc braking lot-tton the
l >on1 ir 1hc rope makrs 3 10r11s around 1hc po11, irthe coefficient offriction of
1hc rope around 1hc po11 isl, and ir a dockworker pullt at the free end of
1hc ropr with a rorrc orlO kg?
2. Consider 1hc mo1ion
� • 1 + 2 sin x
or a pendulum subject 10 a cons1an1 1orquc. Draw phase cu,,...,. or the
pendulum on 1hc ,urfocc or a cylinder. Which mo1ions of the �ndulum
correspond lO the variou.! kinds of curves?
3. Calculate the matrix,..,. where A is a given matrix oforder2 or 3.
4. Draw 1he image of 1hc square lx,I .;; I, lx,I.;; I and the trajectory of
the phase now or 1hc system
x, - 2x2:,
arter time,.
5. Find 1he number of digi1, required 10 write 1hc hundred1h term of the
sequence I, I, 6, 12, 29, 59, , ..(x,.. = x.- 1 + 2x._ 2 + n, x1 = x2 = I).
6.Draw the phase curve oft he sys1em
X= X - y - z, j = X + )',
going through the point ( 1, 0, 0).
7, Find all a, p, y for which the three functions sin a..t, sin /Jt, sin yt att linearly
dependent.
8. Draw the trajectory ofa polnt in the plane (x 1 , x2) cxccutingsmalJoscilla
tions
au U = �(5x/ - 8x 1x 2 + 5-'1),
X; = - Oxr,
subject to the initial conditions
XI = I'
9. A horizontal force of 100 gm lasting I sec acts o n an initially s-uuionary
mathematical pendulum oflength I m and weight I kg.Find the amplitude
(in cm) of the oscillations which result after rhc force ceases 1oacL
269
270 Sample t:x•mina11on Problcrm
I 0 . I nv1•,1lgn IC the l.)'npu nov <UIbilhy orthe null solution or.� ')'lem
{ (, • x1
,
x1 • -w 1, r1 ,
( -{
"'1)
0.4 ror2kn < 1 < (2k + I)•,
0.6 for (2k - l)n < I < 2kn,
k • 0, ±1, ±2, ...
11. Find all the ,ingular points orthcsystem
)' � .r' + 1' - 2).
lnvcstigate the stability and determine th<" type: of <"ach singula, poin1, and
cJraw tlw eorr'cspondi,1g pha-.crurvc�.
12. Find all singular points of the system
j = sin x + sin:,
on 1he toru, (.r mod 2n,.Y mod 2n). Investigate the stability and determine
the type of each singular 1><>in1, and draw the corresponding p� cu,,,cs.
13. It is known from experience that when light is refracted at the iotttfacc
between two media, 1he sin� of the angles formed by the incident and
refracted rays wi1h rhc normal to the interface arc inversely proportional t o
the indices of refr;1ction of the rnc..'1Jia:
sin a 1 n2
sin a: 2 ;;; n, ·
Find 1he form of 1hc light rays in the plane (x,. 1) if the index nfrefraction
is n • n(J).S1udy 1hc case 11(1) • I/)' (the halr,planey 0 with this index
>
of refraction gives a model of Lobachevskian gcomet,r).
14. Dr-aw the rays emanating in different directions from the origin in a
plane with index ofrcfrac1ionn = n()') = ,• -1' + I.
Commrnl. The solution of this problem explains the phenomaion o( the
mirage. The index of refraction of air over a desert has a maximum at a
certain height, since the air is more rarefied a t higher and low-er (hot) layers
and the index of refraction is inversely proportional to the ,,clocityoflighL
The oscillations of theray near the layer with maximum index of rcfract.K>n
is inlerpreted as a mirage.
Another phenomenon explained by the same kind of ray oscilla,;ons is
that of acoustic channels in the ocean, along which sound cao be propagated
for hundreds ofkilome1ers. The reason for this phenomenon is the interplay
of temperature and pressure leading to the formation of a layer of maximum
index of refraction (i.e., minimum soundvelocity) al a depth ofS00-1000 m.
Sample l':xaminalion l'roblcnu 271
Bellman, R., Stal>ilit,y 'f'lt,1t1J .J D18nt11ti"I £qt1•lfMJ, MtCraw 1till, Nf"W Vo,k (19").
4
Index
class of, 35
Andronov, A. A., 94 n
Atlas(es), 235
equivalent, 235
Auto-oscillations, 22
Auto-oscillatory regi111e, 22
B
Base of a tangent bundle, 246
Beats, 184
Cardano's formula, 32
Cauchy sequence, 98 n
Clairaut equation, 67
Class er,§
Con1111utator, 75
Con1plexification
of an operator, 120
of a space, 120
Configuration space, 79
theore111, 212
Convex do,nain, � n
CorioIis force, 66
Curve(s)
tangent, 34
co1nponents of, ll
Deco,nplexiIi cation
of an operator, 120
Derivative
of a curve, 122
of a mapping, 37
Determinacy, l
Detenninant
of a matrix, ill
of an operator, ill
Diffeo1norphism, 6
of a manifold, 241
Differentiability, l ,20 11
Differentiability theore1n,g
Differentiable function,§
co11J1ected,239
diffeo1norphic, 241
finite-di1nensional, 6
parallelizable, 246
parallelized, 246
Page 276
Differentiable n1apping, 6, 241
Differential algebra, 32 n
Differential equation(s)
basic theore1n on, 48
integration of, 32
on 1nanifolds, 233-268
nonautonomous,28-33, 56-59
solution of,�
extended, 1l
Differentiations as mappings, 74
Direct product
of sets, 5
Direction field, ll
Directional derivative, 73
Eigenbasis, I 02
rnultiple, I 67-176
Ernde, F ., 193 n
Energy
kinetic,�
critical, 90
noncritical, fil
potential, 80
total,�
Existence theore1n, 50
Exponential
of a con1plex nu,nber, I 08
of a diagonal operator, I 02
of a 11 ilpotent operator, I 03
Extension of a solution, 53
backward, 53
of equation of order n, 62
forward, 53
up to a subset, 53
First integral, 75
local, 77
ti1ne-dependent, 78
Fixed point
of a t7ow, 2.
of a mapping, 20 I, ill
asy,nptotically stable, 20 I n
theore,n, 257
r-ocus, 127
stable, 127
unstable, 127
Functors, ill
G
Galileo's law, 9
Graph of a ,napping, 5
Page277
Hadamard's lelllna, 85
Half-life, ].Q
Hon1eo1norphisn1, ill
I
hnaginary plane, ill
Index
of an oriented closed curve, 255, 263
lnfinitesi1nal generator, I 07
lnvolutory operator, 12 I
Jacobian matrix,�
Jacobi's identity, 75
Kan1ke, E., 32 n
Kepler potential,�
Khaiki n, S. E., � n
Konstantinov, N. N., l
Level set, 76
Lie algebra, 75
stable, 93
unstable, 93
co111plexified, 129
nonhornogeneous, 179
Linear operator(s)
cornplex conjugate of, 122
nonn of,�
space of, 98
Linearization, 95
Linearized equation, 96
Lipschitz condition,2.1.
Losch, F., I 93 n
Lyapu11ov stability,155,201 ll
compatible, 235
i1nage on a, 234
Page278
Method of Euler lines, I 09
Metric, 98 n
Metric space,� n
co 1nplete, � n
convergence in, 98 n
Morse's lern1na, �
Nali1nov, V. V ., 1§. n
of order n, 193
Norm
of a vector, 215
One-para1neter group
of diffeo1norphis1ns, 2., 20
of transfonnations, 4
Oseillations
forced, 183
free,183
Parallelizability,246
Para1netrized curve,241 n
closed,§ 2 72
-
1naxi1nal, 69 n
Phase tlo,.v(s),I,l, .1
equivalent, 141
differentiably, 141
linearly, 141
topologically, 141
fixed point of, 1
local, iL
Phase points, 3, 4
1notion of, 4
Phase space, l, 2, 3, 1, 12
cylindrical, 21.
extended, 5, ll
Phase velocity, 1
co,nponents of, 1
Poisson bracket, 75
Polyno1nials
Process
detenninistic, l
differentiable, l
evolutionary, l
finite-di,nensional, l
Projection, 245
degree of, I 03
exponent of, I 03
space of, I 03
Page 279
It
Radioactive decay, � 9,
- 1§.
,·-differentiability,§.
Rectifying coordinates, 49
of order k, 119
Resonance, 183
parametric, 204
s
Saddle, 153
Saddle point, 25
Scparatrices, 1!i
Series of functions, I 00
differentiation of, I 00
Set of level C, 76 n
Shilov, G. E., 98
Silvennan, R. A., 98
Singular point(s), 8
si1nple, 258
i n space, 139-140
on a sphere, 260
stationary, ll
Sphere, 233
Stability
asyn1ptotic, 156, 20 I n
Lyapunov, 155, 20 I n
strong, 203
Subrnanifold, 242
T
t-advance ,napping, .1 12.
(I 1, t2)-advance 1napping, 57
Tangent mapping,248
Titne shift,�
Torus
111-din1ensional, 166,267
two-di1nensional,ill, 233
Trace
of a matrix,ill
of an operator, ill
u
Uniformly distributed points, 166
Uniqueness theorem
local, 50
V
Vandermonde determinant, 195
Variation of constants,208-210
Vector field(s)
delinition of, l
linearized, 95
on 1nanifolds, 248
Vector integral, 2I 6
Velocity vector, 33
Vertical fall, 9
deflection fron1, 66
Vitt, A. A., 94 n
w
Weierstrass' test, I 00
Wronskian