1 Definitions Trapping region - A region is trapping if along its

boundary the vector field is pointing inwards or is

Algebraic multiplicity - The algebraic multiplic- tangential to it. Since it is also an invariant set,
ity, A (i ), of an eigenvalue from matrix A is then it must also satisfy V 0.
the largest integer k such that ( i )k divides Attractor - An attracting set, A is defined to have
evenly the polynomial expression for the charac- the following properties:
teristic equation, i.e. it is the number of times a
given i is repeated 1. A is an invariant set.
Geometric multiplicity - The geometric multiplic- 2. A attracts an open set of initial conditions.
ity, A , of of an eigenvalue from matrix A is the There is an open set U containing A such
maximum number of linearly independent eigen- that if x U , then the distance from x(t)
vectors associated with i . It is the dimension of to A tends to 0 as t . A attracts all
the nullspace of (A i I), also called the nullity trajectories that start sufficiently close to it.
of (A i I), which is related to the dimension as: The largest such U is the basin of attraction
of A.
A = n rank (A i I)
3. A is minimal. There is no proper subset of
where n is the dimension A. Also, A A (i ). A that satisfies conditions (1) and (2) above.
If A < A (i ), then the matrix, A, is said to be
degenerate -limit point/set - The -limit point, (x), of the
Equilibrium point - The point x at which x = 0 trajectory, (x), is the point that the trajectory
so that x(t) = x t tends towards as t . The -limit set, (),
is the set of all -limit points.
Hyperbolic equilibrium - An equilibrium point is
called a hyperbolic equilibrium point if none of -limit point/set - The -limit point, (x), of the
the eigenvalues of f (x ) have zero real part. No- trajectory, (x), is the point that the trajectory
tice that the hyperbolic property is defined based tends towards as t . The -limit set, (), is
on the linearised approximation of the non-linear the set of all -limit points.
system. Orbit/Trajectory - An orbit is also known as a
Positively invariant set - An invariant set is a set trajectory, (t), that describes the evolution of a
of values in <n space such that starting within particle starting anywhere in the state space.
the set when t=0 guarantees that the trajectory Cycle - A cycle is a closed orbit. The closed orbit
remains bounded within the set for all t. may be periodic, such as limit cycles and non-
Let U be the invariant set, U <n . If x(0) U , linear centres, or non-periodic, such as homoclinic
then x(t) U t > 0. If an energy-like func- orbits and heteroclinic cycles, since it would take
tion, V (x), describing the system is known, then an infinite amount of time to complete a cycle.
the positively invariant region in state space is the Homoclinic orbit - A trajectory that joins a saddle
region that satisfies V 0. point to itself, by moving out along the unstable
An invariant set is a loose term. For instance, any manifold and returning along the stable manifold.
equilibrium point is an invariant set. A trapping The unstable manifold is the -limit point while
region and domain/region/basin of attraction of the stable manifold is the -limit point.
an equilibrium point are also invariant sets. A Heteroclinic orbit - A trajectory that joins two
trivial invariant set is the whole state space. For different equilibrium points. The unstable equi-
an autonomous system, any trajectory in state- librium point is the -limit point while the stable
space is an invariant set. Since limit cycles are equilibrium point is the -limit point. If a se-
special cases of system trajectories (closed curves quence of heteroclinic orbits form a closed orbit,
in the phase plane), they are also invariant sets. then it is a heteroclinic cycle.
Domain/Region/Basin of attraction - The do- Separatrix cycle - A separatrix cycle partitions
main/region/basin of attraction, U , of an attract- phase space into two regions with different char-
ing set A is the set of initial conditions x0 U such acteristics. Note that a separatrix cycles formed
that x(t) A as t , i.e. any point starting from homoclinic or heteroclinic connections are
in the basin converges towards the attracting set. not periodic since it takes an infinite time to go
It is the union of all points satisfying V 0. from an -limit point to an -limit point.

1
Non-linear centre - A family of closed curves that 4.1. If A < A (i ) for any i (A is degenerate),
form a continuum of periodic cycles. These cycles then try a component solution of the form:
are stable but not asymptotically stable.
xi (t) = tei t vi + ei t w
Limit cycle - An isolated (not a continuum) peri-
odic cycle with a defined radius. It can asymptot- where w is an unknown vector that needs to
ically stable, unstable or semi-stable. For periodic be determined. Substituting this into Eqn.
orbits to be stable, they must satisfy: (2.0.2) and equating like terms gives:
T
(A i I) w = vi


f (t) dt 0
0
which can be solved to find w.
2 General Method 4.2. If i is complex, then vi will also be com-
plex, with the conjugate pairs being i and
vi . The pair of complex eigenvectors can be
For a general non-linear system:
replaced with real eigenvectors.
x = f (x) (2.0.1) If i = i + ji and vi = ai + jbi , then the
component solution is:
1. Solve:
xi (t) = ei t ai cos(i t) bi sin(i t)


x = 0
+ jei t ai sin(i t) + bi cos(i t)

to obtain equilibria, x .
Combining this with the complex conjugate
2. Linearise Eqn. (2.0.1) to obtain: component solution, xi (t), gives two real so-
lutions:
x = Ax (2.0.2)
p(t) = ei t ai cos(i t) bi sin(i t)


where A is the Jacobian matrix of f (x), i.e. A =
q(t) = ei t ai sin(i t) + bi cos(i t)


f (x).
3. Evaluate the Jacobian, A, at an equilibrium 4.3. A can be normalised to obtain its normal
point. form, A0 , by changing coordinates. Consider
The Jacobian, A, represents a new curvilinear co- Eqn. (2.0.2) and let y = V1 x:
ordinate system based on the gradient of the func-
tion, f (x). Evaluating the Jacobian at the equi- x = Ax
librium point therefore gives the gradient of the y = V1 AVy
function locally at the equilibrium point. This is
the tangent space. so:
The new local coordinate system can be decom- A0 = V1 AV
posed to find eigenvalue-eigenvector pairs. This
gives its invariant directions and tells us about 5. Analyse stability of the equilibrium point by
the local stability of the equilibrium point. looking at its eigenvalues.
4. Decompose A to find its eigenvalues and eigen- If the equilibrium is hyperbolic, we can conclude
vectors at the equilibrium point. on the stability of the original non-linear system
using the Hartman-Grobman Theorem. However,
Eqn. (2.0.2) has a solution of the form:
if the equilibrium is non-hyperbolic, the Hartman-
At
x(t) = e x0 Grobman Theorem doesnt apply so we cannot use
P it to conclude on the non-linear systems stability.
where x0 = i ci vi where vi is an eigenvector of
A. In the linearised approximation, a non-hyperbolic
equilibrium is a centre. But the topology of the
This can be re-expressed as:
non-linear system at the equilibrium point is not
X necessarily a centre. We cannot make this conclu-
x(t) = ci ei t vi (2.0.3)
sion since we cannot apply the Hartman-Grobman
i
Theorem - which only works for hyperbolic equi-
see (8.1) for proof. libria.

2
6. To investigate the stability of non-hyperbolic in the linear system. Returning to Eqn. (2.0.2)
equilibria in the original non-linear system, we and changing coordinates:
have the following options:
x = Ax
Polar coordinate transformation = VV1 x
Symmetry V1 x = V1 x
Hamiltonian/Gradient System z = z
Lyapunovs Direct Method where T, , and z are defined as follows:
LaSalles Invariant Set Theorem
. . ..
h .. i h .. i h . i
7. To further analyse the topology of the non-
vju
s
T= vi vkc
linear system, especially when determining the ex- .

.. .. ..
istence limit cycles and separatrix cycles, we have . .
the following options: h i
s
Polar coordinate transformation i 0
h i
0
uj

= 0 0
Poincare-Bendixson Theorem h i
ck

0 0
Bendixson/Dulac Criterion

Gradient Systems zs
ui
Index Theory z = zj
zck
h i
s
i 0 0

3 Invariant Sub-Manifolds zsi h i
s
zui
u
uj

zj = 0 0 zj
h i
This section introduces the concept of invariant zck zc
ck

0 0 k
manifolds and their relation to the stable, unsta-
ble and centre subspaces.
We see that the solution of the stable, unstable
Recall that the general solution to the linear sys-
and centre systems of equations (zsi , zuj , zck respec-
tem is given by Eqn. (2.0.2):
tively) are decoupled from each other. Hence, each
X system of equations is able to completely describe
x(t) = ci ei t vi the behaviour of the flow within its corresponding
i
subspace. Therefore, the subspaces is decoupled
where each component, i, can be grouped under from each other and are therefore invariant.
the categories: stable (s), unstable (u), or centre For example, consider a saddle node, which has
(c). This gives: both stable and unstable directions. The eigenvec-
tors corresponding to the stable and unstable di-
x(t) = ws (t) + wu (t) + wc (t) rections are contained in ws and wu respectively.
These eigenvectors are separatrices which divide
where ws (t) is the stable subspace, E s ; wu (t) is the phase plane into stable and unstable regions.
the unstable subspace, E u ; wc (t) is the centre sub- Trajectories are unable cross separatrices and are
space, E c . therefore contained within either the stable or un-
Each subspace is formed from the linear combina- stable region.
tion of eigenvectors which span said subspace, e.g. The Hartman-Grobman Theorem states that in
ws = E s is formed from the linear combination the case of hyperbolic equilibria, there exists a bi-
of vis - the eigenvectors corresponding to stable continuous function, H, that maps a sub-manifold
eigenvalues - each of which span the subspace, E s . (non-linear surface with curvilinear coordinates)
Having identified the subspaces present in the so- containing the equilibrium, to a subspace contain-
lution, x(t), we must now show that these sub- ing the origin (linear surface with rectilinear co-
spaces are invariant w.r.t. to the flow, eAt . We ordinates), such that trajectories are mapped ex-
therefore need to decouple the system of equations actly and parametrization of time is preserved.

3
Thus, the invariant subspaces E s , E u , and E c map which shows that it is invariant under the transfor-
to invariant sub-manifolds. mation. The system is therefore symmetric about
the x1 axis. Hence, the origin is a non-linear cen-
tre.
4 Stability of Non-Hyperbolic
Equilibria
4.3 Conservative Systems
If an equilibrium point in the linearised approxi-
mation is non-hyperbolic, we cannot conclude that A conservative system possesses an energy-like
this equilibrium point in the original non-linear function, V (x), which is preserved along the tra-
system is a centre, because the Hartman-Grobman jectories of the system. The requirements a con-
Theorem does not apply. servative system needs to satisfy are: there exists
a non-constant V (x) : <n < with dVdt(x) = 0.
Instead, further analysis is required using the tools
below: If the system has an isolated equilibrium at x
(there are no other equilibria in a neighbourhood
around it), and one can construct such a V (x)
4.1 Polar Coordinate Transformation with a local minimum or maximum at x , then
there is a region around x which contains a closed
This transforms the coordinate system of the lin-
orbit, i.e. the trajectories around x evolve on
earised system from Cartesian to Polar Coordi-
level sets of V (x).
nates. Note that this only works for a 2D system,
<2 :
x1 x1 + x2 x2 x1 x2 x2 x1 4.3.1 Hamiltonian Systems
r = =
r r2
where r gives the radius of the trajectory mea- Systems of the form:
sured from the equilibrium point.
p = f (p, q)
4.2 Symmetry q = g (p, q)

A 2D non-linear system is symmetric w.r.t. the where p, q <n for which there exists a twice-
x1 axis if it is invariant under the transformation differentiable function, H(p, q) : <2n < for
(t, x2 ) (t, x2 ) and vice versa. some set U <2n such that:
Likewise, the system is symmetric w.r.t. the x2
axis if it is invariant under the transformation, H(p, q)
(t, x1 ) (t, x1 ) and vice versa. f (p, q) = (4.3.1)
q
If a system is symmetric w.r.t. one of the axes, and H(p, q)
g (p, q) = (4.3.2)
if the origin is a centre in the linearised approxi- p
mation, then the origin is a centre in the original
non-linear system. Consider the system: are called Hamiltonian Systems with n degrees of
3
x1 = x2 x2 , f (x1 , x2 ) freedom. Note that all Hamiltonian Systems are
conservative. See 8.3 for the derivation of Eqn.
x2 = x1 x22 , g (x1 , x2 ) (4.3.1) and (4.3.2) above.
and test it under the transformation (t, x2 ) If it can be shown that the system is a Hamil-
(t, x2 ): tonian System by finding H(p, q), then we have
dx1 also found the possible trajectories of the system
= x1 - since the trajectories evolve along level sets of
d (t)
H(p, q) which form closed orbits.
d (x2 )
= x2 We can therefore conclude on the trajectory and
d (t)
stability of the system. For example, if (p , q ) is
f (x1 , x2 ) = f (x1 , x2 )
an equilibrium and H(p, q) > 0, then this equilib-
g (x1 , x2 ) = g (x1 , x2 ) rium is stable.

4
4.4 Gradient Systems 3. V (x) is negative definite

Systems of the form: then x is asymptotically stable. If additionally,

x = V (x) (4.4.1)
4. V (x) is radially unbounded (U = <n )
where V (x), is a twice-differentiable function de-

fined on some set U <n , are called Gradient then x is globally asymptotically stable.
Systems with n degrees of freedom.
Eqn. (4.4.1) shows that trajectories are perpendic- 4.6 LaSalles Invariant Set Theorem
ular to level sets of V (x) and that the equilibrium
points of Gradient Systems correspond to critical It is difficult to find a a true Lyapunov function
points of V (x). that satisfies condition (3) - strict negative defin-
So if x is a strict local minimum of V (x), then ity - as it is usually only negative semi-definite. So
V (x) V (x ) is a Lyapunov function for x which Lyapunovs Direct Method cannot be used to con-
is asymptotically stable. If x is a strict local max- clude on asymptotic stability. However, Invariant
imum of V (x), then x is unstable. Set Theorem can be used to conclude on asymp-
totic stability.
Note that a Gradient System is the dual of a
Hamiltonian System. A level set in a Gradient Besides often yielding conclusions on asymptotic
System is an orbit in a Hamiltonian System. Like- stability when V (x) is only negative semi-definite,
wise, the gradient of a Hamiltonian System is the the invariant set theorems also allow us to extend
trajectory of a Gradient System. It can be shown the concept of Lyapunov function so as to describe
that the trajectories of both systems are orthogo- convergence to dynamic behaviours more general
nal; see 8.2. than equilibrium, e.g., convergence to a limit cy-
cle.
Note that V = H simply means that the Hamilto-
nian System described by Eqn. 4.3.1 and 4.3.2 is The invariant set theorems reflect the intuition
a dual of the Gradient System described by Eqn. that the decrease of V (x) has to gradually van-
4.4.1. They are not the same systems, i.e. they ish (i.e., V (x) has to converge to zero) because
do not have the same solutions/trajectories. V V (x) is lower bounded.
and H are simply statements of energy at the Let V (x) : <n < be a continuously differen-
same point x (if V = H). The dynamics are then tiable function defined for some set U <n in
described Eqn. 4.3.1 and 4.3.2 for a Hamiltonian the neighbourhood of x . Assume the trapping
System and Eqn. 4.4.1 for a Gradient System. region, D, exists which is defined as:

1. bounded by V (x) V0 for some V0 > 0

4.5 Lyapunovs Direct Method
2. V (x) 0 x(t) D
This involves finding an energy-like function,
called a Lyapunov function, V (x), similar to the
Hamiltonian, H(p, q), except the former exists Let R = {x : V (x) = 0} and M be the largest
for both conservative and dissipative systems, invariant set in R. Then, every solution x(t)
whereas the latter only exists for conservative sys- originating in D tends to M as t .
tems. The Lyapunov function can be used to con-
clude on the non-linear systems stability. In the above theorem, the word largest is un-
To be classified as a Lyapunov function, it must derstood in the sense of set theory, i.e., M is the
fulfil the following criteria. Let V (x) : <n < be union of all invariant sets (e.g., equilibrium points
a continuously differentiable function defined for or limit cycles) within R. In particular, if the set
some set U <n in the neighbourhood of x . If: R is itself invariant (i.e., if once V (x) = 0, then
V (x) for all future time), then M = R.
1. V (x) is positive definite
Note that LaSalles Theorem assumes the exis-
2. V (x) is negative semi-definite tence of the invariant set, D. Note also that V (x)
is not a Lyapunov function. It is not required to
then x is stable. If additionally, be positive definite and V (x) is only required to

5
be negative semi-definite. If V (x) is positive def- which has (scalar) gradient:
inite then it is a Lyapunov function whose level
x2
sets can be used to generate D. m(x) =
x1
The trapping region, D, can be found indirectly
or directly. which can alternatively be derived by considering
the gradient of the trajectory on the phase plane:

dx2
4.6.1 Indirect Trapping Region m (x) =
dx1
dx2 dt
The indirect method is to simply use the function, =
dt dx1
V (x), that satisfies the conditions in Lyapunovs x2
Direct Method (4.5) or LaSalles Invariant Set =
x1
Theorem (4.6). The local region where V (x) sat-

isfies the conditions is the trapping region. The noting that m(x) = m (x) , where (x), is a
condition required on the boundary of the trap- curve representing the trajectorys path in the
ping region is V 0. phase plane.
This is because V (x) = V (x) x - see (8.4) The limiting condition to be satisfied for C to be
for proof - which means that V (x) is the rate of the boundary of a trapping region is:
change of potential, V (x), along the system trajec-
m(x) = m(C)
tory. So V (x) decreases or remains constant over
time. And since V (x) is bounded over this region x2
= m(C)
(it is possible to draw closed curves of V (x) = V0 ), x1
the trajectory must point inward or tangentially x2 mx1 = 0
to the boundary of the region.
where m = m(C). But more generally,
Note that V (x) describes how V (x) changes along
system trajectories (a scalar). It is not the gradi- x2 mx1 0 or x2 mx1 0 (4.6.1)
ent of the potential function (a vector).
where the choice of inequality depends on the ge-
Alternatively, proving that the trajectories point ometry. To determine which inequality applies,
into the trapping region is sufficient to show its draw the tangent to the curve at some point, x,
existence. This can be done in Cartesian using x1 with gradient, m(C). Then draw vectors of x
and x2 or polar coordinates using r. pointing into the trapping region at x. Only one
of the inequalities can be true.
In 3D, the bounded region is defined by some
4.6.2 Direct Trapping Region
closed surface. At any point on the surface, the
trajectory must point inward or lie tangential to
The direct (or graphical) method would be to
the surface. The exact condition required is as
analyse the vector field of the phase space and
follows:
construct some bounded region defined by a char-
acteristic set of vectors (or vector function), y. A Let the outward normal of the surface be, n. Then
trapping region is defined as having a vector field the following must be satisfied:
that is pointing inwards or tangential to it.
x n 0 (4.6.2)
In 2D, the bounded region is defined by some
closed curve, C. At any point along C, the tra- i.e. the trajectory is either tangential to the sur-
jectory must point inward or lie tangential to C. face or points into the region bounded by the sur-
The exact condition required is as follows: face.
Let the gradient of the curve at any point be
m(C). The direction vector of the trajectory at 5 Closed Orbits and Limit Cy-
any point is given by:
cles
!
x1
x = We wish to further analyse the topology of the
x2
region far-from-equilibrium and which may cover

6
a number of equilibrium points. In this scenario, 5.2 Bendixson/Dulac Criterion
the equilibrium points will interact, which gives
rise to behaviours which differ from those of the This criterion is used to show the non-existence of
original equilibrium points. The tools below allow periodic orbits for 2D systems:
us to analyse this behaviour.
x1 = f (x1 , x2 )
x2 = g(x1 , x2 )

5.1 Poincare-Bendixson Theorem The Bendixson Criterion can be stated as:

f g
If x + x 6= 0 and does not change sign over
The Poincare-Bendixson Theorem considers the 1 2
the region, D <2 , then there are no closed
2D phase plane, i.e. <2 , and allows us to deter-
orbits in D.
mine completely and exactly the asymptotic be-
haviour of systems on the plane, i.e. what attrac- The Dulac criterion can be stated as:
tors are present on the plane. The theorem is as
follows: Let B(x1 , x2 ) be a continuously differentiable
Let x M , where M is a positively invariant re- function, defined on a region D <2 . If
Bf Bg
gion of a vector field, containing a finite number x1 + x2 6= 0 and does not change sign over
of equilibria. Consider (x), the -limit set: D, then there are no closed orbits in D.

f g
It may be that x 1
+ x 2
= h(x1 , x2 ), in which case,
f g
1. If M only has stable equilibria, then there it is possible that x1 + x 2
= 0 for some (x1 , x2 )
can only be one, and (x) is a stable equi- and a change of sign occurs across this point. In
librium point this case, the phase plane can be split into different
regions, Di , where each region, Di <2 , covers an
f g
area of the phase plane where x 1
+ x 2
6= 0 and
2. If there are no stable equilibrium points in no change of sign occurs. So each Di is defined by
M , then it must contain a periodic cycle, the boundaries where h(x1 , x2 ) = 0.
and (x) is a periodic cycle
But note that it is possible for a closed orbit to
exist over unions of Di , such that it is not con-
3. If M contains a finite number of equilibria, tained within any one region. Instead, it crosses
then (x) is a connected set composed of over them.
a finite number of equilibrium points con-
nected with homoclinic and heteroclinic or- 5.3 Gradient Systems
bits. Mathematically, it will have orbits, ,
with () = xi and () = xj Gradient systems which have the form given by
Eqn. (4.4.1):

where condition (3) is read as: the alpha and x = V (x)

omega limit points are the equilibrium points. cannot have closed orbits. This can be seen by
Note that by considering the -limit set rather taking the dot product of both sides with x:
than -limit set, by definition, the attractors in
x x = V (x) x
the -limit set are necessarily stable, i.e. sta-
ble equilibrium points, stable spirals, stable limit |x|2 = V (x)
cycles and the stable manifolds of saddle nodes. dV (x)
= |x|2
Non-linear centres, unstable nodes, unstable spi- dt
rals, unstable limit cycles and the unstable mani- which can be integrated to give:
folds of saddle nodes form the -limit set. T T
dV (x)
Applying this theorem relies on the existence of dt = |x|2 dt
0 dt 0
the positively invariant region, M . So M must T
first be defined by finding the points that satisfy

V x(T ) V x(0) =


|x|2 dt
V 0. 0

7
and if a closed orbit existed, then x(T ) = x(0), so: 3. Let I(C) encircle an equilibrium point, x .
If the equilibrium point is a saddle node,
T
then I(C) = 1, otherwise I(C) = 1. We
|x|2 dt = 0
0 identify the index of an equilibrium point as
x = 0 t [0, T ] I(x ).

which means that the point cant have moved, so 4. If I(C) follows a closed orbit of the system,
x(0) was a fixed point, not an orbit. then I(C) = 1

5. If I(C) encircles a number of fixed points,

5.4 Index theory then
X
I(C) = I(xi )
Index theory is used for determining far-from-
i
equilibrium behaviour of a system, and is partic-
ularly useful for 2D systems, i.e. <2 .
6. If I(C) follows a closed orbit which encloses
It makes use of the fact that, for a 2D system: a number of equilibrium points, then
X
x1 = f (x1 , x2 ) I(C) = I(xi ) = 1
x2 = g(x1 , x2 ) i

if such a closed orbit exists, e.g. a limit cy-

we can write:
cle, separatrix cycle, etc. This follows from
x2 g(x1 , x2 ) properties (4) and (5).
=
x1 f (x1 , x2 )
dx2 g(x1 , x2 ) Index theory can therefore be used to identify the
= existence of closed orbits by letting C follow the
dx1 f (x1 , x2 )
proposed closed orbit, using the properties above,
which is the gradient of the vector field at location and the additional property that trajectories can-
(x1 , x2 ). The angle that the vector field makes to not overlap (unless at an equilibrium point) so C
the x1 axis is given by: cannot cross over other existing trajectories.
!
g(x1 , x2 )
(x1 , x2 ) = arctan 5.5 Poincare Map
f (x1 , x2 )
Poincare maps are useful for studying swirling
and taking the line integral of the change in flows and its stability as they convert problems
along some closed curve, C, gives the index of the about closed orbits into problems about equilib-
curve, I(C): rium points, i.e. a 2D problem is reduced to a 1D
problem.
1
I(C) = d(x1 , x2 )
2 Consider an n-dimensional system given by Eqn.
(2.0.1). Let S be an n 1-dimensional hyperplane
which can be read as: the total change in angle which is transverse to the flow, i.e. all trajectories
that the vector field along the curve, C, makes starting on S flow through it, not parallel to it.
with the x1 axis. From its definition above, the
index has the following properties: The Poincare map, P , is a mapping obtained by
following the intersection of trajectories with S. If
xk S represents the kth intersection, then the
1. I(C) is always an integer.
Poincare map is given by:
This is because varies continuously since
the vector field is smooth, and the angle at xk+1 = P (xk )
the beginning and end of C is the same. So
the net change in is an integer multiple of and suppose that a periodic orbit passes through
2. S at the point x . Then a trajectory starting any-
where on the periodic orbit will always intersect
2. If there are no equilibria enclosed by C, then the point x . So the periodic orbit in <n is an equi-
I(C) = 0 librium point of P (i.e. P (x ) = x ) on S <n1 .

8
The stability of the periodic orbit (or equilibrium For = 0, the two points collide to form a
point of P ) can be analysed by looking at the be- saddle point. For < 0, the saddle point
haviour of P near x . The condition to be satisfied vanishes.
for a stable limit cycle is:
2. Transcritical bifurcation - This has the nor-
mal form:

dP 
<1 (5.5.1)
dr x=x
x = x x2

whereby, for > 0 there are two equilibrium

6 Bifurcations points, one stable and one unstable. For
= 0, the two points collide to form a sad-
A bifurcation is a change in stability properties, dle point. For < 0, the stable and unstable
e.g. number and/or type of equilibrium points as equilibrium points emerge, but the stabili-
the governing system equations are modified by ties are swapped, i.e. an exchange of stabil-
means of a bifurcation parameter, . The general ity properties has occurred between the two
non-linear system can now be rewritten as: equilibrium points.
x = f (x, ) (6.0.1) 3. Pitchfork bifurcation - This has the normal
form:
where x <n and <m . Note that is a
parameter, not a variable. It is constant for a set x = x x3
of differential equations. Changing changes the
type differential equations and therefore changes whereby, for < 0 there is only one equi-
the nature of the system and its solution librium point. For = 0, the equilibrium
breaks into 3 equilibrium points.
6.1 One-Dimensional Bifurcations
Note that these bifurcations can occur for a sys-
A bifurcation in 1D space can only occur when the tem of any number of dimensions. The bifurca-
conditions below are met: tions will occur on a one-dimensional subspace of
the higher dimensional system under study. Fur-

f (x , 0 ) = 0 (6.1.1) thermore the condition imposed by Eqn. (6.1.2) is

f  generalised as:
=0 (6.1.2)
x x ,0  !
f 
det =0
x x ,0
where x is the equilibrium point and 0 is the
value of for which the real part of

the eigenvalue i.e. the determinant of the Jacobian evaluated at
of the system is zero, i.e. < (f ) = 0.
the equilibrium point is equal to zero.
Eqn. (6.1.1) states that a bifurcation can only oc-
To see why this is true, consider Eqn. (6.0.1),
cur at an equilibrium point and at a specific value
which has eigenvalues given by:
of = 0 . Eqn. (6.1.2) states that a bifurcation
occurs when the eigenvalues of the system (assum- det (A v) = 0
ing that they are real) are equal to zero, which
occurs when = 0 . Graphically, this means that where A is the Jacobian of f (x , 0 ). But at
bifurcations occur at the turning points of f (x, ) = 0 , = 0 by definition of 0 .
as a result of varying the system equations.
There are different types of bifurcations: 6.2 Multi-Dimensional Bifurcations

1. Saddle-node bifurcation - This system has While the bifurcations listed in 6.1 can exist for
the normal form: a system of any number of dimensions, a Hopf bi-
furcation can only exist in 2D systems.
x = x2
Hopf bifurcations occur for systems which have
whereby, for > 0 there are two equilib- complex eigenvalues whose real part is a function
rium points, one stable and one unstable. of the bifurcation parameter. Hopf bifurcations

9
are therefore characterised by a shift in the loca- Deterministic means that the system has no
tion of the eigenvalues from the LHP (stable) to random or noisy (probabilistic) inputs or parame-
the RHP (unstable) and vice versa, with zero real ters. The irregular behaviour arises from the sys-
part when = 0 . tems non-linearity, rather than from probabilistic
driving factors.
1. Supercritical Hopf bifurcation - Consists of a Sensitive dependence on initial conditions
stable spiral when < 0 , bifurcating when means that nearby trajectories separate exponen-
= 0 to become weakly stable, then be- tially fast, i.e. the system has a positive Lyapunov
coming an unstable spiral with a stable limit exponent.
cycle when > 0 .
Chaotic systems exist in both discrete-time sys-
2. Subcritical Hopf bifurcation - Consists of a tems and continuous-time systems. Note that
stable spiral surrounded by an unstable limit discrete-time systems can be chaotic even if it is
cycle when < 0 . As approaches 0 , the 1D.
limit cycle collapses onto the stable spiral,
rendering it unstable. For > 0 , the equi-
librium point is now an unstable spiral. 7.1 Discrete-Time Systems

3. Degenerate Hopf bifurcation - Consists of a 7.1.1 Maps

stable spiral when < 0 , which becomes
a non-linear centre when = 0 , and then See handwritten notes on finding equilibrium
an unstable spiral when > 0 . There is no points, determining their stability and drawing bi-
limit cycle at either < 0 or > 0 , hence furcation diagrams and orbit diagrams
the name degenerate.
7.1.2 Cobwebs
6.3 General Method
See Lecture 8 of C24 Dynamical Systems. Need to
1. Identify equilibrium points and limit cycles be able to navigate around a cobweb and develop
as functions of . a qualitative understanding of why some systems
can exhibit more exotic behaviour such as chaos.
2. Calculate eigenvalues of the linearised ap-
proximation at x as functions of . If the
eigenvalues are real, then the the bifurcation 7.1.3 Lyapunov Exponent
will be similar to one of cases (1) - (3) above.
If the eigenvalues are complex, then it will How do we know whether the aperiodic behaviour
undergo a Hopf bifurcation. of the logistic map is chaotic? This requires sen-
sitivity to initial conditions. We can describe this
3. Classify the type of bifurcation based on the mathematically using the concept of Lyapunov
tangency conditions; see Lecture 7 of C24 Exponents.
Dynamical Systems
Let the k-th separation from a previous initial con-
4. Describe behaviour of the system over a dition is denoted as k , if |k | |0 |ek for some
range of for which the eigenvalues are neg- > 0, then the system is chaotic. If this is the
ative, zero and positive. case, the Lyapunov exponent can be calculated as
depicted in the notes; see Lecture 8 of C24 Dy-
namical Systems.
7 Chaos

Chaos is defined as: the aperiodic long-term be- 7.2 Continuous-Time Systems
haviour in a deterministic system that exhibits
Characteristics of the Lorenz Equations are:
sensitive dependence on initial conditions.
Aperiodic long-term behaviour means that 1. Non-linearity
there are trajectories which do not settle down
to equilibrium points, periodic orbits or quasi- 2. Symmetry - Mapping (x, y) (x, y)
periodic orbits as t . gives the same equations. Hence, if

10


x(t), y(t), z(t) is
a solution, so is
x(t), y(t), z(t) . So all solutions are
either symmetric themselves, or have a sym-
metric partner.

3. Volume contraction - The Lorenz system is

said to be dissipative, i.e. volumes in phase
space contract under the flow. To show that
a system is indeed dissipative, it must fulfil:

V (t) < 0

where V (t) is the volume of some region in

phase space, not the Lyapunov function.
To obtain an expression for V (t), pick an
arbitrary closed surface S(t) with volume
V (t) in phase space.

Now consider an infinitesimal surface, dA

with outwards normal, n. Since f is the
instantaneous velocity of the points, f n is
the outward normal component of velocity
from dA. Therefore in time dt, a patch of
area dA sweeps out a volume, (f ndt)dA.

Hence,

V (t + dt) = V (t) + (f n) dtdA
S
V (t + dt) V (t)
= (f n) dA
dt
S
V (t) = (f n) dA
S

and rewriting using the divergence theorem,

V (t) = ( f ) dV
V

so V ( f ) dV < 0

11
8 Miscellaneous Proofs and Using both Eqn. (8.1.1) and (8.1.2):
Derivations x(t) = eAt x0
 X
1 2 2
= I + At + A t + . . . ci vi
2
8.1 Proof that x(t) =P eAt x0 can be i
rewritten as x(t) = i ci ei t vi X X t2 X
= ci vi + t ci Avi + ci A2 vi
2
i i i

eA t
X X t2 X
Taylors expansion of gives: = ci vi + t ci Avi + ci A2 vi
2
i i i
X X t2 X
= ci vi + t ci i vi + ci 2i vi
2
i i i
 
1
X 1
eAt = I + A + A2 t2 + . . . (8.1.1) = ci vi 1 + i t + 2i t2
2 2
i
X
= ci ei t vi
i

Also, x0 can be written as a linear combination of

8.2 Proof that trajectories in Hamilto-
the eigenvectors of A: nian Systems are orthogonal to tra-
jectories in Gradient Systems

8.3 Derivation of Hamiltonian relation

in Eqn. (4.3.1) and (4.3.2)
X
x0 = ci vi (8.1.2) 8.4 Proof that V (x) = V (x) x
i

12