D26e.

D26e.
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Dynamical Systems
M. R. E. Proctor
DAMTP, University of Cambridge
Michaelmas Term 2011
c
M.R.E.Proctor
2010
(These notes may be copied for the personal use of students on the course)
Dynamical Systems: Course Summary
Informal Introduction
Need for geometric and analytic arguments. Fixed points, periodic orbits and stability.
Structural instability and bifurcations. Chaos.
Chapter 1: Basic Definitions
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1.1 Notation
Phase space, order of a system, autonomous ODEs, maps.
1.2 Initial value problems
Examples of non-uniqueness and finite-time blow-up.
1.3 Trajectories and flows
1.4 Trajectories, orbits, invariant & limit sets
Orbit/trajectory, forwards orbit, invariant set, fixed pt, periodic pt, limit cycle, homoclinic
& heteroclinic orbits, ω- & α-limit sets, N -cycles.
1.5 Topological equivalence and structural stability of flows
Chapter 2: Fixed Points in R2
2.1 Linearization
The Jacobian A.
2.2 Classification of fixed points
Canonical forms. Saddle point, stable/unstable nodes, stable/unstable foci, stellar node,
improper node, centre, line of fixed pts. Hamiltonian systems. Effect of linear perturbations.
Hyperbolic and nonhyperbolic fixed points. Sources, sinks & saddles.
2.3 Effect of nonlinear terms
2.3.1 Stable, unstable and centre subspaces. Stable and unstable manifolds. Stable manifold
theorem. 2.3.2 Nonhyperbolic cases.
2.4 Sketching phase planes/portraits
General procedure. Worked example.
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Chapter 3: Stability
3.1 Definitions
Lyapunov, quasi-asymptotic and asymptotic stability of fixed pts and invariant sets.
3.2 Lyapunov functions
Lyapunov function. Lyapunov’s 1st theorem, La Salle’s invariance principle. Domain of
stability, global stability. General method for Lyapunov functions.
3.3 Bounding functions
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Chapter 4: Existence and Stability of Periodic Orbits in R2
4.1 Poincaré index test
4.2 Dulac’s Criterion (& the divergence test)
4.3 Poincaré–Bendixson Theorem
**Proof** and examples
4.4 Near-Hamiltonian flows
Energy balance (Melnikov) method
4.5 Stability of periodic orbits
Floquet theory. Hyperbolic and nonhyperbolic orbits. Use of ∇ · f to find stability in R2 .
4.6 Example: van der Pol oscillator
Chapter 5: Bifurcations of Flows
5.1 Introduction
Bifurcation, bifurcation point, bifurcation diagram.
5.2 Centre manifold and extended centre manifold
Centre manifold theorem. Evolution on the centre manifold. Extended system.
5.3 Stationary bifurcations (λ = 0)
Normal forms: saddle-node, transcritical, subcritical pitchfork & supercritical pitchfork bifurcations. Classification and reduction to normal form. Structural instability of the transcritical and pitchfork cases.
5.4 Oscillatory/Hopf bifurcation (λ = ±iω)
Normal form for subcritical and supercritical Hopf bifurcations.
5.5 **Bifurcations of periodic orbits**
2
Chapter 6: Fixed Points and Bifurcations of Maps
6.1 Introduction
Logistic, tent, rotation and sawtooth (shift) maps
6.2 Fixed points, cycles and stability
Definitions. Stability. Nonhyperbolicity for maps.
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6.3 Local bifurcations in 1D maps
6.3.1 λ = 1: saddle-node, transcritical, subcritical pitchfork & supercritical pitchfork bifurcations. 6.3.2 λ = −1: period-doubling bifurcation.
6.4 Example: The logistic map
Chapter 7: Chaos
7.1 Introduction
Definitions of SDIC and TT. Chaos (Devaney). Horseshoe. Chaos (Glendinning).
7.2 The sawtooth map (Bernoulli shift)
7.3 Horseshoes, symbolic dynamics and the shift map
7.4 Period-3 implies chaos
7.5 The existence of N -cycles
Period-3 implies all periods. Sharkovsky’s theorem.
7.6 The tent map
Attracting set. Rescaling of subtents for F 2 . Chaos for 1 < µ ≤ 2.
7.7 The logistic map
n
Period-doubling cascades and chaos. **7.7.1 Unimodal maps. Periods 22 only or chaos.
7.7.2 Rescaling and Feigenbaum’s constant.**
3
Introduction
Books
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There are many excellent texts.
• P.A.Glendinning Stability, Instability and Chaos [CUP].
A very good text written in clear language.
• D.K.Arrowsmith & C.M.Place Introduction to Dynamical Systems [CUP].
Also very good and clear, covers a lot of ground.
• R.Grimshaw An Introduction to Nonlinear Ordinary Differential Equations [CRC Press].
Very good on stability of periodic solutions. Quite technical in parts.
• P.G.Drazin Nonlinear Systems [CUP].
Covers a great deal of ground in some detail. Good on the maps part of the course. Could
be the book to go to when others fail to satisfy.
• D.W.Jordan & P.Smith Nonlinear Ordinary Differential Equations [OUP].
A bit long in the tooth and not very rigorous but has some very useful material especially
on perturbation theory.
• S.H.Strogatz Nonlinear Dynamics and Chaos [Perseus Books, Cambridge, MA.]
An excellent informal treatment, emphasising applications. Inspirational!
Motivation
Why do we study nonlinear phenomena? Because almost all physical phenomena are describable
by of nonlinear dynamical systems (either differential equations or maps). Of course some things
like simple harmonic motion can be understood on linear theory: but to know how the period
of a pendulum changes with amplitude we must solve a nonlinear equation. Solving a linear
system usually requires a simple set of tasks, like determining eigenvalues. Nonlinear systems
have an amazingly rich structure, and most importantly they do not in general have analytical
4
solutions, or at least none expressible in terms of elementary functions. Thus in general we
rely on a geometric approach, which allows the determination of important characteristics of
the solution without the need for explicit solution. Much of the course will be taken up with
such ideas.
We also study the stability of various simple solutions. A solution (steady or periodic state) is
not of much use if small perturbations destroy it (an example: a pencil balanced exactly on its
point). So we need to know what happens to the solution near such a special solution. This
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involves linearizing, which allows the classification of fixed and periodic points(corresponding
to oscillations). We shall also develop perturbation methods, which allow us to find good approximations to solutions of systems that are near simple solutions that we understand.
Many nonlinear systems depend on one or more parameters. Examples include the simple
equation ẋ = µx − x3 , where the parameter µ can take positive or negative values. If µ > 0
there are three stationary points, while if µ < 0 there is only one. The point µ = 0 is called
a bifurcation point, and we shall see that we can classify bifurcations and develop a general
method for determining the solutions near such points.
Most of the systems we shall look at are of second order, and we shall see that these have
relatively simple long-time solutions (fixed and periodic points, essentially). In the last part
of the course we shall look at some aspects of third order (and time-dependent second order)
systems, which can exhibit “chaos”. These systems are usually treated by the study of maps
(of the line or the plane) which can be identified with the dynamics if the differential system.
Maps can be treated in a rigorous manner and there are some remarkable theorems (such as
Sharkovsky’s on the order of appearance of periodic orbits in one-humped maps of the interval)
that can be proved.
A simple example of a map is the Lotka-Volterra system describing two competing populations
(e.g. r=rabbits, s=sheep):
ṙ = r(a − br − cs),
ṡ = s(d − er − f s)
where a, b, c, d, e, f are (positive in this example) constants. This is a second order system which
is autonomous (time does not appear explicitly). The system lives in a state space or phase
space (r, s) ∈ [0, ∞) × [0, ∞). We regard r, s as continuous functions of time: some people call
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this a flow (as opposed to a map where numbers vary discontinuously - which might be better
for this example!)
Typical analysis looks at fixed points. These are at (r, s) = (0, 0), (r, s) = (0, d/f ), (r, s) =
(a/b, 0) and a solution with r, s 6= 0 as long as bf 6= ce. Assuming, as can be proved, that at
long times the solution tends to one of these, we can look at local approximations near the fixed
points. Near (0, d/f ), write u = s−d/f , then approximately ṙ = r(a−cd/f ), u̇ = −du−der/f ,
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so the solution tends to this point (r = u = 0) if s < d/f (so this fixed point is stable), but
not otherwise. The concept of stability if more involved than naive ideas would suggest and so
we will be investigating the nature of stability. In fact there are three different phase portraits
depending on the parameters. The solutions follow lines in the phase space called trajectories.
We use bifurcation theory to study the change in stability as parameters are varied.
Other Lotka-Volterra models have different properties, for example the struggle between sheep
s and wolves w:
ẇ = w(−a + bs),
ṡ = s(c − dw)
This system turns out to have periodic orbits.
In two dimensions periodic orbits are common for topological reasons, so we also look at their
stability. Consider the system
ẋ = −y + x(µ − x2 − y 2 ),
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ẏ = x + y(µ − x2 − y 2 )
In polar coordinates r, θ, x = r cos θ, y = r sin θ we have
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ṙ = µr − r3 ,
θ̇ = 1
The case µ = 0 is special since there are infinitely many periodic orbits. This is non-hyperbolic
or not structurally stable. Any non-zero value of µ makes a change to the nature of the system.
The stability of periodic orbits can be cast in terms of maps. If an orbit (e.g. in a threedimensional phase space) crosses a plane at a point xn and then crosses again at xn+1 this
defines a map of the plane into itself (the Poincaré map).
Maps also arise naturally as approximations to flows,e.g. the equation ẋ = µx − x3 can be
approximated using Euler’s method (with xn = x(n dt)) to give xn+1 = xn (µdt + 1) − x3n dt.
Poincaré maps arising from flows in 2D are dull, but for 3D flows they become maps of the
plane and can have many interesting properties including chaos. Even one dimensional maps
can have chaotic behaviour in general. Examples include the Lorenz equations (ẋ = σ(y − x),
ẏ = rx − y − xz, ż = −bz + xy, and the logistic map xn+1 = µxn (1 − xn ).
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1
1.1
Nonlinear Differential Equations
Elementary concepts
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We need some notation to describe our equations.
Define a State Space (or Phase Space) E ⊆ Rn (E is sometimes denoted by X). Then the
state of the system is denoted by x ∈ E. The state depends on the time t and the (ordinary)
differential equation gives a rule for the evolution of x with t:
ẋ ≡
dx
= f (x, t) ,
dt
(1)
where f : E × R → E is a vector field.
∂f
≡ 0 the equation is autonomous. The equation is of order n. N.B. a system of n first
∂t
order equations as above is equivalent to an nth order equation in a single dependent variable.
If
If dn x/dtn = g(x, dx/dt, . . . , dn−1 x/dtn−1 ) then we write y = (x, dx/dt, . . . , dn−1 x/dtn−1 ) and
ẏ = (y2 , y3 , . . . , yn , g).
Non-autonomous equations can be made (formally) autonomous by defining y ∈ E × R by
y = (x, t), so that ẏ = g(y) ≡ (f (y), 1).
Example 1 Second order system ẍ+ ẋ+x = 0 can be written ẋ = y, ẏ = −x−y, so (x, y) ∈ R2 .
1.2
Initial Value Problems
Typically, seek solutions to (1) understood as an initial value problem:
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Given an initial condition x(t0 ) = x0 (x0 ∈ E, t0 ∈ I ⊆ R), find a differentiable function x(t)
for t ∈ I which remains in E for t ∈ I and satisfies the initial condition and the differential
equation.
For an autonomous system we can alternatively define the solution in terms of a flow φt :
Definition 1 (Flow) φt (x) s.t. φt (x0 ) is the solution at time t of ẋ = f (x) starting at x0
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when t = 0 is called the flow through x0 at t = 0. Thus φ0 (x0 ) = x0 , φs (φt (x0 )) = φs+t (x0 )
etc. (Continuous semi-group). We sometimes write φft (x0 ) to identify the particular dynamical
system leading to this flow.
Does such a solution exist? And is it unique?
Existence is guaranteed for many sensible functions by the **Cauchy-Peano theorem**:
Theorem 1 (Cauchy-Peano). If f (x, t) is continuous and |f | < M in the domain D : {|t −
t0 | < α, |x − x0 | < β}, then the initial value problem above has a solution for |t − t0 | <
min(α, β/M ).
But uniqueness is guaranteed only for stronger conditions on f .
Example 2 Unique solution: ẋ = |x|, x(t0 ) = x0 . Then x(t) = x0 et−t0 (x0 > 0), x(t) = x0 et0 −t
(x0 < 0), x(t) = 0 (x0 = 0). Here f is not differentiable, but it is continuous.
1
Example 3 Non-unique solution: ẋ = |x| 2 , x(t0 ) = x0 . We still have f continuous. Solving
gives x(t) = (t + c)2 /4 (x > 0) or x(t) = −(c − t)2 /4 (x < 0). So for x0 > 0 we have
√
x(t) = (t − t0 + 4x0 )2 /4 (t > t0 ). However for x0 = 0 we have two solutions: x(t) = 0 (all t)
and x(t) = (t − t0 )|(t − t0 )|/4 (t ≥ t0 ).
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1
Why are these different? Because in second case derivatives of |x| 2 are not bounded at the
origin. To guarantee uniqueness of solutions need stronger property than continuity; function
to be Lipschitz.
Definition 2 (Lipschitz property). A function f defined on a subset of Rn satisfies a Lipschitz condition at a point x0 with Lipschitz constant L if ∃(L, a) such that ∀x, y with
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|x − x0 | < a, |y − x0 | < a, |f (x) − f (y)| ≤ L|x − y|.
Note that Differentiable → Lipschitz → Continuous.
We can now state the result (discussed in Part IA also):
Theorem 2 (Uniqueness theorem). Consider an initial value problem to the system (1) with
x = x0 at t = t0 . If f satisfies a Lipschitz condition at x0 then the solution φt−t0 (x0 ) exists
and is unique and continuous in a neighbourhood of (x0 , t0 )
Note that uniqueness and continuity do not mean that solutions exist for all time!
Example 4 (Finite time blowup). ẋ = x3 , x ∈ R, x(0) = 1. This is solved by x(t) =
√
1/ 1 − 2t, so x → ∞ when t → 21 .
This does not contradict earlier result [why not?] .
From now on consider continuous functions f unless stated otherwise.
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1.3
Trajectories and Flows
Consider the o.d.e. ẋ = f (x) with x(0) = x0 , or equivalently the flow φt (x0 )
Definition 3 (Orbit). The orbit of φt through x0 is the set O(x0 ) ≡ {φt (x0 ) : −∞ < t < ∞}.
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This is also called the trajectory through x0 .
Definition 4 (Forwards orbit). The forwards orbit of φt through x0 is O+ (x0 ) ≡ {φt (x0 ) :
t ≥ 0}; backwards orbit O− defined similarly for t ≤ 0.
Note that flows and maps can be linked by considering xn+1 = φδt (xn ) In this course we adopt a
geometric viewpoint: rather than solving equations in terms of “elementary” (a.k.a. tabulated)
functions, look for general properties of the solutions. Since almost all equations cannot be
solved in terms of elementary functions, this is more productive!
1.4
Invariant and Limit Sets
Work by considering the phase space E, and the flow φt (x0 ), considered as a trajectory
(directed line) in the phase space. we are mostly interested in special sets of trajectories, as
long-time limits of solutions from general initial conditions. These are called invariant sets.
Definition 5 (Invariant set). A set of points Λ ∈ E is invariant under f if x ∈ Λ ⇒
O(x) ∈ Λ. (Can also define forward and backward invariant sets in the obvious way).
Clearly O(x) is invariant. Special cases are;
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Definition 6 (Fixed point). The point x0 is a fixed point (equilibrium, stationary point,
critical point) if f (x0 ) = 0. Then x = x0 for all time and O(x0 ) = x0 .
Definition 7 (Periodic point). A point x0 is a periodic point if φT (x0 ) = x0 for some
T > 0, but φt (x0 ) 6= x0 for 0 < t < T . the set {φt (x0 ) : 0 ≤ t < T } is called a periodic orbit
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through x0 . T is the period of the orbit. If a periodic orbit C is isolated, so that there are
no other periodic orbits in a sufficiently small neighbourhood of C, the periodic orbit is called a
limit cycle.
Example 5 (Family of periodic orbits). Consider
ẋ
ẏ
=
−y 3
x3
This has solutions of form x4 + y 4 = const., so all orbits except the fixed point at the origin are
periodic.
Example 6 (Limit cycle). Now consider
ẋ
ẏ
=
−y + x(1 − x2 − y 2 )
x + y(1 − x2 − y 2 )
Here we have ṙ = r(1 − r2 ), where r2 = x2 + y 2 , and there are no fixed points except the origin,
so there is a unique limit cycle r = 1.
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Definition 8 (Homoclinic and heteroclinic orbits). If x0 is a fixed point and ∃y 6= x0 such
that φt (y) → x0 as t → ±∞, then O(y) is called a homoclinic orbit. If there are two fixed
points x0 , x1 and ∃y 6= x0 , x1 such that φt (y) → x0 (t → −∞), φt (y) → x1 (t → +∞) then
O(y) is a heteroclinic orbit. A closed sequence of heteroclinic orbits is called a heteroclinic
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cycle (sometimes also called heteroclinic orbit!)
When the phase space has dimension greater than 2 can get more exotic invariant sets.
Example 7 (2-Torus). let θ1 , θ2 be coordinates on the surface of a 2-torus, such that θ˙1 = ω1 ,
θ˙2 = ω2 . If ω1 , ω2 are not rationally related the trajectory covers the whole surface of the torus.
Example 8 (Strange Attractor). Anything more complicated than above is called a strange
attractor. Examples include the Lorenz attractor.
13
We have to be careful in defining how invariant sets arise as limits of trajectories. Not enough
to have definition like “set of points y s.t. φt (x) → y as t → ∞”, as that does not include e.g.
periodic orbits. Instead use the following:
Definition 9 (Limit set). The ω-limit set of x, denoted by ω(x) is defined by
ω(x) ≡ {y : φtn (x) → y for some sequence of times t1 , t2 , . . . , tn , . . . → ∞}. Can also define
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α-limit set by sequences → −∞.
The ω-limit set ω(x) has nice properties when O(x) is bounded: In particular, ω(x) is:
(a) Non-empty [every sequence of points in a closed bounded domain has at least one accumulation point] (b) Invariant under f [Obvious from definition] (c) Closed [think about not being
in ω(x)] and bounded (d) Connected [if disconnected, ∃ an sequence of times for which x(t)
does not tend to any of the disconnected parts of ω(x)]
1.5
Topological equivalence and structural stability
What do we mean by saying that two flows (or maps) have essentially the same (topological/geometric) structure? Or that the structure of a flow changes at a bifurcation?
Definition 10 (Topological Equivalence). Two flows φft (x) and φgt (y) are topologically equivalent if there is a homeomorphism h(x) : E f → E g (i.e. a continuous bijection with continuous
inverse) and time-increasing function τ (x, t) (i.e. a continous, monotonic function of t) with
φft (x) = h−1 ◦ φgτ ◦ h(x) and τ (x, t1 + t2 ) = τ (x, t1 ) + τ (φft1 , t2 )
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i.e. it is possible to find a map h from one phase space to the other, and a map τ from time
in one phase space to time in the other, in such a way that the evolution of the two systems
are the same. Clearly topological equivalence maps fixed points to fixed points, and periodic
orbits to periodic orbits – though not necessarily of the same period.
Example 9 The dynamical systems
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ṙ = −r
θ̇ = 1
and
ρ̇ = −2ρ
ψ̇ = 0
are topologically equivalent with h(0) = 0, h(r, θ) = (r2 , θ + ln r) for r 6= 0 in polar coordinates,
and τ (x, t) = t. To show this, integrate the ODEs to get
φft (r0 , θ0 ) = (r0 e−t , θ0 + t),
φgt (ρ0 , ψ0 ) = (ρ0 e−2t , ψ0 )
and check
h ◦ φft = (r02 e−2t , θ0 + ln r0 ) = φgt ◦ h
Example: The dynamical systems
ṙ = 0
ṙ = 0
and
θ̇ = 1
θ̇ = r + sin2 θ
are topologically equivalent. This should be obvious because the trajectories are the same and
so we can put h(x) = x. Then stretch timescale.
Definition 11 **(Structural Stability)** .The vector field f [system ẋ = f (x) or flow φft (x)]
is structurally stable if ∃ǫ > 0 s.t. f + δ is topologically equivalent to f
P
i |∂δ/∂xi | < ǫ.
∀δ(x) with |δ| +
Examples: System (1) is structurally stable. System (2) is not (since the periodic orbits are
destroyed by a small perturbation ṙ 6= 0).
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Flows in R2
2
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2.1
Linearization
In analyzing the behaviour of nonlinear systems the first step is to identify the fixed points.
Then near these fixed points, behaviour should approximate linear. In fact near a fixed point
∂fi 2
is the
x0 s.t. f (x0 ) = 0, let y = x − x0 ; then ẏ = Ay + O(|y| ), where Aij = ∂xj x=x0
linearization of f about x0 . The matrix A is also written Df , the jacobian matrix of f
at x0 . We hope that in general the flow near x0 is topologically equivalent to the linearized
problem. This is not always true, as shown below.
2.2
Classification of fixed points
Consider general linear system ẋ = Ax, where A is a constant matrix. We need the eigenvalues λ1,2 of the matrix, given by λ2 − λTrA + DetA = 0. This has solutions λ = 12 TrA ±
q
1
(TrA)2 − DetA. We can then classify the roots into classes.
4
• Saddle point(DetA < 0). Roots are real and of opposite sign.
λ1 0
0 3
−2 0
: λ1 λ2 < 0.)
; (canonical form
,
E.g.
0 λ2
1 0
0 1
• Node ((TrA)2 > 4DetA > 0). Roots are real and either both positive (TrA > 0: unstable,
repelling node), or both negative (TrA < 0: stable, attracting node).
λ1 0
0
1
2 0
: λ1 λ2 > 0.)
;(canonical form
,
E.g.
0 λ2
−1 −3
0 1
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• Focus (Spiral) ((TrA)2 < 4DetA). Roots are complex and either both have positive real
part (TrA > 0: unstable, repelling focus), or both negative real part (TrA < 0: stable,
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attracting focus).
λ ω
2 1
−1 1
: eigenvalues λ ± iω.)
;(canonical form
,
E.g.
−ω λ
−2 3
−1 −1
Degenerate cases occur when two eigenvalues are equal ((TrA)2 = 4DetA 6= 0) giving either
1 1
1 0
.
or Improper nodes e.g.
Star/Stellar nodes, e.g.
0 1
0 1
In all these cases the fixed point is hyperbolic.
Definition 12 (Hyperbolic fixed point). A fixed point x of a dynamical system is hyperbolic
iff all the eigenvalues of the linearization A of the system about x have non-zero real part.
This definition holds for higher dimensions too.
Thus the nonhyperbolic cases, which are of great importance in bifurcation theory, are those
for which at least one eigenvalue has zero real part. These are of three kinds:
• A = 0. Both eigenvalues are zero.
• DetA = 0. Here one eigenvalue is zero and we have a line of fixed points. e.g.
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0 0
0 −1
• TrA = 0, DetA > 0 (Centre). Here the eigenvalues are ±iω and trajectories are closed
0 2
.
curves, e.g.
−1 0
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All this can be summarized in a diagram
To find canonical form, find the eigenvectors of A and use as basis vectors (possibly generalised
if eigenvalues equal), when eigenvalues real. For complex eigenvalues in R2 we have two complex
eigenvectors e, e∗ so use (Re(e), Ime) as a basis. This can help in drawing trajectories. But
note classification is independent of basis.
Centres are special cases in context of general flows; but Hamiltonian systems have centres
generically. Consider ẋ = (Hy , −Hx ) for some H = H(x, y). At a fixed point ∇H = 0, and the
matrix
A=
Hxy
Hyy
−Hxx −Hxy
⇒ TrA = 0.
Thus all fixed points are saddles or centres. Clearly also ẋ · ∇H = 0, so H is constant on all
trajectories.
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2.3
Effect of nonlinear terms
For a general nonlinear system ẋ = f (x), we start by locating the fixed points, x0 , where
x0 = 0. Then what does the linearization of the system about the fixed point x0 tell us about
the behaviour of the nonlinear system?
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We can show (e.g. Glendinning Ch. 4) that if
(i) x0 is hyperbolic; and
(ii)the nonlinear corrections are O(|x − x0 |2 ),
Then the two systems are topologically conjugate and that nodes ⇒ nodes, foci ⇒ f oci.
(Without (ii) nodes and foci cannot be distinguished by topological conjugacy).
We thus discuss separately hyperbolic and non-hyperbolic fixed points.
2.3.1
Stable and Unstable Manifolds
For the linearized system we can separate the phase space into different domains corresponding
to different behaviours in time.
Definition 13 (Invariant subspaces). The stable, unstable and centre subspaces of the
linearization of f at the fixed point x0 are the three linear subspaces E u , E s , E c , spanned by
the subsets of (possibly generalised) eigenvectors of A whose eigenvalues have real parts < 0,
> 0, = 0 respectively.
Note that a hyperbolic fixed point has no centre eigenspace. This concept can be extended
simply for hyperbolic fixed points into the nonlinear domain. We suppose that f.p. is at the
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origin and that f is expandable in a Taylor series. We can write ẋ = Ax + F (x), where
F = 0(|x|2 ). We need the Stable (or Invariant) Manifold Theorem.
Theorem 3 (Stable (invariant) Manifold Theorem). Suppose 0 is a hyperbolic fixed point of
ẋ = f (x), and that E u , E s are the linear unstable and stable subspaces of the linearization
u
s
of f about 0. Then ∃ local stable and unstable manifolds Wloc
(0), Wloc
(0), which have
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the same dimension as E u , E s and are tangent to E u , E s at 0, such that for x 6= 0 but in a
sufficiently small neighbourhood of 0,
u
Wloc
= {x : φt (x) → 0 as t → −∞}
s
Wloc
= {x : φt (x) → 0 as t → +∞}
Proof: rather involved; see Glendinning, p.96. The trick is to produce a near identity change of
coordinates. Suppose that x, y span the unstable(stable) subspaces E u (E s ), so the linearized
stable manifold is x = 0. We look for the (e.g.) stable manifold in the form x = S(y); then
make a change of variable ξ = x − S(y) so that the transformed equation has ξ = 0 an an
invariant manifold. The function S(y) can be expanded as a power series, and the idea is to
check that the expansion can be performed to all orders, giving a finite radius of convergence.
The local stable (unstable) manifold can be extended to a global invariant manifold W s (W u )
s
u
by following the flow backwards (forwards) in time from a points in Wloc
(Wloc
).
It is easy to find approximations to the stable and unstable manifolds of a saddle point in R2 .
The stable(say) manifold must tend to the origin and be tangent to the stable subspace E s
(i.e. to the eigenvector corresponding to the negative eigenvalue. (It is often easiest though not
necessary to change to coordinates such that x = 0 or y = 0 is tangent to the manifold). Then
for example if we want to find the manifold (for 2D flows just a trajectory) that is tangent to
y = 0 at the origin for the system ẋ = f (x, y), ẏ = g(x, y), write y = p(x); then
g(x, p(x)) = ẏ = p′ (x)ẋ = p′ (x)f (x, p(x)),
which gives a nonlinear ODE for p(x). In general this cannot be solved exactly, but we can find
a (locally convergent) series expansion in the form p(x) = a2 x2 + a3 x3 + . . ., and solve term by
term.
20
Example 10 Consider
ẋ
ẏ
=
x
−y + x2
. This can be solved exactly to give x = x0 et ,
y = 31 x20 e2t + (y0 − 13 x20 )e−t or y(x) = 31 x2 + (y0 − 13 x20 )x0 x−1 . Two obvious invariant curves are
x = 0 and y = 13 x2 , and x = 0 is clearly the stable manifold. The linearization about 0 gives
1
0
and so the unstable manifold must be tangent to y = 0; y = 31 x2
the matrix A =
0 −1
fits the bill. To find constructively write y(x) = a2 x2 + a3 x3 + . . .. Then
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dy
dy
= ẋ
= (2a2 x + 3a3 x2 + . . .)x = −a2 x2 − a3 x3 + . . . + x2
dt
dx
Equating coefficients, find a2 = 13 , a3 = 0, etc.
Example 11 Now
ẋ
ẏ
=
x − xy
−y + x2
; there is no simple form for the unstable manifold
(stable manifold is still x = 0). The unstable manifold has the form y = ax2 + bx3 + cx4 + . . .,
where [exercise] a = 13 , b = 0, c =
2
,
45
etc. Note that this infinite series (in powers of x2 ) has
a finite radius of convergence since the unstable manifold of the origin is attracted to a stable
focus at (1, 1).
21
2.3.2
Nonlinear terms for non-hyperbolic cases
We now suppose that there is at least one eigenvalue on the imaginary axis. Concentrate on
R2 , generalization not difficult. There are two possibilities:
(i) A has eigenvalues ±iω. The linear system is a centre. The nonlinear system have different
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forms for different r.h.s.’s
• Stable focus:
• Unstable focus:
−y − x3
x − y3
• Nonlinear centre:
−y + x3
x + y3
−y − 2x2 y
x + 2y 2 x
22
(ii) A has one zero eigenvalue, other e.v.non-zero, e.g. (a) ẋ = x2 , ẏ = −y [Saddle-node], (b)
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ẋ = x3 , ẏ = −y [Nonlinear Saddle].
(iii) Two zero eigenvalues. Here almost anything is possible. Change to polar coords. Find
lines as r → 0 on which θ̇ = 0. Between each of these lines can have three different types of
behaviour. (See diagram).
23
2.4
Sketching phase portraits
This often involves some good luck and good judgement! Nonetheless there are some guidelines
which if followed will give a good chance of success. The general procedure is as follows:
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• 1. Find the fixed points, and find any obvious invariant lines e.g.x = 0 when x = xh(x, y) etc.
• 2. Calculate the jacobian and hence find the type of fixed point. (Accurate calculation of
eigenvalues etc. for nodes may not be needed for sufficiently simple systems - just find the
type.) Do find eigenvectors for saddles.
• 3. If fixed points non-hyperbolic get local picture by considering nonlinear terms.
• 4. If still puzzled, find nullclines, where x or y (or r or θ) are zero.
• 5. Construct global picture by joining up local trajectories near fixed points (especially saddle
separatrices) and put in arrows.
• 6. Use results of Ch. 4 to decide whether there are periodic orbits.
1 − y −x
x(1 − y)
.
. Jacobian A =
=
Example 12 (worked example). Consider
2x −1
−y + x2
0 ∓1
. TrA2 = 1 < DetA so stable foci.
Fixed points at (0, 0) (saddle) and (±1, 1); A =
±2 −1
ẋ
ẏ
x = 0 is a trajectory, ẋ = 0 on y = 1 and ẏ ≶ 0 when y ≷ x2 .
24
3
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3.1
Stability
Definitions of stability
If we find a fixed point, or more generally an invariant set, of an o.d.e. we want to know what
happens to the system under small perturbations. We also want to know which invariant sets
will be approached at large times. If in some sense the solution stays “nearby”, or the set is
approached after long times, then we call the set stable. There are several differing definitions
of stability; different texts are not consistent. I shall define stability of whole invariant sets
(and not just of points in those sets). This shortens the discussion.
Consider an invariant set Λ in a general (autonomous) dynamical system described by a flow
φt . t (This could be a fixed point, periodic orbit, torus etc.) We need a definition of points
near the set Λ:
Definition 14 (Neighbourhood of a set Λ). For δ > 0 the neighbourhood Nδ (Λ) = {x : ∃y ∈
Λ s.t. |x − y| < δ}
We also need to define the concept of a flow trajectory tending to Λ.
Definition 15 (flow tending to Λ). The flow φt (x) → Λ iff min|φt (x) − y| → 0 as t → ∞
y∈Λ
Definition 16 (Lyapunov stability)[LS]. The set Λ is Lyapunov stable if ∀ǫ > 0 ∃δ >
0 s.t. x ∈ Nδ (Λ) ⇒ φt (x) ∈ Nǫ (Λ) ∀t ≥ 0. (“start near, stay near”).
Definition 17 (Quasi-asymptotic stability)[QAS]. The set Λ is quasi-asymptotically stable if ∃δ > 0 s.t.x ∈ Nδ (Λ) ⇒ φt (x) → Λ as t → ∞. (“get close eventually”).
25
Definition 18 (Asymptotic stability)[AS]. The set Λ is asymptotically stable if it is both
Lyapunov stable and quasi-asymptotically stable.
Example 13 (LS but not QAS).
ẋ
ẏ
=
−y
x
=
r(1 − r2 )
sin2 2θ
.
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All limit sets are circles or the origin.
Example 14 (QAS but not LS).
ṙ
θ̇
.
Point r = 1, θ = 0 is a saddle-node.
Example 15
ẋ
ẏ
=
y − ǫx
−2ǫy
(ǫ > 0). Here y = y0 e−2ǫt , x = x0 e−ǫt + y0 ǫ−1 (e−ǫt − e−2ǫt ).
Thus for t ≥ 0, |y| ≤ |y0 |, |x| ≤ |x0 | + 41 ǫ−1 |y0 |, and so x2 + y 2 ≤ (x20 + y02 )(1 + ǫ−1 + ǫ−2 /16).
This proves Lyapunov stability. Furthermore the solution clearly tends to the origin as t → ∞.
This example is instructive because for ǫ sufficiently small the solution can grow to large values
before eventually decaying. To require that the norm of the solution decays monotonically is a
stronger result, only applicable to a small number of problems.
26
If an invariant set is not LS or QAS we say it is unstable (or according to some books,
nonstable).
There may be more than one asymptotically stable limit set. In that case we want to know
what parts of the phase space lead to which limit sets being approached. Then we define the
basin of attraction(or domain of stability):
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Definition 19 If Λ is an asymptotically stable invariant set the basin of attraction of Λ,
B(Λ) ≡ {x : φt (x) → Λ as t → ∞}. If B(Λ) = Rn then Λ is globally attracting(or globally
stable). Note that B(Λ) is an open set.
When there are many fixed points the basin of attraction can be quite complicated. (See
handout).
When Λ is an isolated fixed point (x0 , say) we can investigate its stability by linearizing the
system about x0 (see previous chapter).
Theorem 4 (Stability of hyperbolic fixed points). If 0 is a hyperbolic sink then it is asymptotically stable. If 0 is a hyperbolic fixed point with at least one eigenvalue with ℜλ > 0, then it is
unstable.
27
3.2
Lyapunov functions
We can prove a lot about stability of a fixed point (which can be taken to be at the origin)
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if we can find a positive function V of the independent variables that decreases monotonically
under the flow φt . Then under certain reasonable conditions we can show that V → 0 so that
the appropriately defined modulus of the solution similarly tends to zero. This is a Lyapunov
function, defined precisely by
Definition 20 (Lyapunov function). Let E be a closed connected region of Rn containing
the origin. A function V : Rn → R which is differentiable except perhaps at the origin is a
Lyapunov function for a flow φ if (i) V(0) = 0, (ii) V is positive definite (V(x) > 0 when
0 6= x), and if also (iii) V(φt (x)) ≤ V(x) ∀x ∈ E (or equivalently if V̇ ≤ 0 on trajectories).
Then we have the following theorems:
Theorem 5 (Lyapunov’s First Theorem [L1]). Suppose that a dynamical system ẋ = f (x) has
a fixed point at the origin. If a Lyapunov function exists,as defined above, then the origin is
Lyapunov stable.
Theorem 6 (Lyapunov’s Second Theorem [L2]). If in addition V̇ < 0 for x 6= 0 then V is a
Strict Lyapunov function and the origin is asymptotically stable.
Proof of First Theorem: We want to show that for any sufficiently small neighbourhood U of
the origin, there is a neighbourhood V s.t. if x0 ∈ V , φt (x0 ) ∈ U for all positive t. We start by
choosing δ sufficiently small that |x| < δ ⇒ x ∈ U . Let α = min{V : |x| = δ}. Clearly α > 0
from the definition of V. Now consider the set U1 = {x : V(x) < α and |x| ≤ δ}. Then since
for x0 ∈ U1 , V(x0 ) < α and V does not increase along trajectories, |φt (x0 )| can never reach δ, as
if this happened V would be ≥ α. Thus clearly x ∈ U1 at t = 0 ⇒ |x(t)| < δ ⇒ x(t) ∈ U ∀t ≥ 0,
as required.
28
Proof of Second Theorem: [L1] implies that x remains in the domain U . For any initial point
x0 6= 0, V(x0 ) > 0 and V̇ < 0 along trajectories. Thus V(φt (x0 )) decreases montonically and is
bounded below by 0. Then V → α ≥ 0; suppose α > 0. Then |x| is bounded away from zero
(by continuity of V), and so W ≡ V̇ < −b < 0. Thus V(φt (x0 )) < V(x0 ) − bt and so is certainly
negative after a finite time. Thus there is a contradiction, and so α = 0. Hence V → 0 as
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t → ∞ and so |x| → 0 (again by continuity). This proves asymptotic stability.
To prove results about instability just reverse the sense of time. Sometimes we can demonstrate
asymptotic stability even when V is not a strict Lyapunov function. For this need another
theorem, La Salle’s Invariance Principle:
Theorem 7 (La Salle’s Invariance Principle). If V is a Lyapunov function for a flow φ then
∀x ∃c s.t. ω(x) ∈ Mc ≡ {x : V(φt (x)) = c ∀t ≥ 0}.(Or, φt (x) → an invariant subset of the
set {y : V̇(y) = 0}.)
Proof : choose a point x and let c = inf t≥0 V(φt (x)). If c = −∞ then φt (x) → ∞ and ω(x)
is empty. So suppose φt (x) remains finite and that ω(x) is not empty. Then if y ∈ ω(x), ∃ a
sequence of times tn s.t. φtn (x) → y as n → ∞, then by continuity of V and since V̇ ≤ 0 on
trajectories we have V(y) = c. We need to prove that y ∈ Mc (i.e. that V(y) = c for t ≥ 0).
Suppose to the contrary that ∃s s.t. V(φs (y)) < c. Thus for all z sufficiently close to y we
have V(φs (z)) < c. But if z = φtn (x) for sufficiently large n we have φtn +s (x) < c, which is a
contradiction. This proves the theorem.
As a corollary we note that if V is a Lyapunov function on a bounded domain D and the only
invariant subset of {V̇ = 0} is the origin then the origin is asymptotically stable.
Examples of the use of the Lyapunov theorems.
Example 16 (Finding the basin of attraction). Consider
ẋ
ẏ
=
−x + xy 2
−2y + yx2
. We can
ask: what is the best condition on x which guarantees that x → 0 as t → ∞? Consider
29
V(x, y) = 21 (x2 + b2 y 2 ) for constant b.Then V̇ = −(x2 + 2b2 y 2 ) + (1 + b2 )x2 y 2 We can then show
easily that
√
V̇ < 0 if V < (3 + 2 2)b2 /2(1 + b2 ) [Check by setting z = by and using polars for (x, z)]. The
domain of attraction of the origin certainly includes the union of all these sets (see handout).
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Example 17 (Damped pendulum). Here
ẋ
ẏ
=
y
−ky − sin x
with k > 0. We can
choose V = 21 y 2 + 1 − cos x; clearly V is positive definite provided we identify x and x + 2π,
and V̇ = −ky 2 ≤ 0. So certainly the origin is Lyapunov stable by [L1]. But we cannot use [L2]
since V̇ is not negative definite. Nonetheless from La Salle’s principle we see that the set Mc is
contained in the set of complete orbits satisfying y = 0. The only such orbits are (0, 0) (c = 0)
and (π, 0) (c = 2) So these points are the only possible members of ω(x). Since the origin is
Lyapunov stable we conclude that for all points x s.t. V < 2 the only member of ω(x) is the
origin and so this point is asymptotically stable.
Special cases are gradient flows.
Definition 21 A system is called a gradient system or gradient flow if we can write
ẋ = −∇V (x).
In this case we have V̇ = −|∇V |2 ≤ 0, with V̇ < 0 except at the fixed points which have |∇V | =
0. Thus we can use La Salle’s principle to show that Ω = {ω(x) : x ∈ E} consists only of the
fixed points. Note that this does NOT mean that all the fixed points are asymptotically stable
(see diagram). These ideas can be extended to more general systems of the form ẋ = −h∇(V ),
where h(x) is a strictly positive continuously differentiable function. [Proof: exercise].
30
3.3
Bounding functions
Even when we cannot find a Lyapunov function in the exact sense, we can sometimes find
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positive definite functions V s.t. V̇ < 0 outside some neighbourhood of the origin. We call
these Bounding functions. They are used to show that x remains in some neighbourhood of
the origin.
Theorem 8 Let V : E ⊆ Rn → R be a positive definite continuously differentiable function and
let D be a region s.t. V̇(x) < −δ (δ > 0) if x ∈
/ D, and let α = sup{V(x) : xon boundary of D}.
Then all orbits eventually enter and remain inside the set Vα = {x : V(x) ≤ α}.
Proof: exercise (see diagram).
31
4
Existence and stability of periodic orbits in R2
Example 18 Damped pendulum with torque Consider the system θ̈ + k θ̇ + sin θ = F , k > 0,
F > 0. We would like to know whether there is a periodic orbit of this equation. We can find a
bounding function of the form V = 21 p2 + 1 − cos θ (p = θ̇). Then V̇ = pṗ + θ̇ sin θ = p(F − kp).
Thus V̇ < 0 unless 0 < p < F/k. Maximum value of V on boundary of this domain is
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Vmax = 12 ( Fk )2 + 2. By previous result on bounding functions we see that all orbits eventually
enter and remain in the region V < Vmax .
What happens within this region? We can look for fixed points: these are at p = 0, sin θ = F .
So there are 2 f.p.’s if F < 1, no f.p.’s if F > 1. In the first case one fixed point is stable (node
or focus depending on k) and the other is a saddle. In the second case what can happen? Either
there is a closed orbit or ?possibly? a space filling curve. There are some nontrivial theorems
that we can use to answer this.
4.1
The Poincaré Index
Closed curves can be distinguished by the number of rotations of the vector field f as the curve
is traversed. This property of a curve in a vector field is very useful in understanding the phase
portrait.
Theorem 9 (Poincaré Index). Consider a vector field f = (f1 , f2 ). At any point the direction
32
of f is given by ψ = tan−1 (f2 /f1 ) (with the usual conventions). Now let x traverse a closed
curve C. ψ will increase by some multiple (possibly negative) or 2π. This multiple is called the
Poincaré Index IC of C.
This index can be put in integral form. We have
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1
IC =
2π
I
1
dψ =
2π
C
I
d tan
−1
C
f2
f1
1
=
2π
I
C
f1 df2 − f2 df1
f12 + f22
In fact the index is most easily worked out by hand. There are several results that are easily
proved about Poincaré indices that makes them easier to calculate.
1. The index takes only integer values, and is continuous when the vector field
has no zeroes. It therefore is the same for two curves which can be deformed into each
other without crossing any fixed point.
2. The index of any curve not enclosing any fixed point is zero. This is because it
can be shrunk to zero size.
3. The index of a curve enclosing a number of fixed points is the sum of the
indices for curves enclosing the fixed points individually. This is because the curve
can be deformed into small curves surrounding each fixed point together with connecting
lines along which the integral cancels.
33
4. The index of a curve for ẋ = f (x) is the same as that of the system ẋ = −f (x).
Proof: Consider effect on integral representation of the change f → −f .
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5. The index of a periodic orbit is +1. The vector field is tangent to the orbit at every
point.
6. The index of a saddle is −1, and of a node or focus +1. By inspection, or by
noting that in complex notation a saddle can be written, in suitable coordinates as ẋ = x,
ẏ = −y or ż = z̄. For a node or focus, curve can be found such that trajectories cross in
same direction.
7. Indices of more complicated, non hyperbolic points can be found by adding
the indices for the simpler fixed points that may appear under perturbation.
This is because a small change in the system does not change the index round a curve
where the vector field is smooth. e.g. index of a saddle-node ẋ = x2 , ẏ = −y is zero,
since indices of saddle and node cancel.
The most important non-trivial result that can be proved is that any periodic orbit contains at
least one fixed point. In fact the total number of nodes and foci must exceed the total number
of saddles by one. Proof: simple exercise.
34
4.2
Poincaré-Bendixson Theorem
This remarkable result, which only holds in R2 , is very useful for proving the existence of
periodic orbits.
Theorem 10 (Poincaré-Bendixson). Consider the system ẋ = f (x), x ∈ R2 , and suppose
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that f is continuously differentiable. If the forward orbit O+ (x) remains in a compact (closed
and bounded) set containing no fixed points then ω(x) contains a periodic orbit.
We can apply this directly to the pendulum equation with F > 1 to show that there is at least
one (stable) periodic orbit. However we cannot rule out multiple periodic orbits.
Proof: not trivial!! We first note that since O+ (x) remains in a compact set, then ω(x) is
non-empty. Recall that ω(x) is invariant under f . Choose y ∈ ω(x), and pick z ∈ ω(y). Since
z is not a fixed point, will show first that O+ (y) is a periodic orbit. Pick a curve Σ transverse
to flow. Then by continuity of f trajectories cross Σ in the same direction everywhere in a
neighbourhood of z. Since z ∈ ω(y), O+ (y) comes arbitrarily close to z as t → ∞, and so
makes intersections with Σ arbitrarily close to z. If in fact these are all at the same point, we
have a periodic orbit.
But suppose not. Then by uniqueness of the flow all intersections with Σ must be on the same
side of z, and distinct. This latter follows from uniqueness of flow, the former from the Jordan
35
curve lemma (see diagrams). If y 1 6= y 2 (first two intersections distinct) then the orbit O+ (x)
cannot return to the nbd. of y i (again,see picture + use Jordan curve lemma). But y i ∈ ω(x),
so we have a contradiction. Thus in fact y 1 = y 2 and O+ (y) is a periodic orbit, and since
O+ (y) ⊂ ω(x) we have that ω(x) contains a periodic orbit. Now in fact O+ (y) = ω(x). To
see this take a transverse section Σ through the orbit at y, say, and look at intersections of
close parts of φt (x) with Σ. If an intersection is at y then x is on the orbit and ω(x) = O+ (y).
Otherwise as before we have a monotonic sequence of intersections tending to y and so again
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ω(x) = O+ (y).
Example 19 Consider the system
x − y + 2x2 + axy − x(x2 + y 2 )
ẋ
. In polar coordinates we get
=
y + x + 2xy − ax2 − y(x2 + y 2 )
ẏ
√
√
ṙ = r + 2r2 cos θ − r3 , θ̇ = 1 − ar cos θ. Then ṙ > 0 for r < 2 − 1, and ṙ < 0 for r > 2 + 1.
√
√
Thus the trajectories enter the annulus 2 − 1 ≤ r ≤ 2 + 1. For any fixed points we have
1 + 2r cos θ − r2 = 0, 1 − ar cos θ = 0. Hence x = 1/a, r2 = 1 + 2/a. So there are no fixed
√
√
points in the annulus, and so no periodic solutions, if 1/a > 1 + 2 or if 1/a < 1 − 2.
4.3
Dulac’s criterion and the divergence test
Suppose that ẋ = f (x) has a periodic orbit C. Then f is tangent to C at every point and since
there are no fixed points on C we have that ρ(x)f (x) · n = 0 everywhere on C, where ρ is a C 1
non-negative function and n is the unit outward normal. Then
I
C
ρf · ndℓ =
Z
A
∇ · (ρf )dA = 0
Thus unless ρ and f are such that ∇ · (ρf ) is not all of one sign within the periodic orbit, we
have a contradiction. This is is a special case of
Theorem 11 (Dulac’s negative criterion). Consider a dynamical system ẋ = f , x ∈ Rn and
a non negative scalar function ρ(x), so that ∇ · (ρf ) < 0 everywhere in some domain E ⊆ Rn ,
36
or if the same expression is > 0 everywhere in E. Then there are no invariant sets of finite
volume V lying wholly within E.
Proof: suppose there is such a set V , and consider ρ as a “density”; then
R
V
ρdV is the total
“mass” within V and since V is invariant this is independent of time. Thus the total “mass
R
R
flux” ∂V ρf · ndS out of V is zero, and this equals V ∇ · (ρf )dV from the divergence theorem.
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Thus we have a contradiction.
Often it is only necessary to take ρ = 1. If conditions of the theorem satisfied in R2 then there
are no periodic orbits in E, if in R3 there are no invariant 2-tori, etc.
Example 20 (Lorenz system). Here we are in R3 and equations are

  
σy − σx
ẋ
 ẏ  =  rx − y − zx  .
−bz + xy
ż
Hence ∇ · f = −σ − 1 − b < 0. Thus this system has no invariant sets (including tori) with
non-zero volume. If there is a strange attractor it must have zero volume (typically it will be a
fractal set). Note that periodic orbits (of zero volume) are not excluded.
Example 21 (Damped pendulum with torque). We have seen using Poincaré-Bendixson that
when F > 1 there are no fixed points and that there must be a periodic orbit encircling the
cylinder. IN this case ∇ · f = −k < 0 and so there are no periodic orbits not encircling the
cylinder. We cannot rule out orbits that do encircle the cylinder as they have no ‘interior’ and
so theorem does not apply. But suppose there are two periodic orbits (both necessarily encircling
the cylinder). Then the region between them on the cylinder surface must be invariant, which is
not allowed by a simple extension of the criterion above. Thus we have a unique stable periodic
orbit of rotational type when F > 1.
Example 22 (Predator-Prey equations). These take the general form
37
ẋ
ẏ
=
x(A − a1 x + b1 y)
y(B − a2 y + b2 x)
, where a1 , a2 > 0. The lines x = 0, y = 0 are invariant. Are
there periodic orbits in the first quadrant? Consider ρ = (xy)−1 . Then ∇ · ρf =
a1 x + b1 y) +
∂
(x−1 (B
∂y
∂
(y −1 (A
∂x
−
− a2 y + b2 x) = −a1 y −1 − a2 x−1 < 0. So no periodic orbits in this
case. When a1 = a2 = 0 we have an infinite number of periodic orbits, which are contours of
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U (x, y) = A ln y − B ln x + b1 y − b2 x, provided these are closed curves [when is this true??]
Another result that is sometimes useful relies on the fact that if there is a periodic orbit the
vector field f is everywhere tangent to the orbit and is never zero.
Theorem 12 (Gradient criterion). Consider a dynamical system ẋ = f where f is defined
throughout a simply connected domain E ⊂ R2 . If there exists a positive function ρ(x) such
that ρf = ∇ψ for some single valued function ψ, then there are no periodic orbits lying entirely
within E.
Proof: if ρf = ∇ψ then for a periodic orbit C,
as ρf · dℓ has the same sign everywhere on C.
Example 23 Consider the system
ẋ
ẏ
=
H
C
ρf · dℓ =
H
C
∇ψ · dℓ = 0. But this is impossible
2x + xy 2 − y 3
−2y − xy 2 + x3
. Hard to apply Dulac. But
in fact exy f = ∇(exy (x2 − y 2 )) and so there are no periodic orbits.
4.4
Near-Hamiltonian flows
Many systems of importance have a Hamiltonian structure; that is they may be written in the
form
Hy
ẋ
=
−Hx
ẏ
38
Then it is easy to see that
dH
dt
= ẋHx + ẏHy = 0 so that the curves H = const. are invariant.
If these curves are closed then they are periodic orbits, but not limit cycles since they are not
Hxy
Hyy
. Thus the trace is always
isolated. The jacobian of f at the fixed point is
−Hxx −Hxy
zero and fixed points are either saddles or (nonlinear) centres.
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Example 24 Consider ẍ + x − x2 . Writing y = ẋ, we have an equation in hamiltonian form,
with H = 21 (y 2 + x2 ) − 31 x3 . There are two fixed points, at y = 0, x = 0, 1, which are a centre
and a saddle respectively. Using symmetry about the x-axis we can construct the phase portrait.
If a system can be written in Hamiltonian form with perturbations, we can find conditions for
periodic orbits. Suppose we have the system
ẋ
ẏ
=
Hy + g1 (x, y)
−Hx + g2 (x, y)
(2)
then we can see that
dH
= g2 ẋ − g1 ẏ .
dt
If there is a closed orbit C, we have
(3)
(g1 Hx + g2 Hy )dt =
C
H
y
−x + x2 + ǫy(a − x)
H
C
g2 dx − g1 dy = 0. If we can show
that these line integrals cannot vanish, we have shown that there can be no periodic orbits.
Example 25 Consider
ẋ
ẏ
=
(ǫ > 0). Choose the same value
of H as in the previous example; then for a periodic orbit C we must have
H
C
y 2 (a − x) dt = 0.
We can see from the equation that the extrema of x are reached when y = 0, and that the fixed
points of the system are still at (0, 0) (focus/node) and (1, 0) (saddle) (Exercise: prove this).
The index results show that no periodic orbit can enclose both fixed points, so the maximum
value of x on a periodic orbit is 1. Thus if a > 1 there are no periodic orbits.
39
If the flow is nearly Hamiltonian, and the value of H is such that the Hamiltonian orbit is
closed,we can derive an approximate relation for the rate of change of H. From equation
(3) we have exactly
dH
dt
= g2 ẋ − g1 ẏ. If g1 , g2 are very small, then H scarcely changes and
the trajectories are almost closed. If we average over many periods of the Hamiltonian flow,
then the rapid oscillations in the r.h.s. are averaged out and we find an equation for the slow
variation of H:
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dH
≈ F(H) = hg2 ẋ − g1 ẏi
dt
where the brackets denote an average over a period of the Hamiltonian flow, with the quantities
to be averaged evaluated for the Hamiltonian flow itself. Looking for fixed points of this reduced
system is equivalent to finding periodic orbits of the nearly-Hamiltonian flow. The error in
assuming that the gi = 0 when calculating ẋ as a function of x (and H) leads to errors of order
|gi |2 . Alternatively one can integrate (3) around a single period of the Hamiltonian flow, giving
RP
a map Hn+1 = Hn + ∆(H), where ∆(H) ≈ 0 g2 ẋ − g1 ẏdt, where to leading order the integral
is calculated over a period and path of the Hamiltonian flow. Fixed points of the map give
the value of H whose value approximates to a periodic orbit of the perturbed system. This
method is known as the energy balance or Melnikov method though the latter is usually applied
to non-autonomous perturbations to Hamiltonian flows.
Example 26 Consider the same example as above but now with ǫ ≪ 1. Then for the Hamilq
tonian flow we have y = ± 2H − x2 + 32 x3 . The equation for the slow variation of H is
then
dH
ǫ
=
dt
P (H)
I
2
(a − x)y dt =
(Alternately we have ∆H = 2ǫ
R xmax
xmin
2ǫ
R xmax
1
(a − x)(2H − x2 + 32 x3 ) 2 dx
xmin
R xmax
(2H
2 xmin
1
− x2 + 23 x3 )− 2 dx
,
1
(a − x)(2H − x2 + 32 x3 ) 2 dx,) where xmin , xmax are the
extrema of x on the orbit. We can write the integrals in terms of x by noting that ydt = dx.
Setting the l.h.s.=0 gives the relationship between H and a. The integrals cannot be done
exactly, except in the special case where H takes the value 1/6, for which the orbit is very close
to the homoclinic orbit of the Hamiltonian flow. Then the integrals can be done exactly and we
find that the critical value of a is 1/7. Thus we see that the bound a = 1 is not very good, at
least at small ǫ!
40
4.5
4.5.1
Stability of Periodic Orbits
Floquet multipliers and Lyapunov exponents
While individual points on a periodic orbit are not stable, we can look at the stability of the
whole orbit as an invariant set. We can develop a theory (Floquet theory) for determining
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whether an orbit is asymptotically stable to infinitesimal disturbances. Consider again ẋ =
f (x), and suppose there is a periodic orbit x = x̂(t). Letting x = x̂ + ξ(t), and linearizing, we
get
ξ̇ = A(t)ξ , where A = Df x=x̂(t) .
(4)
This is a linear equation with periodic coefficients, and there is much theory concerning it. We
want to know what happens to an initial disturbance ξ(0) after one period P of the original
orbit. Integrating equation (4) from t = 0 to t = P , we find that due to linearity we will have
the relation ξ(P ) = Fξ(0), where F is a matrix that depends only on the dynamical system
and on x̂, not on ξ(0). The eigenvalues of this matrix are called Floquet multipliers. One
of them is always unity for an autonomous system because equation (4) is always solved by
˙
ξ(t) = x̂(t).
Another way to find the exponents is via a map. We construct a local transversal subspace Σ
through x̂(0). Then all trajectories close enough to the periodic orbit intersect Σ in the same
direction as the periodic orbit. Successive intersections of trajectories with Σ define a map (the
Poincaré [Return] Map Φ : Σ → Σ. If z 0 is the intersection of x̂ with Σ then Φ(z 0 ) = z 0
so that z 0 is a fixed point. Linearizing about this point we have Φ(z) = z 0 + (DΦ)(z − z 0 ),
where DΦ is an (n − 1) × (n − 1) matrix. Then the Floquet multipliers can be defined as
the eigenvalues of DΦ. This method suppresses the trivial unit eigenvalue described above.
It is easy to prove that the choice of intersection with the periodic orbit does not affect the
eigenvalues of DΦ [Exercise].
We can define the stability of the orbit to small perturbations in terms of the multipliers
µi , i = 1, 2, . . . , (n − 1).
41
Definition 22 (Hyperbolicity). A periodic orbit is hyperbolic if none of the µi lie on the unit
circle.
Then we have the theorem on stability analogous to that for fixed points:
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Theorem 13
(i) A periodic orbit x̂(t) is asymptotically stable (a sink) if all the µi satisfy |µi | < 1.
(ii) If at least one of the µi has modulus greater than unity then the orbit is unstable (i.e. not
Lyapunov stable).
Proof: very similar to that for fixed points: simple exercise.
Definition 23 (Lyapunov exponents (Floquet exponents)). The Lyapunov Exponents λi of
a periodic orbit are defined as λi = P −1 log |µi |, where the µi are the Floquet multipliers and P
is the period. These are a measure of the rate of separation of nearby orbits.
In R2 there is only one non-trivial µ, which must be real and positive (see proof of PoincaréBendixson Theorem).
In this case we have µ = exp
R
P
0
∇ · f (x̂(t)) dt .
Proof: consider infinitesimal rectangle of length δs and width δξ at x̂(0). Then A(0) = δξδs.
R
By standard result for conservation of particles, Ȧ = ∂A f · ndS ∼ ∇ · f × A. Thus
R
P
log(A(P )/A(0)) = µ = exp 0 ∇ · f (x̂(t) dt , as required.
42
*Remark*: when we are in Rn , n ≥ 3 the µi may be complex; this leads to a wide variety of
possible ways in which periodic orbits can lose stability:
• µ passes through +1. This is similar to bifurcations of fixed points (saddle-node, pitchfork
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etc.)
• µ passes through e±2ikπ/n (k, n coprime). This leads to a new orbit that has a period
approximately n times the original. IN particular when n = 2k we have twice the period.
• µ passes through e±2iνπ , ν irrational. the new solution is a 2-torus.
Example 27 (Damped Pendulum with Torque) Here we have ∇ · f = −k, and so µ < 1 and
the periodic orbit that we have already shown to exist is thus stable.
4.6
Example – the Van der Pol oscillator
This much studied equation can be derived from elementary electric circuit theory, incorporating
a nonlinear resistance. It takes the form
Ẍ + (X 2 − β)Ẋ + X = 0
If β > 0 then we have negative damping for small X, positive damping for large X. Thus we
would imagine that oscillations grow to finite amplitude and the stabilize. It’s convenient to
√
scale the equation by writing X = βx, so that
ẍ + β(x2 − 1)ẋ + x = 0 .
43
(5)
This equation is a special case of the Liénard equation ẍ + f (x)ẋ + g(x) = 0. For analysis it
Rx
is convenient to use the Liénard Transformation. We write y = ẋ + F (x); F (x) = 0 f (s)ds.
Then in terms of x, y we have
ẋ
ẏ
=
y − F (x)
−g(x)
(6)
For the Van der Pol system, we have g(x) = x, F (x) = β( 13 x3 − x) If β > 0 then it is easily seen
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that the (only) fixed point, at the origin, is unstable. It is hard to apply Poincaré-Bendixson,
however, since the distance from the origin does not decrease monotonically at large distances.
However, we expect that there is a stable limit cycle. Can use qualitative methods to show
this, but first we look at the cases of small and large β.
(a) β ≪ 1. In this case the system is nearly Hamiltonian, with H = 21 (x2 +y 2 ), and g1 = −F (x),
√
√
g2 = 0. Hence the Hamiltonian flow with Hamiltonian H is x = 2H sin t, y = 2H cos t, and
the period P is 2π. Thus the map relation yields
∆H = −
Z
2π
ẏF (x)dt = β
0
Z
2π
x4
(x − )dt = β
3
2
0
Z
2π
0
(2H sin2 t −
4H 2
H2
sin4 t)dt = 2πβ(H −
)
3
2
Clearly ∆H > 0 (< 0) if H < 2 (H > 2) and so the map has a stable fixed point at H = 2,
corresponding to a periodic orbit with x ≈ 2 sin t.
(b) β ≫ 1. If x2 is not close to unity then the damping term is very large, so we might expect
ẋ = O(β −1 ); in fact if we write Y = yβ −1 we get
ẋ
Ẏ
=
β(Y − 13 x3 + x)
−xβ −1
so Ẏ is very small and so Y only varies slowly. Either ẋ ≫ 1 or else Y ≈ 31 x3 − x.
44
So the trajectory follows this curve (the slow manifold) until it runs out, and then moves quickly
(with ẋ ≫ Ẏ ) to another branch of the curve. Oscillations of this kind are called relaxation
oscillations. Assuming that there is such an orbit (prove below) we can find an approximation
to the period. Almost all the time is taken on the slow manifold, so
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P =2
Z
B
dt = 2
A
Z
2
3
− 32
dY
=2
Ẏ
Z
2
3
− 32
β
− dY = 2β
x
Z
−1
−2
(1 − x2 )
dx = β(3 − 2 ln 2)
x
In fact we can prove that there is a unique periodic orbit for all positive β. F (x) has three zeroes
√
√
at − 3, 0, 3 and F (x) → ±∞ as x → ±∞. The same methods can be used for a variety of
Rx
different forms of g, F (see various old Tripos questions). If we define G(x) = 0 g(s)ds = 21 x2
and H = 21 (y 2 + x2 ) then Ḣ = −βx2 (1 − x2 /3). Consider the nullclines ẋ = 0 (y = F (x)) and
ẏ = 0 (x = 0) . We show that trajectories enter the four regions of the plane bounded by the
nullclines, and that we can define a return map for H which has at least one stable fixed point.
45
First note that if x > 0, y > F (x) x is increasing, y is decreasing. Thus trajectory must cross
the curve y = F (x). Now both x and y are decreasing, and ẏ decreases in magnitude as the
origin is approached. Thus trajectory must cross into x < 0 where now ẏ > 0. Repeating the
argument we see that trajectory must return to x > 0, y > F (x) eventually. For sufficiently
small orbits around the origin Ḣ > 0 and so we see that for sufficiently small orbits H increases,
in fact by looking at the time reversed problem with β → −β we can show that eventually
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x2 + y 2 > 3 and remains so.
Need to show that large enough initial conditions lead to H decreasing. Consider diagram.
Then
Z √3
Z B
Ḣ
βx2 (1 − x2 /3)
dx = √
dx
H(B) − H(A) =
2
A ẋ
− 3 y − βx(x /3 − 1)
This is positive but a monotonically decreasing function of y. On the other hand
H(C) − H(B) =
Z
C
B
Ḣ
dy =
ẏ
Z
C
B
−βx(1 −
x2
)dy
3
√
This is negative and gets more negative as y( 3) increases. So H(C) − H(A) decreases mono-
tonically with y(A) and so there is a value of y(A) for which y(C) = y(A). By symmetry this
leads to a unique closed orbit for some y(A).
46
5
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5.1
Bifurcations
Introduction
We return now to the notion of dynamical systems depending on one or more parameters
µ1 , µ2 , . . .. We are interested in parameter values for which the system is not structurally
stable. Recall the definitions:
Definition 24 (Topological equivalence). Two vector fields f g and associated flows φf , φg are
topologically equivalent if ∃ a homeomorphism (1-1, continuous, with continuous inverse)
h : Rn → Rn , and a map τ (t, x) → R, strictly increasing on t, s.t.
τ (t + s, x) = τ (s, x) + τ (t, φfs (x)) , and φgτ (t,x) h(x) = h(φft (x))
(Structural stability). The vector field f is structurally stable if for all twice differentiable
vector fields v ∃ǫv > 0 such that f is topologically equivalent to f + ǫv for all 0 < ǫ < ǫv .
It turns out that if we change parameters for a given f (x, µ) then we will have structural stability in general except on surfaces where f is not structurally stable. We define a bifurcation
point as a point in parameter space where f is not structurally stable. A bifurcation (change
in structure of the solution) will occur when the parameters are varied to pass through these
points. If we plot e.g the amplitude of the fixed points and periodic orbits etc as the parameters
are varied, as functions of the parameters, this is called a bifurcation diagram.
47
5.2
5.2.1
Stationary bifurcations in R2
One-dimensional bifurcations
Stationary bifurcations occur when one eigenvalue of the Jacobian for a given fixed point is zero.
These are best understood initially in one dimension. Suppose we have an o.d.e. ẋ = f (x, µ),
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and that when µ = 0 the equation has a fixed point at the origin, which is non-hyperbolic.
Thus we have f (0, 0) = ∂fx (0, 0) = 0 (subscripts denote partial derivatives). There are three
possible types of bifurcation involving one parameter. The first is the generic case, and the
others will be found when there are particular restrictions on the system.
√
√
1. Saddle-Node Bifurcation. ẋ = µ − x2 . We have x = + µ (stable) and x = − µ
(unstable) when µ > 0, a saddle-node at 0 when µ = 0 and no fixed points for µ < 0.
2. Transcritical Bifurcation. ẋ = µx − x2 . There are fixed points at x = 0, µ which
exchange stability at µ = 0.
√
3. Pitchfork Bifurcation. ẋ = µx ∓ x3 . Fixed point at x = 0, also at x = ± ±µ when
±µ > 0. The (. . . − x3 ) case is called supercritical, the other case subcritical; in the
supercritical(subcritical) case the bifurcating solutions are both stable(unstable).
More insight is gained by considering two-parameter families. Supppose we have
ẋ = µ1 + µ2 x − x2 .
48
This includes both families (1) and (2) as special cases. Fixed points are at
x=
µ2 ±
p
µ22 + 4µ1
provided that µ22 + 4µ1 > 0
2
There ia a single non-hyperbolic fixed point on the parabola µ22 + 4µ1 = 0, and none to the left
Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission.
of this curve.
Clearly passing through any point of the parabola yields a saddle-node bifurcation. To see a
transcritical bifurcation, it is necessary for the path in parameter space to be tangential to the
parabola. e.g. If we vary µ2 at fixed µ1 then in general only bifurcations are saddle-nodes.
We say that the saddle-node bifurcation is codimension 1 (i.e. bifurcation set has a dimension
one less than that of the parameter space). The o.d.e. ẋ = µ − x2 is a universal unfolding
of the saddle node ẋ = −x2 . We can also say that ẋ = µ − x2 is the normal form for the
saddle-node bifurcation, in the sense that generic vector fields near the saddle-node can be
reduced to this form by a near-identity diffeomorphism.
Example 28 (Reduction to standard form.) Consider a more general two-parameter family
ẋ = µ1 + µ2 x − x2
49
and try a change of variable y = x − α, so that
ẏ = (µ1 + µ2 α − α2 ) + y(µ2 − 2α) − y 2
Choosing α = µ2 /2 gives ẏ = (µ1 + µ22 /4) − y 2 , which is in the standard form. The threeparameter system
ẋ = µ1 + µ2 x − µ3 x2
can be reduced scaling time so that T = µ3 to
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µ1 µ2
dx
=
+ x − x2
dT
µ3 µ3
and so to the standard form.
We can now treat the case of general f (x, µ) provided that (as we assume) f can be expanded
in a Taylor series in both x and µ near the non-hyperbolic (bifurcation) point (0, 0). At this
point we already have f (0, 0) = 0 = fx (0, 0). Now suppose, what is generally true, that fxx 6= 0,
fµ 6= 0 at this point. Now we expand f in a double Taylor series about (0, 0):
f (x, µ) = f (0, 0) + xfx (0, 0) + µfµ (0, 0)
x2
µ2
+ fxx (0, 0) + xµfxµ (0, 0) + fµµ (0, 0) + O(3) .
2
2
Rearranging, we have
f (x, µ) = (µfµ + O(µ2 )) + x(µfxµ ) +
x2
fxx + O(3)
2
and this is in the correct form to be reduced to the standard saddle-node equation.
There are important special cases, non-generic in the space of all problems, that nonetheless
are of physical importance.
• Transcritical bifurcation. If the system is such that f (0, µ) = 0 for all µ (or can be
put into this form by a change of variable), then fµ (0, 0) = 0, and we have instead of the
above
f (x, µ) = µ2 x + µ3 x2 + O(3) ,
with all the higher order terms vanishing when x = 0. This is in the standard form for a
transcritical bifurcation.
50
• Pitchfork bifurcation. If the system has a symmetry; that is if the equation is unchanged under an operation on the space variables whose square is the identity, then
simple bifurcations are pitchforks. In R the only such operation is x → −x; for the
equations to be invariant f must be odd in x. Then expanding in the same way we get
f (x, µ) = µ2 x + µ4 x3 + O(x5 ). (In higher dimensions symmetries can take more compli-
cated form. For example, the system ẋ = µx − xy , ẏ = −y + x2 has symmetry under
Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission.
x → −x, y → y).
The saddle-node bifurcation is robust under small changes of parameters as shown above. But
transcritical and pitchfork bifurcations depend on the vanishing of terms, and therefore change
under small perturbations.
ẋ = ǫ + µx − x2
ẋ = ǫ + µx − x3
ẋ = µx + ǫx2 − x3
The most general ’unfolding’ of the pitchfork bifurcation takes two parameters:
ẋ = ǫ1 + µx + ǫ2 x2 − x3
51
(Diagram: exercise)
In R2 , The fixed point is non-hyperbolic if there is at least one purely imaginary (or zero)
eigenvalue.
Consider now the situation where ẋ = f (x, µ) has a non-hyperbolic fixed point at the origin,
Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission.
for some value µ0 of µ. There are four cases:
• (i) λ1 = 0, ℜλj 6= 0, j 6= 1. This is a simple, or steady-state bifurcation, and is
essentially the same as the 1D examples shown above. WE show this below when we
discuss the centre manifold.
• (ii) λ1,2 = ±iω, ℜλj 6= 0, j 6= 1, 2. This is an oscillatory or Hopf bifurcation, and leads
to the growth of oscillations.
• (iii) and (iv) There are two zero eigenvalues. Canonical form of matrix A is either
0 1
0 0
(Takens-Bogdanov bifurcation).
(double-zero bifurcation), or
0 0
0 0
They have quite different properties and are not seen generically as they need two parameters to be varied.
Note that there are extra technical requirements on the way the eigenvalues change with µ (e.g.
for (i) we must have dλ1 /dµ 6= 0 at µ = µ0 ). We shall only look at (i) and (ii) in detail.
5.3
The Centre Manifold
Consider first the simple bifurcation. By analogy with the hyperbolic case, when µ = µ0 the
linear system has a subspace on which the solutions decay (or grow) exponentially, and another
in which the dynamics is non-hyperbolic (the centre eigenspace as defined below). E.g. for
a saddle node we have ẋ = x2 , ẏ = −y. By analogy with the stable and unstable manifolds
and their relation to the stable and unstable subspaces in the hyperbolic case, we might expect
52
a manifold to exist (the centre manifold), which is tangent to the centre eigenspace at the
origin, and on which the dynamics correspond to that in the centre eigenspace.
Example 29 Consider the non-hyperbolic system
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ẋ
ẏ
=
x2 + xy + y 2
−y + x2 + xy
.
The linear system has ẏ = −y and so trajectories approach y = 0, which is the centre eigenspace.
In this space ẋ = x2 . Thus hope there is an invariant manifold tangent to y = 0 at the origin
on which ẋ ∼ x2 , (the Centre Manifold or CM) (see handout)
It turns out that this is true (see Centre Manifold Theorem below), though CM’s are not like
stable and unstable manifolds in that they are not unique (see example sheet 2).
We can find the CM in this example, assuming existence, by expansion. Suppose it is of form
y = p(x) ≡ a2 x2 +a3 x3 +a4 x4 +. . .. Then ẏ = −p+x2 +xp = ẋpx = (x2 +xp+p2 )px . This o.d.e.
for p(x) cannot be solved in general, but substitute the expansion for p, equate coefficients and
get a2 = 1, a3 = −1, a4 = 0. Thus the CM is given by p(x) = x2 − x3 + O(5). The dynamics
on the CM is then given by replacing y by p in the equation for ẋ:
ẋ = x2 + xp + p2 = x2 + x3 − x4 + O(6) + x4 − 2x5 + O(6) = x2 + x3 − 2x5 + O(6) ,
and so close to the origin we have ẋ ∼ x2 as expected so that we have a saddle-node for the
nonlinear system.
The existence of a centre manifold is guaranteed under appropriate conditions by the Centre
Manifold Theorem:
Theorem 14 (Centre Manifold Theorem). Given a dynamical system ẋ = f (x) in Rn with
a non-hyperbolic fixed point at the origin, let E c be the (generalised) linear eigenspace corresponding to eigenvalues of A = Df |0 with zero real part (the centre subspace), and E h the
53
complement of E c (the hyperbolic subspace). Choose a coordinate system (c, h), c ∈ E c ,
h ∈ E h and write the o.d.e. in the form
ċ
ḣ
=
C(c, h)
H(c, h)
.
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Then ∃ a function p : E c → E h with graph h = p(c), called the centre manifold which has
properties:
• (i) is tangent to E c at 0;
• (ii) is locally invariant under f ;
• (iii) dynamics is topologically equivalent to
ċ
ḣ

C(c,p(c))

=  ∂H h
∂h

0
• (iv) p(c) can be approximated by a polynomial in c in some neighbourhood of 0.
Thus in Example 29, c = x, h = y and p(x) = x2 − x3 + . . .; local dynamics is equivalent to
ẋ = x2 + xp(x) + p(x)2 , ẏ = −y
This is very helpful for non-hyperbolic systems, but we believe that such reductions ought to
be possible near, and not just at, bifurcation points. We can use the centre manifold theorem
by means of a trick:
To our original system ẋ = f (x, µ), which now has a hyperbolic fixed point at 0, adjoin the
equations µ̇ = 0. We now have a system in R(n+m) , where m is the number of parameters. In
this new system the terms giving the linearized growth rates as functions µ will be nonlinear,
and so we are at a non-hyperbolic fixed point of the extended system, and can use the CM
theorem to reduce the system.
54
Example 30 Consider the system
ẋ
ẏ
=
µ + x2 + xy + y 2
2µ − y + x2 + xy
.
Regarding µ as a variable we have
 
 
x
0 1
0
ẋ


 µ̇  =  0 0
µ  + nonlinear terms.
0
y
0 2 −1
ẏ
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
Thus the centre eigenspace is y = 2µ, which is a plane in R3 , and the CM is of the form
y = p(x, µ) = 2µ + a20 x2 + a11 xµ + a02 µ2 + . . . ,
we have
ẏ = 2µ − p + x2 + xp = −
∂p
∂p
(µ + x2 + xy + y 2 ) +
· 0 , or
∂x
∂µ
2µ + (2µ + a20 x2 + a11 xµ + a02 µ2 + . . .)(x − 1) + x2
= µ + x2 + x(2µ + a20 x2 + a11 xµ + a02 µ2 + . . .)+
.
+ (2µ + a20 x2 + a11 xµ + a02 µ2 + . . .)2 (2a20 x + a1 µ + . . .)
Equating coefficients, we find [x2 ] : a20 = 1, [xµ] : a11 = 0, [µ2 ] : a02 = 0. Thus the CM is
given by y = p(x, µ) = 2µ + x2 + O(3), and the dynamics on the extended CM is given by
ẋ =µ + x2 + x(2µ + . . .) + 4µ2 + . . .
µ̇ =0
This is clearly (cf. the one-parameter families above) a saddle-node bifurcation when µ = 0.
Of course none of this is really necessary if only the local behaviour is desired, but it gives one
confidence to proceed.
Thus we have the important result that for a dynamical system in Rn , with a simple bifurcation
at µ0 (i.e. the system has a single zero eigenvalue at µ = µ0 , and (wlog) all eigenvalues are in
+
the l.h. half plane for µ = µ−
0 , and there is just one eigenvalue in the r.h. half-plane for µ = µ0 ),
there is a centre manifold, the dynamics is essentially one-dimensional and the bifurcation is
generically a saddle-node.
55
5.4
Oscillatory/Hopf Bifurcations in R2
The simplest model of a Hopf (oscillatory) bifurcation is given by the system (in polar coordinates) ṙ = µr − r3 , θ̇ = 1. For µ < 0 we have a stable focus, and for µ > 0 an unstable
√
focus and a stable periodic orbit r = µ. This is a supercritical Hopf. If instead we have
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ṙ = µr + r3 then the periodic orbit exists when µ < 0, and is unstable (subcritical Hopf).
It turns out that generically all dynamical systems can be put into this form in the neighbourhood of a Hopf bifurcation. We need a technical definition of this bifurcation.
Definition 25 (Hopf bifurcation). Suppose we have a dynamical system ẋ = f (x, µ) =
(f (x, µ), g(x, µ)) which at µ = 0 has a fixed point at 0, and has linearization A satisfying detA > 0, TrA = 0 [so that the linearization has eigenvalues λ1,2 (0) = ±iω], and that
d(ℜ(λ1,2 )/dµ > 0 at µ = 0. Then provided a constant γ, defined as the value at µ = 0 of
1
fxxx + gyyy + fxyy + gxxy + ω −1 (fxy (fxx + fyy ) − gxy (gxx + gyy ) − fxx gxx + fyy gyy )
16
is not equal to zero, then there is a Hopf bifurcation at µ = 0 and there is a stable limit cycle
for µ = 0+ if γ < 0 (supercritical Hopf bifurcation) and an unstable limit cycle for µ = 0−
if γ > 0 (subcritical Hopf bifurcation).
Remark: because A is not singular at µ = 0, there is no change in the number of fixed points
for small |µ|. Most questions will not demand a knowledge of the exact form of γ, but will
usually be in recognizable form. Rather than derive the formula (see Glendinning if interested)
we show how the equation can be brought into standard, i.e. normal form at µ = 0 by a near
identity diffeomorphism.
56
Suppose then that µ = 0 and choose canonical coordinates so that A =
0 −ω
ω
0
. Then
writing (in these coordinates) z = x + iy, the linear part of the problem is ż = iωz. Then the
full equation must take the form
ż = iωz + α1 z 2 + α2 zz ∗ + α3 z ∗2 + O(3) .
Define a new complex variable ξ by ξ = z + a1 z 2 + a2 zz ∗ + a3 z ∗2 . We try to choose a1,2,3 so
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that ξ˙ = iωξ + O(3). In fact
ξ˙ = ż(1 + 2a1 z + a2 z ∗ ) + z˙∗ (a2 z + 2a3 z ∗ ) so correct to O(3),
iω(z + a1 z 2 + a2 zz ∗ + a3 z ∗2 ) = (iωz + α1 z 2 + α2 zz ∗ + α3 z ∗2 )(1 + 2a1 z + a2 z ∗ )
+ (−iωz ∗ + α1∗ z ∗2 + α2∗ zz ∗ + α3∗ z 2 )(a2 z + 2a3 z ∗ )
Equating coefficients of the quadratic terms we get
iωa1 = α1 + 2iωa1 ; iωa2 = α2 + iωa2 − iωa2 ; iωa3 = α3 − 2iωa3
and clearly these equations can be solved. Thus in the transformed system there are no
quadratic terms. Attempting to perform the same trick at cubic order, we find that all the cubic
terms can be removed except the term ∝ z 2 z ∗ [Exercise]. Thus after all the transformations
have been completed we are left with the equation
ż = iωz + νz 2 z ∗ + h.o.t., or ṙ = ℜ(ν)r3 , θ̇ = ω + ℑ(ν)r2
and γ ∝ ℜ(ν).
To find the canonical equation when µ 6= 0 we can either use the same ideas on the extended
CM that we used for the simple bifurcation, or just add the relevant linear terms; which can
be shown to yield the same result in non-degenerate cases.
Note that the normal form is appropriate only when r is sufficiently small. Because the constant
part of θ̇ is of order unity there will be no fixed points in a neighbourhood of the origin.
57
Example 31 Find the nature of the Hopf bifurcation for the system ż = (µ+iω)z +αz 2 +β|z|2 .
Clearly bifurcation point is at µ = 0, so choose this value and, guided by above analysis, choose
ξ = z + az 2 + b|z|2 . Then
ξ˙ = (iωz + αz 2 + β|z|2 )(1 + 2az + bz ∗ ) + (−iωz ∗ + α∗ z ∗2 + β ∗ |z|2 )bz
= iωz + (α + 2ia)z 2 + β|z|2 + |z|2 z(2aβ + αb + bβ ∗ ) + other cubic
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= iω(ξ − az 2 − b|z|2 ) + (α + 2ia)z 2 + β|z|2 + |z|2 z(2aβ + αb + bβ ∗ ) + other cubic
Now choose iωa = −α : iωb = β. Then quadratic terms vanish, and
= iωξ + (iω)−1 (|β|2 − αβ)|ξ|2 ξ + other cubic + O(4)
so that ℜ(ν) = −ℑ(αβ)/ω.
In fact it can be shown by successive transformations that the canonical form for the dynamics
on the CM for a Hopf bifurcation is ż = zF (|z|2 ), where F is a complex valued function with
F (0) = iω. This allows the treatment of degenerate situations for which ℜ(ν) = 0.
For higher dimensional systems the CM theorem can be invoked to show that in the neighbourhood of a Hopf bifurcation there is a 2-dimensional (extended) CM on which the dynamics can
be put into the above form.
58
5.5
*Bifurcations of periodic orbits*
(a) Homoclinic bifurcation
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This is the simplest ‘global’ bifurcation and occurs when the stable and unstable manifolds of a
saddle point intersect at a critical value µ0 of the parameter. There are two ‘flavours’ possible:
(b) ’Andronov bifurcation’
A very similar type of bifurcation arises when a saddle-node develops on a periodic orbit
(“Andronov bifurcation”).]
In each case, periodic orbits are produced as a result if the bifurcation, and we would like to
know the stability of these orbits.
59
6
Bifurcations in Maps
6.1
Examples of maps
We have seen that the study of the dynamics of flows near periodic orbits can be naturally
expressed in terms of a map (the Poincaré map). This helps to motivate the study of maps in
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their own right. There are other motivations too.
• Maps of the interval. These originally arose as discrete versions of 1D flows, but
have much richer structure. Suppose we have the ode ẋ = f ; if we have discrete time
intervals t0 , t1 , . . . , tn ≡ t0 + n∆t, . . ., with x(tn ) = xn then Euler’s method gives xn+1 =
xn + ∆tf (xn ). Alternatively can think of xn as a population at the nth generation.
Generalising gives the general nonlinear map xn+1 = g(xn ). Famous example is the
Logistic (quadratic) map :
xn+1 = µxn (1 − xn ) 0 < xn ≤ 1 :
1 ≤ µ ≤ 4.
Another important map (since calculations are easy!) is the
Tent (piecewise linear) map. An example is
xn+1 =
µxn
0 ≤ xn ≤ 21
µ(1 − xn ) 21 ≤ xn ≤ 1
60
0 < xn ≤ 1 :
1≤µ≤2
• Circle Maps. These are motivated by Poincaré maps for the instability of periodic
orbits. Use units in which the circumference of the circle is unity.
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Rotation. xn+1 = xn + ω [mod 1]. This is periodic if ω rational, aperiodic if ω irrational.
Standard Map. xn+1 = xn + µ sin 2πxn [mod 1]. When µ small this is almost the
identity, for larger µ get more interesting behaviour.
Sawtooth Map (Bernoulli Shift). xn+1 = 2xn [mod 1]. We can find the solution for
any x0 by expressing xn in binary form: xn = 0.i1 i2 i3 . . . in . . . where the ij are zero or
unity. Then xn+1 = 0.i2 i3 . . .. So if x0 = 0, xn = 0, if x0 is rational then binary expansion
repeats and so xn = x0 for some n (periodic) while if x0 is irrational solution is aperiodic.
This map is a prototype of chaotic behaviour.
The shift map can be seen as a special case of the logistic map. Put xn = sin2 πθn , with
θn satisfying the shift map. Then xn+1 = sin2 2πθn = 4xn (1 − xn ) so that xn satisfies the
logistic map with n = 4. In fact the these maps are topogically conjugate:
Definition 26 (Topological conjugacy for maps.) Two maps F and G are topologically
conjugate if there exists a smooth invertible map H such that F = H −1 · G · H.
61
• Maps of the Plane. These come from Poincaré maps of flows in R3 . As such they
should be invertible, or at least have unique inverse when an inverse exists.
Hénon Map
xn+1
yn+1
=
1 + yn − ax2n
bxn
For appropriate choice of a, b this has ‘strange’ behaviour with fractal structure.
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Baker Map. Composed of two maps:
xn+1
yn+1
=
 2x

n


1

y

2 n


2 − 2xn



1 − 12 yn
0 < xn < 21 , 0 < yn < 1
1
2
< xn < 1, 0 < yn < 1
So this is a map of [0 < x < 1, 0 < y < 1] into itself.
Horseshoe Map (Smale). Best seen in terms of diagram:
Set of points that do not leave box form a fractal Cantor-type set.
62
6.2
Fixed points, cycles and stability
We can define in a way analogous to flows fixed points and periodic points (cycles)
of a map:
Definition 27 Given a map xn+1 = f (xn ), a fixed point x0 satisfies x0 = f (x0 ), while
a periodic point with period n if x0 = f n (x0 ), x0 6= f m (x0 ), m < n. Here f 2 ≡ f ◦ f
(composition), f n = f ◦ f n−1 . A set of periodic points {x1 , x2 , . . . , xn = x1 } is called
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cycle.
We can define ω-points (and α-points for invertible maps with inverses) as points of
accumulation of iterates of xn as n → ∞ (−∞ for α-points).
We can define notions of Lyapunov stability and quasi-asymptotic stability just as for
flows. The notions can be combined into the notion of an attractor.
Definition 28 Suppose A is a closed set mapped into itself by f . This could be a fixed
point, cycle or a more exotic set. We suppose that f is continuous at points of A. Then
A is an attractor if
(i) For any nbd U of A ∃ a nbd V of A such that for any x ∈ V, f n (x) ∈ U ∀n ≥ 0
(Lyapunov stability);
(ii) ∃ an nbd W of A such that for any x ∈ W and any set A′ containing A ∃ n0 such
that f n (x) ∈ A′ ∀n > n0 (quasi-asymptotic stability).
Consider a fixed point (chosen to be at the origin) of xn+1 = f (xn ), where f has continuous first derivative at and near the origin. Then we can form the jacobian matrix
A = Df 0 . If we suppose that eigenvalues of A are distinct or there is a complete set of
eigenvectors, then we can show that if all the eigenvalues λ of A satisfy |λ| < 1, then the
origin is an attractor.
P
Proof: Let z j be the left eigenvectors of A (possibly complex). Then let Vn =
vi |z i ·xn |2 ,
P
for some positive set of numbers vi . Then Vn+1 =
vi |z i · Axn |2 + O(|xn |3 ). The first
P
term on the rhs is
vi |λi |2 |z i · xn |2 < a2 Vn , where a2 = max |λi |2 < (1 − ǫ), ǫ > 0. We
can choose Vn sufficiently small that the cubic remainder term is less than ǫVn /2, say,
and so Vn → 0, n → ∞. Conversely if any eigenvalue λ has |λ| > 1 the fixed point is a
repellor (neither QAS nor Lyapunov stable). Proof: exercise.
For a cycle of least period r each point of the cycle is a fixed point of the map f r , so
stability is determined by the eigenvalues of the jacobian of this map. If we write for each
63
point x(j) of the cycle A(j) ≡ Df (x(j) ) the linearization of the map about the cycle can
be written ξ n+1 = A(n) ξ n and so the linearization of f r is just A(r) A(r−1) . . . A(1) (can also
see this from the chain rule).
6.3
Local bifurcations in 1-dimensional maps
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Now the jacobians are just real numbers, and bifurcations must occur when eigenvalues
of a map passes through the unit circle, and so through ±1. We can classify these
bifurcations just as for 1-dimensional flows.
• Saddle-Node. If we suppose as before that f = f (x, µ) and that f (0, 0) = 0, fx (0, 0) = 1
then expanding in x, µ as before get
1
xn+1 = xn + µfµ + x2n fxx + xn µfxµ + . . .
2
and truncating, shifting the origin and rescaling we get the canonical form
xn+1 = xn + µ − x2n
which has no fixed points when µ < 0 and two fixed points when µ > 0, for which
√
x = ± µ and the jacobian is 1 − 2x. So for 0 < µ < 1 one of the fixed points is stable
and the other unstable. (something new happens for µ > 1, but the normal form is
supposed to be valid for sufficiently small µ).
64
• Transcritical bifurcation. Now suppose in addition that fµ (0, 0) = 0; then we have
1
xn+1 = xn + (x2n fxx + µ2 fµµ ) + xn µfxµ + . . .
2
Truncating and seeking fixed points, we need (x2 fxx + µ2 fµµ + 2xµfxµ ) = 0 and this is
2
only possible if fxµ
> fxx fµµ . Otherwise there are no fixed points and so no bifurcation.
If satisfied can write truncated system in canonical form
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xn+1 = xn − (xn − aµ)(xn − bµ)
for some a, b. There are two lines of fixed points x = aµ, x = bµ crossing at origin when
µ = 0, each exchanges stability as µ passes through zero. If we write yn = xn − aµ, say
get even simpler form yn+1 = yn − yn (yn − cµ).
• Pitchfork bifurcation. This is achieved as before when the eigenvalue is unity, and if
there is a symmetry (equivariance) under x → −x, in which case only odd terms occur
in the expansion of f . Then we have
xn+1 = xn + µxn ± x3n + O(5)
We also get a pitchfork when fµ = fxx = 0, fµx , fxxx 6= 0, in which case the correction is
O(4).
65
• Period-doubling bifurcation. The remaining case has eigenvalue −1 at the bifurcation
point. Thus in the general case
xn+1 = −xn + a′ + b′ xn + c′ x2n + d′ x3n + O(4)
where a′ , b′ , c′ , d′ are functions of µ, with a′ , b′ = O(µ), c′ = 21 fxx +O(µ), d′ = 16 fxxx +O(µ).
There is a fixed point of this system at x = x∗ , where 2x∗ ≈ a′ + b′ x∗ + c′ x∗2 + d′ x∗3 .
Changing variable so that y = x − x∗ , we get
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yn+1 = −yn + byn + cyn2 + dyn3
where b = b′ + a′ c′ + O(µ2 ) and c, d = c′ , d′ + O(µ). now f.p. is unstable when b < 0,
2
3
stable if b > 0. Consider the map f 2 . Then yn+2 = −yn+1 (1 − b) + cyn+1
+ dyn+1
, so that
yn+2 = (1 − b)((1 − b)yn − cyn2 − dyn3 ) + c(yn2 (1 − b)2 − 2c(1 − b)yn3 ) − d(1 − b)3 yn3 + . . .
simplifying and keeping only leading terms in b we get
yn+2 = (1 − 2b)yn − bcyn2 − 2(c2 + d)yn3
If we make a change of variable of the form zn = yn + αb, then correct to order b we
can remove the quadratic term to give zn+2 = (1 − 2b)zn − 2(c2 + d)zn3 . Thus there is
a pitchfork bifurcation of f 2 at b = 0. The nonzero fixed points of f 2 correspond to a
2-cycle of f . It can be shown that these cycles are stable if the origin is unstable and vice
verca.
66
7
7.1
Chaos
Introduction
What do we mean by chaos? Two main concepts (a) in a chaotic system, initially nearby
orbits separate, and also we have some sort of mixing of even the smallest sets by iterating
the map.
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Consider a continuous map f : I → I of a bounded interval I ⊂ R into itself, and with
Λ ⊂ I a set invariant under f :
Definition 29 Sensitive dependence on initial conditions. f has sensitive dependence
on initial conditions[SDIC] on Λ if ∃ δ > 0 s.t. for any x ∈ Λ and ǫ > 0 ∃y ∈ Λ and
n > 0 s.t. |y − x| < ǫ |f n (x) − f n (y)| > δ
Note that not all points in the neighbourhood have to separate in this way, and nothing
said about exponential divergence.
Definition 30 Topological Transitivity. The map f above is topologically transitive[TT] on Λ if for any pair of open sets K1 , K2 s.t. Ki ∩ Λ 6= ∅, i = 1, 2, ∃ n > 0
s.t. f n (K1 ) ∩ K2 6= ∅. This means that there are orbits that are dense in Λ, i.e. come
arbitrarily close to every point of Λ, and so Λ cannot be decomposed into disjoint invariant
sets
Example 32 (TT but not SDIC).The rotation map xn+1 = xn + ω[mod 1] is TT on [0, 1]
if ω is irrational (though not SDIC).
Example 33 (SDIC but not TT). The map xn+1 = 2xn (|xn | < 21 ), xn+1 = 2(sign(xn ) −
xn ) ( 12 < |xn | < 1) has SDIC on [−1, 1] since |f ′ | = 2, but is not TT on [−1, 1] as x = 0
is invariant.
There are two apparently quite different definitions of chaos, though they are more similar
than they look.
Definition 31 (Chaos [Devaney]). f : I → I is chaotic on Λ if (i) f is SDIC on Λ;
(ii) f is TT on Λ; (iii) periodic points of f are dense in Λ.
The second definition depends on the Horseshoe property.
67
Definition 32 (Horseshoe property). f : I → I has a horseshoe if ∃ J ⊆ I and disjoint
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open subintervals K1 , K2 of J s.t. f (Ki ) = J for i = 1, 2.
If f has a horseshoe it can be shown that (i) f n has at least 2n periodic points; (ii) f has
periodic points of every period; (iii) f has an uncountable number of aperiodic orbits.
Definition 33 (Chaos[Glendinning]) A continous map f : I → I is chaotic if f n has a
horseshoe for some n ≥ 1.
This last definition allows for maps with stable fixed points, but demands exponential
divergence of nearby trajectories. It will be shown that Chaos[G]⇒Chaos[D].
7.2
The Sawtooth Map (Bernoulli shift)
As an example consider the sawtooth map f (x) = 2x[mod 1].
(a) f clearly has a horseshoe with K1 = (0, 12 ), K2 = ( 21 , 1). (Note, open sets). So f is
chaotic[G].
(b) (i) As before use binary expansion so if xn = 0.a1 a2 a3 . . . then xn+1 = 0.a2 a3 . . .. So
if expansion repeats after n terms we have a cycle of period n. There are 2n of these.
Clearly also periodic points are dense in [0, 1].
If x 6= y then suppose x and y differ first in the (n + 1)th place; then for r ≤ n |f r (x) −
f r (y)| = 2r |x − y|[mod 1].
Choose δ < 12 . Then given any x ∈ [0, 1) and ǫ > 0 choose n so that 2−n−1 < ǫ and choose
y to differ from x only in the n + 1 binary place. Then |f n (y) − f n (x)| =
SDIC.
68
1
2
> δ, so f is
(ii) Choose any point x = 0.a1 a2 . . .; then for another point z = 0.b1 b2 . . . choose
yN =
0.b1 b2 . . . bN a1 a2 . . .. Then f N (yN ) = x and we can make yN arbitrarily close to z, by
taking N sufficiently large, so map is TT.
(iii) Now define xN = 0.a1 a2 . . . aN a1 a2 . . . aN a1 . . . (or 0.a˙1 a2 . . . a˙N . This is certainly a
periodic point and we can make xN arbitrarily close to x. So periodic points are dense in
[0, 1).
Thus f is chaotic[D].
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In fact f is a very effective mixer. Define f ( 12 ) = 1 and suppose x is not a preimage of
1/2, i.e. that there is no n s.t. f n (x) = 21 . Thus x does not end with a infinite sequence
of 0’s (or 1’s). Choose m s.t. am+1 = 0 Then it is easy to see that y = 0.a1 . . . am 0 0 . . . <
x < z = 0.a1 . . . am 1 0, . . ., and that f n (y) = 0, f n (z) = 21 . Thus f n+1 ((y, z)) = (0, 1),
and so any arbitrarily small neighbourhood of x can be mapped into the whole interval.
(this is because the map is 1-1 on each half range and by construction y, z lie in the same
half range and so do f (y), f (z).)
7.3
Horseshoes, symbolic dynamics and the shift map
Aim to show that a map with the horseshoe property acts on a certain invariant set Λ in
the same way that as the shift map.
Suppose a continous map f has a horseshoe on an interval J ⊂ R and define the closed
intervals I = J, Ii = Ki , i = 1, 2.
Assume that f is monotonic on Ii , I1 ∩ I2 = ∅ and that f (x) ∈ I ⇒ x ∈ I1 or I2 . Define
Λ = {x : f n (x) ∈ I, ∀n ≥ 0}. Clearly x ∈ Λ ⇒ f (x) ∈ Λ, and x ∈ Λ ⇒ x = f (y) for some
y ∈ I (Intermediate value theorem). Thus f (Λ) = Λ so Λ is invariant.
For each x ∈ Λ f n (x) ∈ I ⇒ f n−1 (x) ∈ I1 or I2 . Define an = 0 if f n−1 (x) ∈ I1 , an = 1
if f n−1 ∈ I2 . Thus x corresponds to the sequence a1 a2 . . . while f (x) corresponds to
a2 a3 . . .. This is essentially the same as the action of the shift map on binary expansions
of numbers in [0, 1]. There are small differences as 0.111 . . . and 1.000 . . . are the same
69
number but different symbol sequences; however this does not affect the proofs of TT and
SDIC. Thus chaos[G]⇒horseshoe⇒chaos[D].
What does Λ look like? The set f −1 (I) ≡ {x ∈ I : f (x) ∈ I} comprises two disjoint
T
−n
closed intervals. So f −n (I) has 2n closed intervals and Λ = ∞
(I). Limit gives a
n=1 f
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closed set with an uncountable number of points but length zero, cf. the middle third
Cantor set.
7.4
Period 3 implies chaos
Recall the Intermediate Value Theorem: If f : [a, b] → r is continuous, and f (a) =
c, f (b) = d, then ∀y ∈ [c, d]∃x ∈ [a, b] s.t. f (x) = y. In particular if f (x) − x changes
sign on [a, b] then ∃x0 ∈ [a, b] then ∃x0 ∈ [a, b] s.t. f (x0 ) = x0 . We can now prove the
remarkable theorem:
Theorem 15 (Period 3 implies chaos). If a continuous map f on I ⊆ R has a 3-cycle
then f 2 has a horseshoe and so f is chaotic.
Proof: let x1 < x2 < x3 be the elements of the 3-cycle. wlog suppose f (x1 ) = x2 , f (x2 ) =
x3 , f (x3 ) = x1 (otherwise consider instead −f (−x)).
f (x2 ) = x3 > x2 , f (x3 ) = x1 < x2 ⇒ ∃z ∈ (x2 , x3 ) s.t. f (z) = z
f (x1 ) = x2 < z, f (x2 ) = x3 > z ⇒ ∃y ∈ (x1 , x2 ) s.t. f (y) = z
Thus f 2 (y) = f 2 (z) = z > y and f 2 (x2 ) = x1 < y, so ∃ a smallest r ∈ (y, x2 ) s.t. f 2 (r) =
y, and ∃ largest s ∈ (x2 , z) s.t. f 2 (s) = y. Thus f 2 has a horseshoe with K1 = (y, r),
K2 = (s, z) and J = (y, z).
70
7.5
Existence of N-cycles
Have shown that f 2 has a horseshoe if there is a 3-cycle, which implies existence of cycles
for f 2 of all periods. In fact can show that f has cycles of all periods.
Lemma. Recall that if f s continuous and V ⊆ f (U ), where U, V are closed intervals,
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the ∃ a closed interval K ⊆ U s.t. F (K) = V .
Theorem 16 If a continuous map f on ⊆ R has a 3-cycle the it has an N -cycle ∀ N ≥ 1.
Take setup as before.
N = 1: f (x3 ), x2 < x3 = f (x2 ) ⇒ there is a fixed point of the map.
N > 1: let IL = [x1 , x2 ], IR = [x2 , x3 ]. Then f (IL ) ⊇ IR , f (IR ) ⊇ IL ∪ IR .
Choose JN = IR . Then define F (JN −1 ) = JN , JN −1 ⊆ IL by the Lemma. Similarly define
JN −2 , . . . , J0 ⊆ IR by f (Ji ) = Ji+1 .
f N (J0 ) = IR ⇒ ∃a, b ∈ J0 s.t. f N (a) = x2 , f N (b) = x3 . But J0 ⊆ IR so a ≥ f N (a), b ≤
f N (b) Thus by IVT there is a fixed point z ∈ [a, b] of f N . f N −1 (z) ∈ IL . so only possibility
is if f N −1 (z) = x2 ⇒ f N (z) = x3 = z.
But f 2 (x3 ) = x2 6= x3 . Also f (x3 ) = x1 6∈ IR so violating construction. Thus f N −1 (z) 6=
x2 so must not be in IR This shows that one iterate is definitely different from all others
so an N -cycle.
The statements f (Il ) ⊇ IR , f (IR ) ⊇ IL ∪ IR can be shown as a directed graph:
Cycles exist when there are closed paths in the diagram.
Example 34 Suppose there is a 4-cycle x1 = f (x4 ) < x3 = f (x2 ) < x2 = f (x1 ) < x4 =
f (x3 ). Let IA = [x1 , x3 ], IB = [x3 , x2 ], IC = [x2 , x4 ]. Only fixed points in IC and 2-cycles
between IA , IB possible.
71
A remarkable result due to Sharkovsky can be proved (proof not in course) by similar
methods to the above.
Theorem 17 (Sharkovsky.) If f : R → R is continous, f has a k-cycle and l ⊳ k in the
following ordering, then f also has an l-cycle:
1 ⊳ 2 ⊳ 22 ⊳ 23 ⊳ 24 ⊳ . . .
...
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. . . ⊳ 23 · 9 ⊳ 23 · 7 ⊳ 23 · 5 ⊳ 23 · 3
. . . ⊳ 22 · 9 ⊳ 22 · 7 ⊳ 22 · 5 ⊳ 22 · 3
... ⊳ 2 · 9 ⊳ 2 · 7 ⊳ 2 · 5 ⊳ 2 · 3
... ⊳ 9 ⊳ 7 ⊳ 5 ⊳ 3
This has many implications, not least that if f has a cycle of period 3 then it has cycles
of all periods, as proved separately above.
Note that the theorem says nothing about the stability of the cycles. However for the
logistic equation at least we know that all cycles either arise from a period-doubling
bifurcation, or in the case of the odd-period cycles as a saddle-node bifurcation so they
are all stable in some range!!
The period-three orbit for the logistic map. Shown is the map f 3 where f (x) = µx(1 − x).
Top picture: µ = 3.81, bottom picture; µ = 3.84.
72
7.6
The Tent Map
A more interesting map, because it depends on a parameter, is the tent map f (x) =
Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission.
µ[ 12 − |x − 21 |]
Fixed point at 0 stable for µ < 1. Choose µ ∈ (1, 2]. In this range the origin is unstable
and the interval [0, 1] is mapped into itself. There is a f.p. x0 = µ/(1 + µ), which is always
unstable (exercise).
We show that the map is chaotic[G] when 1 < µ ≤ 2.
Step 1. All non-zero orbits eventually enter and stay in the interval A = [f 2 ( 21 ), f ( 12 )] =
[µ(1 − µ/2, µ/2]. Note that if the preimage of x0 , [x−1 = 1/(1 + µ)] ∈ A; i.e. if 1/(1 + µ) >
√
µ(1 − µ/2) then µ > 2.
Step 2. Now consider f 2 (x):
1
0≤x≤ µ
2
1
1
µ≤x≤
2
2
1
1
≤x≤1− µ
2
2
1
1− µ≤x≤1
2
f 2 (x) = µ2
µ(1 − µx)
µ(1 − µ(1 − x))
µ2 (1 − x)
Let x−2 be the preimage under f of x−1 (Or the preimage under f 2 of the fixed point x0 )
in x > 21 . Then x−2 = (µ2 + µ − 1)/µ(µ + 1).
73
Step 3. By a change of coordinates we can see that f 2 acts like a tent map with parameter
µ2 on the two intervals JL = [x−1 , x0 ] and JR = [x0 , x−2 ].
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Two different cases:
0.7
F2
1
0.6
3
4
F
x–2
0.5
2
x–1
0.4
0.4
For µ <
√
x0
0.5
0.6
0.7
2, (see picture for µ = 1.3), f 2 gives tent maps with parameter µ2 on the inter-
vals [x−1 , x0 ] and [x0 , x−2 ], and the attracting set (shaded) has two components defined
by f i ( 21 ), i = 1, . . . , 4 (circles).
74
0.8
F2
1
0.7
4
0.6
x–2
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3
0.5
0.4
0.3
0.3
For µ >
√
F
x–1
0.4
x0
0.5
0.6
2
0.7
0.8
2, (see picture for µ = 1.5), f 2 has horseshoes on the intervals [x−1 , x0 ] and
[x0 , x−2 ], and the attracting set (shaded) has one component since f 4 ( 21 ) > f 3 ( 12 ).
√
n+1
has a horseshoe on 2n intervals that
Step 4. Now suppose that 2 ≤ µn < 2. Now f 2
are permuted by f . Proof: apply above arguments inductively.
The union of all these intervals is the complete range 1 < µ ≤ 2.
75
The following images of the (µ, x) plane were produced by interating the tent map for
O(1000) iterations to allow the orbit to settle towards the chaotic attractor, and then
plotting the next O(500) points. (The vertical stripiness is an annoying artefact of the
printer resolution!)
n+1
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The attracting set contains 2n intervals in 21/2
76
n
≤ µ < 21/2 .
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7.7
The Logistic Map
77
Here xn+1 = µxn (1−xn ), 0 < µ ≤ 4. There is one non-trivial fixed point at x̄ = (µ−1)/µ.
jacobian is µ(1 − 2x̄) = 2 − µ, so there is a period-doubling bifurcation at µ = 3. By
√
looking at iterate of map we find a further bifurcation (to period 4) at µ = 1 + 6; to
period 8 at µ ≈ 3.544. Call µk point of bifurcation to cycle of period 2k ; then it was
shown by Feigenbaum (1978) that
δ = lim
Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission.
k→∞
µk − µk−1
µk+1 − µk
= 4.6692 . . . Feigenbaum’s constant; µk → µ∞ = 3.5699 . . .
This ratio turns out to be a universal constant for all one-humped (unimodal) maps with
a quadratic maximum (taken at x = 21 ). First we give some general results about such
maps.
7.7.1
Unimodal Maps
Definition 34 A unimodal map on the interval [a, b] is a continuous map F : [a, b] into
[a, b] such that (i) F (a) = F (b) = a and (ii) ∃c ∈ (a, b) such that F is strictly increasing
on [a, c] and strictly decreasing on [c, b]. i.e.
Note: a map of the form
is effectively unimodal under x 7→ −x and F 7→ −F .
Definition 35 An orientation reversing fixed point (ORFP) of a unimodal map F
is a fixed point in the interval (c, b) where F is decreasing.
Lemmas
(1) If F (c) ≤ c then all solutions tend to fixed points, which lie in [a, F (c)].
(2) If F (c) > c then there is a unique ORFP x0 ∈ (c, F (c)).
(3) If F (c) > c then orbits either tend to fixed points in [a, F 2 (c)] or are attracted into
[F 2 (c), F (c)].
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Proof
(1) F ([a, c]) = F ([c, b]) = [a, F (c)] ⊆ [a, c]. So after one iteration x1 ∈ [a, c], where
Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission.
x < y ⇐⇒ F (x) < F (y).
If x1 < F (x1 ) then xi increases monotonically to the nearest fixed point.
If x1 > F (x1 ) then xi decreases monotonically to the nearest fixed point.
(2) Apply the IVT to F (x) − x on [c, F (c)] noting that F (c) > c ⇒ F 2 (c) < F (c).
(3) Exercise. (Cases split on whether F 3 (c) < F 2 (c) or vice versa.)
Theorem 18 If F has an ORFP x0 then ∃ x−1 ∈ (a, c) and x−2 ∈ (c, b) such that
F (x−2 ) = x−1 and F (x−1 ) = x0 . Moreover,
– either (i) F 2 has a horseshoe on JL ≡ [x−1 , x0 ] and JR ≡ [x0 , x−2 ]
– or (ii) all solutions tend to fixed points of F 2
– or (iii) F 2 is a unimodal map with an ORFP on both JL and JR .
79
Proof
IV T
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x0 ∈ (c, b) ⇒ F (c) > F (x0 ) = x0 > F (b) = F (a) =⇒ ∃ x−1 ∈ (a, c) such that F (x−1 ) =
x0 .
IV T
x−1 ∈ (a, c) ⇒ F (b) = a < x−1 < x0 = F (x0 ) =⇒ ∃ x−2 ∈ (x0 , b) such that F (x−2 ) = x−1 .
Thus F 2 (x−2 ) = F 2 (x−1 ) = F 2 (x0 ) = x0 . Also x ∈ [x−1 , x0 ] ⇒ F 2 (x) ∈ [F 2 (c), x0 ] and
x ∈ [x0 , x−2 ] ⇒ F 2 (x) ∈ [x0 , F (c)] i.e. F 2 has the graph
(i) If F 2 (c) < x−1 (equivalent to F (c) > x−2 ) then F 2 has horseshoes.
(ii) If F 2 (c) > c then all solutions on JL ∪ JR tend to fixed points of F 2 . All solutions
on [a, x−1 ] ∪ [x−2 , b] either tend to fixed points of F or are attracted into [F 2 (c), F (c)] ⊂
JL ∪ JR .
(iii) If x−1 < F 2 (c) < c then F 2 is a unimodal map on JL and JR with ORFPs that give
a 2-cycle of F , and the attracting set consists of two disjoint subintervals.
Applying Theorem 1 successively to F 2 , F 4 , F 8 , . . . we deduce that
80
Theorem 19 If F has an ORFP then
N
– either (i) ∃ N such that F 2 has a horseshoe and F is chaotic
N
– or (ii) ∃ N such that all solutions tend to fixed points of F 2
and F has 2m -cycles
for 0 ≤ m ≤ N − 1
– or (iii) there are (mostly unstable) 2m -cycles ∀m, and the attracting set is a Cantor
m
set formed by the infinite intersection of the attracting subintervals of F 2 .
Copyright © 2015 University of Cambridge. Not to be quoted or reproduced without permission.
Proof
Induction (except for the comment on stability which depends on the following).
7.7.2
Scaling Invariance and Feigenbaum’s Constant
k
If we write Gk = F 2 then in situation (iii) of Theorem 19 the successive subgraphs of
Gk+1 , Gk+2 , . . . all seem to look the same after renormalisation and all seem to have the
same properties. This suggests that we look for a graph that is invariant under iteration
and renormalisation:
Suppose w.l.o.g that c = 0 and after renormalisation Gk (0) = 1 ∀k. (This is easier than
scaling the sub-interval so that the end x0 is always at 1.)
Suppose, for simplicity, that G(x) = G(−x) = 1 + ax2 + bx4 + . . . . The graphs of Gk and
G2k look like
Gk(0)=1
1
Gk2
0
Gk
λ
Gk(1)=λ
–λ
–1
–1
0
1
81
Let λ be the value of G2k (0) = Gk (1).
Renormalise G2k so that Gk+1 (0) = 1 by defining
Gk+1 (y) =
G2k (λy)
≡ T [Gk ] say, where λ = G2k (0)
λ
To get an approximation to G try a truncated series expansion:
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e.g. Gk = 1 + ak x2 + o(x2 ) with Gk (1) = 1 + ak = λk
⇒
Gk+1 = T [Gk ] =
1 + ak {1 + ak [(1 + ak )x]2 }2
= 1 + 2a2k (1 + ak )x2 + o(x2 )
1 + ak
i.e. ak+1 = 2a2k (1 + ak ), which has an unstable fixed point a = − 21 (1 +
√
λ = −0.37, where the Jacobian is 4 + 3 = 5.73
√
3) = −1.37 ⇒
e.g. Gk = 1 + ak x2 + bk x4 + o(x4 ) with Gk (1) = 1 + ak + bk = λk gives the 2D map
ak+1 = 2ak (ak + 2bk )λk
bk+1 = (2ak bk + a3k + 4b2k + 6a2k bk )λ3k
which has a fixed point a = −1.5222, b = 0.1276, λ = −0.3946, where the Jacobian has
eigenvalues 4.844 and −0.49.
In fact, numerical solution shows that the functional map T has a fixed point G = T [G],
where
G = 1 − 1.52736x2 + 0.10482x4 − 0.02671x6 + . . . ,
λ = G(1) = −0.3995
Numerical linearisation about T [G] = G gives a single eigenvalue δ = 4.6692016 . . .
outside the unit circle, and an infinite spectrum of eigenvalues inside the unit circle.
Hence situation (iii) of Theorem 19 (renormalisation possible infinitely often) is unstable
in one direction; the stable manifold occupies ‘all but one dimension’ of the possible space
of functions.
The map G0 = µ∞ x(1 − x), µ∞ = 3.5700 . . . , is on the stable manifold of G, but a small
perturbation grows to give situation (i) if µ > µ∞ (GN has a horseshoe for some N ) or
situation (iii) if µ < µ∞ (GN has no ORFP for some N and cycle lengths divide 2N ).
If µ∞ − µ = O(δ −N ) then it takes O(N ) renormalisations for the perturbation to grow to
O(1) and eliminate the ORFP, thus explaining why µ∞ − µk ∼ Aδ −k as k → ∞.
82
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