Dynamics of Differential Equations Prof. R.S. Mackay Lent 1997

These notes are maintained by Paul Metcalfe. Comments and corrections to [email protected].

Revision: 2.6 Date: 2004/07/26 07:41:24

The following people have maintained these notes. – date

Paul Metcalfe

Contents Introduction 1

2

v

Differential Equations 1.1 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Traditional Approach . . . . . . . . . . . . . . . . . . . . . . 1.3 The Geometric Approach . . . . . . . . . . . . . . . . . . . . . . 1.4 Basic Theorem of ODE’s . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Existance, uniqueness and continuity wrt initial conditions 1.5 Interval of definition of solutions . . . . . . . . . . . . . . . . . .

. . . . . .

. . . . . .

1 1 1 2 3 3 4

Basic Examples and Ideas 2.1 The Ideal Pendulum . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Damped Pendulum . . . . . . . . . . . . . . . . . . . . . 2.2.1 The effect of nonlinearity on stability of equilibria . . 2.2.2 Equivalence of “attracting” and “asymptotically stable” 2.3 Damped Pendulum with torque . . . . . . . . . . . . . . . . . 2.3.1 The Poincar´e - Bendixson Theorem . . . . . . . . . . 2.4 The van der Pol equation . . . . . . . . . . . . . . . . . . . . 2.4.1 Topics arising from the van der Pol equation . . . . .

. . . . . . . .

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7 7 8 9 10 12 13 20 21

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3

Bifurcations of 2D flows 3.1 Structural stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Saddle-node equilibria . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Other cases of degenerate equilibria . . . . . . . . . . . . . . . . . . 3.3.1 Imperfections . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Hopf Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Homoclinic Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Finding homoclinic bifurcations in nearly Hamiltonian systems 3.6 Saddle-node of periodic orbits . . . . . . . . . . . . . . . . . . . . .

25 25 26 29 29 30 32 34 36

4

Introduction to dynamics in 3D 4.1 Time-T map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Existence of periodic orbits . . . . . . . . . . . . . . . . . . . . . . . 4.3 Linear stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 The Mathieu Equation . . . . . . . . . . . . . . . . . . . . . 4.4 Near-identity maps and averaging . . . . . . . . . . . . . . . . . . . 4.4.1 Application of averaging to the periodically forced van der Pol equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 37 38 39 40 41

iii

42

iv

CONTENTS 4.5

Differences from autonomous systems . . . . . . . . . . . . . . . . .

43

Introduction These notes are based on the course “Dynamics of Differential Equations” given by Prof. R.S. Mackay in Cambridge in the Lent Term 1997. These typeset notes are totally unconnected with Prof. Mackay. Note that these notes are based on the first year of the course. More material was lectured in 1998. Other sets of notes are available for different courses. At the time of typing these courses were: Probability Analysis Methods Fluid Dynamics 1 Geometry Foundations of QM Methods of Math. Phys Waves (etc.) General Relativity Combinatorics

Discrete Mathematics Further Analysis Quantum Mechanics Quadratic Mathematics Dynamics of D.E.’s Electrodynamics Fluid Dynamics 2 Statistical Physics Dynamical Systems Bifurcations in Nonlinear Convection

They may be downloaded from http://www.istari.ucam.org/maths/.

v

vi

INTRODUCTION

Chapter 1

Differential Equations 1.1

The Problem

In this course, differential equation means a system of equations of the form dxi = vi (x(t)), dt with x ∈ Rn , t ∈ R and vi : Rn 7→ R.1 In this course, we only study ordinary differential equations, as opposed to partial differential equations. More precisely, we study autonomous ODE’s, those with no explicit time dependence in v. (Or the equation is invariant under a shift of the origin in time.) A restriction to autonomous DE’s is not that much of a restriction, since the nonautonomous system x˙ = f (x, t) can be reduced to the autonomous system x˙ = f (x, τ ), τ˙ = 1. We also restrict to coupled first-order ODE’s. This again is little of a restriction – many higher order systems can be reduced to the first-order case, since if x ¨ = f (x, x), ˙ then introduce y = x˙ to get the coupled system y˙ = f (x, y), x˙ = y. We also only consider explicit systems, but if we have the implicit system g(x, x) ˙ =  0, and x0 , y0 such that g(x0 , v0 ) = 0, and the matrix ∂∂gx˙ ij is invertible at (x0 , v0 ) then ∃v : U 7→ Rn on a neighbourhood U of x0 such that for (x, x) ˙ near (x0 , v0 ), g(x, x) ˙ = 0 ⇔ x˙ = v(x).

1.2

The Traditional Approach

Given v in terms of “elementary” functions, look for a general solution of x˙ = v(x) in terms of elementary functions. This has some drawbacks, not least of which is that it is frequently hard to do (read impossible). We also need some valid definition of elementary function. And in any case the nicest definition of some elementary functions is as the solution of certain ODE’s, for example 1. exp(t) is the solution of x˙ = x with x(0) = 1. The property exp(s + t) = exp(s) exp(t) is easy to derive – the ODE is autonomous so exp(s+t) is the solution after time t starting at exp(s). But the ODE is linear, so this is exp(s) exp(t). 1 Underlining

of vectors will be sporadic in these notes.

1

2

CHAPTER 1. DIFFERENTIAL EQUATIONS 2. 

sin t cos t



is the solution of     x˙ y = y˙ −x with the initial conditions x(0) = 0, y(0) = 0. Periodicity is easy to derive just note that H = (x2 + y 2 )/2 is conserved. Thus solutions remain on the unit circle. The direction of rotation is constant, at constant speed. So (x(t), y(t)) must return to its initial conditions at some T > 0 (choose the least such). Then by autonomy, (x(t + T ), y(t + T )) = (x(t), y(t)). Traditionally, set T = 2π, as a definition of π. So the traditional approach is (ahem) circular. Even if you allow solutions of (algebraic) equations in elementary functions and integrals of elementary functions, there are still some equations which cannot be solved (e.g. y˙ = y 2 − t). Thus we try a different approach — to see what happens to solutions.

1.3

The Geometric Approach

This is due to Poincar´e. It has two senses.

pictorial Let v define a vector field on a state space X = Rn (or more generally on some differentiable manifold, eg S 1 × R, S 2 , Π2 = S 1 × S 1 ). Under suitable conditions, x˙ = v(x) defines a flow φ : R × X 7→ X, (t, x0 ) 7→ φt (x0 ), the solution at time t with x(0) = x0 . A picture showing how “all” the solutions move in X is called a phase portrait. These are hard to draw when n > 3, but we shall restrict to n = 2 or n = 3.2

Kleinian For Klein, the geometrical approach to a set of objects is to identify a group of transformations of the objects which we regard as taking each object to an equivalent one and then studying the properties which are invariant under the group action. The objects we will look at are the vector fields v on X with their associated differential equation x˙ = v(x). The transformations we will use are change of variables and time rescaling. For change of variables, let y = h(x), with h a diffeomorphism. In this case, ∂hi . y˙ = Dhh−1 (y) v(h−1 (y)), where Dh is the matrix of partial derivatives ∂x j dx For time rescaling, let ds dt = α(x) > 0, which implies that ds = v(x)/α(x). We will study the features of x˙ = v(x) which are preserved by the group generated by these kinds of transformations, for instance existance of a periodic orbit (but not its formula or period). The geometric approach does not try to obtain as much as the traditional approach, but can get useful results where the traditional approach gets stuck. 2I

have problems when n = 3, let alone anything higher.

3

1.4. BASIC THEOREM OF ODE’S

1.4

Basic Theorem of ODE’s

We would like to justify the idea of a flow with some sort of existance and uniqueness results. Definition. A map f : M1 7→ M2 between metric spaces with metrics d1 and d2 is Lipschitz if ∃ L ∈ R such that ∀ x, y ∈ M1 , d2 (f (x), f (y)) ≤ Ld1 (x, y). L is said to be a Lipschitz constant.

Example A map f : U 7→ Rm , U open in Rn is C 1 (continuously differentiable) if 1. it is differentiable, that is ∃ a linear map Dfx : Rn 7→ Rm such that |f (x + h) − f (x) − Dfx h| = o(|h|). 2. Dfx depends continuously on x with respect to the operator norm kDfx k = x h| suph6=0 |Df |h| . Then f is Lipschitz on any convex compact set K ⊂ U , with Lipschitz constant L = kDfx k. Proof. Given x, y ∈ K, let z(s) = x + s(y − x) for 0 ≤ s ≤ 1. Then Z 1 Z 1 d Dfx(s) (y − x) ds ≤ K |y − x| . f (x(s)) ds = |f (y) − f (x)| = 0 0 ds

1.4.1

Existance, uniqueness and continuity wrt initial conditions

Theorem (Basic Theorem of ODE’s). If v : U 7→ Rn , U open in Rn , is locally Lipschitz, then ∀ y ∈ U , ∃ a neighbourhood V of y and τ > 0 such that ∀ x0 ∈ V , ∃ unique differentiable function x : [−τ, τ ] 7→ U such that x(0) = x0 and x(t) ˙ = v (x(t)) ∀ t ∈ [−τ, τ ]. Furthermore, if the solution x(t) is denoted by φt (x0 ), then φ : [−τ, τ ] × V 7→ U is Lipschitz. Proof. 3 Given y ∈ U , choose a neighbourhood N = {x : |y − x| ≤ b} on which v is Lipschitz. Let L be the Lipschitz constant for v and K = supx∈N |v(x)|. Now choose τ , b0 > 0 such that τ ≤ 1/L and b0 +Kτ ≤ b. Let V = {x : |x − y| ≤ b0 }. Let B = {Lipschitz functions x ˜ : [−τ, τ ] 7→ Rn of constant K and x ˜(0) ∈ V } . Given functions x ˜, y˜ ∈ B, let d(˜ x, y˜) = supt∈[−τ,τ ] |˜ x − y˜|. Then (B, d) is a complete metric space. Now, for x0 ∈ V , define a map Px0 : B 7→ B by Z t (Px0 x ˜) (t) = x0 + v(˜ x(s)) ds. 0

Fixed points of Px0 correspond to solutions of x˙ = v(x) with x(0) = x0 . Now Px0 maps B into itself and contracts d (by at least Lτ < 1). So by the contraction mapping principle, Px0 has a unique fixed point x ˜ that depends Lipschitz continuously on x0 . It follows that φ : [−τ, τ ] × V 7→ U is Lipschitz. 3 This

should be familiar from Analysis IB.

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CHAPTER 1. DIFFERENTIAL EQUATIONS

Remarks 1. If v is C 1 , we can prove that φ is C 1 . Similarly, if v is C 1+Lip , C 2 , . . . , C ∞ , analytic, then so is φ. 2. x(t) is always one derivative smoother that v(x), since x˙ = v(x). 1/2

3. If v is not locally Lipschitz then x may fail to be unique, for instance x˙ = |x| has solutions  −(2−t)2  0≤t≤2  4 x(t) = 0 for any T ≥ 2. 2≤t≤T   (t−T )2 t≥T 4

This is the ODE for a particle sliding down a viscous slope of height h(x) = 3/2 − 23 |x| sgn(x). 4. We cannot take τ = ∞, and τ may depend on x0 . For instance, take x˙ = x2 ,
−1 x(0) = x0 > 0. Thus x(t) = x−1 → ∞ as t % x−1 0 −t 0 . This is called “finite time blowup”. It is not just a pathology: it is suspected that for many solutions of the Euler equations for a 3D fluid (and maybe even the Navier ∂ui → ∞ in finite time. Stokes equations), supx ∂x j

1.5

Interval of definition of solutions

There are several circumstances under which the solution x(t) starting from x0 is defined for all positive and negative time. Theorem (Extension Theorem). Let K ⊂ X be compact and v a C 1 vector field on a neighbourhood of K. Then φt (x0 ) is defined and C 1 for all x0 ∈ K and all t such that φs (x0 ) ∈ K∀ 0 ≤ s ≤ t, plus a little more. Proof. For all y ∈ K there exists a neighbourhood Vy and τy > 0 such that φt (x0 ) is defined and C 1 for all x0 ∈ Vy , |t| ≤ τ . The sets {Vy : y ∈ K} form an open cover of K, so by compactness ∃ a finite subcover {Vyi : i = 1, . . . , N }. Let τ = mini τyi . Then ∀ x0 ∈ K, φt (x0 ) exists for at least |t| ≤ τ . If φτ (x0 ) ∈ K then repeat argument. Thus φt (x0 ) is defined and C 1 as long as it remains in K, and for time τ more. Corollary. The only way that a solution of a C 1 ODE can cease to exist is to leave every compact set in bounded time. We can use these results to get a couple of examples. Example. If |v(x)| ≤ A |x| + B for some A > 0, B ≥ 0, then φt (x) exists for all t and |φt (x)| ≤ (|x| + B/A) exp(A |t|). Proof. Compare with r˙ = Ar + B on R+ . Example. If x˙ = v(x, s), s˙ = 1 then no solution remains in any compact set since s is unbounded. But for any compact K ⊂ X, solutions can be extended as long as they remain in K × R.

1.5. INTERVAL OF DEFINITION OF SOLUTIONS

5

Proof. Otherwise, let T1 be the infimum of positive times that solutions starting in K × {0} which do not cross ∂K × R but which cannot be continued to t = T . By assumption T1 < ∞. Then K × [−T1 , T1 ] is compact, so we can apply the extension theorem and continue all solutions which do not cross ∂K × [−T1 , T1 ] to t = T1 + τ1 . This is a contradiction. We can often find a compact K which all solutions enter and remain inside. We could also in principle rescale time to prevent finite-time blowup. Thus we shall henceforth assume that φt (x0 ) is defined for all t.

6

CHAPTER 1. DIFFERENTIAL EQUATIONS

Chapter 2

Basic Examples and Ideas We will introduce the theory of nonlinear systems by means of various examples.

2.1

The Ideal Pendulum

This has the ODE θ¨ + sin θ = 0. (With appropriate rescaling of time and θ to remove constants.) Writing as a linked first order system, we obtain     p θ˙ = . − sin θ p˙ Write x = (θ, p) ∈ X, where X is an infinite cylinder. Notice the conservation of energy, that is, let H = 12 p2 − cos θ. Then H˙ = 0. Now, given E ∈ R, define ΣE = {x ∈ X : H(x) = E} . Note that x(0) ∈ ΣE ⇒ x(t) ∈ ΣE ∀t. We say that ΣE is invariant under the flow. Now, if E < −1 then ΣE = ∅. If E = −1 then ΣE = {(0, 0)} - it is an equilibrium point. If −1 < E < 1, then ΣE is a closed curve. Now v 6= 0 on ΣE . v is continuous and ΣE is compact. Thus ∃ δE > 0 such that |v| ≥ δE on ΣE . Also, the length of ΣE is finite, so given x ∈ ΣE , ∃T > 0 such that φT (x) = x. Thus ΣE is a periodic orbit. Physically, this corresponds to oscillations. If E > 1 then ΣE is two closed curves. v is non-zero, so we have two periodic orbits. Physically, this is rotations. If E = 1 then ΣE consists of the equilibrium point x = (π, 0) and two orbits which are asymptotic to the equilibrium in both positive and negative time. They are called homoclinic orbits to the equilibrium. An orbit which is asymptotic to one equilibrium in positive time and another equilibrium in negative time is called a heteroclinic orbit. Finding all the equilibria is easy — you just have to set v(x) = 0. The equilibrium θ = 0 and each of the periodic orbits are stable. Definition. An invariant set Λ for a flow φ : R × X 7→ X is (Lyapunov) stable if ∀ neighbourhoods U of Λ, ∃ a neighbourhood V of Λ such that x0 ∈ V ⇒ φt (x0 ) ∈ U ∀t ≥ 0. 7

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CHAPTER 2. BASIC EXAMPLES AND IDEAS

Notes. It’s useful to spot conserved quantities (if any exist). Some types of system for which there exist useful conserved quantities are frictionless mechanical systems (in which the Hamiltonian is conserved) and chemical reactions — numbers of atoms of each kind of element must be constant.

2.2

The Damped Pendulum

This has ODE θ¨ + k θ˙ + sin θ = 0, where k > 0. As a linked system, we obtain     p θ˙ = . −kp − sin θ p˙ Now, H = 12 p2 − cos θ is no longer conserved, but H˙ = −kp2 . Thus H decreases along solutions (except when p = 0). H decreases strictly along all solutions except on two equilibria at θ = 0, π. Theorem. If H (φt (x)) < H (φs (x)) for t > s and x is not an equilibrium then φt (x) tends to an equilibrium as t → ∞. Proof. Given x ∈ X, then P = {y : H(y) ≤ H(x)} is compact. Therefore (φt (x))t→∞ has a limit point (in P ). Define ω(x) = {Limit points of (φt (x))t→∞ } . Now ω(x) 6= ∅ (by compactness)1 . Note that d(φt (x), ω(x)) → 0, otherwise ∃ > 0 and tj → ∞ such that 2 ≥ d(φtj (x), ω(x)) ≥ . Now Q = {y : 2 ≥ d(y, ω(x) ≥ } is compact, so φtj (x) has a limit point in Q. This is a contradiction. Now H must be constant on ω(x). Otherwise, suppose ∃ξ, η ∈ ω(x) with H(ξ) < H(η). Let  = H(η) − H(ξ) > 0. Now ∃t1 such that φt1 (x) is close enough to ξ that H(φt1 (x)) is within /3 of H(ξ). Now, ∃t2 > t1 such that φt2 (x) is close enough to η that H(φt2 (x)) is within /3 of H(η). Thus H(φt2 (x)) − H(φt1 (x)) ≥ /3 > 0 and t2 > t1 . This is a contradiction. Now ω(x) is invariant, since if φtj (x) → y, then φtj +τ (x) → z by the continuity of φτ . Since H is constant on ω(x), ω(x) consists only of equilibria. In fact, ω(x) has to be only a single equilibrium, as H(π, 0) = 1 and H(0, 0) = −1. More generally, we have La Salle’s Invariance Principle. Theorem (La Salle’s Invariance Principle). If H : X 7→ R is continuous, decreasing along all solutions of a flow except at equilbria, and x0 ∈ X has a bounded orbit then φt (x0 ) converges as t → ∞ to a connected subset of the set of equilibria on which H is constant.2 The proof is left as an exercise. We now need to examine the way that the orbits approach the equilibria and which orbits converge to which equilibrium. The first step is to perform linear stability analysis of the equilibria. Linearise the system about θ = 0 or π, p = 0. We obtain 1 ω(x) is called the ω-limit set of x. Similarly, as t → −∞ there is the α-limit set. For the advanced student — why α and ω? 2 Such a function H is called a Lyapunov function.

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2.2. THE DAMPED PENDULUM

     ˙ δθ 0 1 δθ = ˙ −c −k δp δp √

where c = cos θ = ±1.

2

This matrix has eigenvalues − k2 ± k 2−4c . At θ = π, c = −1, so we have real eigenvalues, one positive and one negative. Thus we have a saddle point. At θ = 0, c = 1 so if 0 < k < 2 we have complex conjugate eigenvalues with negative real part. This is a focus. If k > 2 we have a pair of negative real eigenvalues. This is a node. We wish now to ask which features of the linearised system survive the nonlinear remainder term. In particular, does a sink attract all nearby orbits and does a saddle have precisely two invariant curves through it?

2.2.1

The effect of nonlinearity on stability of equilibria

Definition. A sink is an equilibrium all of whose eigenvalues lie in the open left half plane <(λ) < 0. Definition. An invariant T set Λ is attracting if it has a neighbourhood U such that φt (U ) ⊂ U for t > 0 and t>0 φt (U ) = Λ. Lemma. If A is a linear operator on a finite-dimensional real vector space E, and all eigenvalues λi of A have α < <(λi ) < β, then there is an inner product on E such that 2 2 α |x| ≤ hx, Axi ≤ β |x| . Proof. We choose a basis for E such that A has real JNF with blocks of the forms [λ1 ]  α2 ω2

−ω2 α2

  λ3     

 .. .

 ..

.

..

.

     λ3  α4  ω4    

−ω4 α4



 0

0 

α4

−ω4

..

.

α4

..

.

ω4

    

Now  can be chosen arbitrarily small (by rescaling of basis vectors), so that λ3 ±  ∈ [α, β] for all non-trivial blocks. Let xi be the co-ordinates with respect to this P basis and choose the standard inner product hx, yi = i xi yi . The blocks contribute additively to hx, Axi and the contributions are respectively λ1 x21 2

α2 |x|

2

X

xi xi+1 ∈ [λ3 − , λ3 + ] |x|

2

X

xi xi+2 ∈ [α4 − , α4 + ] |x| ,

λ3 |x| +  α4 |x| + 

2 2

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CHAPTER 2. BASIC EXAMPLES AND IDEAS

where |x| refers only to components in the appropriate block and the inclusion is obtained using Cauchy-Schwarz. Summing over the blocks we obtain the result 2

2

α |x| ≤ hx, Axi ≤ β |x| .

Theorem. Every sink of a C 1 vector field is attracting. Proof. If all eigenvalues have <(λi ) < 0 then ∃b > 0 such that <(λi ) < −b. The 2 2 d |x| = 2hx, xi ˙ ≤ lemma gives us an inner product with hx, Axi ≤ −b |x| . So dt 2 2 −2b |x| + o(|x| ), as x˙ = Ax + o(|x|) as |x| → 0. Now given c ∈ (0, b), there 2 2 d exists δ > 0 such that dt |x| ≤ −2c |x| for |x| ≤ δ. Let U = {x : |x| ≤ δ}. If 2 x(0) ∈ U then |x(t)| is non-increasing so that x(t) remains in U . Furthermore, for 2 d x ∈T U \ {0}, dt log |x| ≤ −2c, so that |x(t)| ≤ |x(0)| e−ct . Thus φt (U ) ⊂ U ∀t ≥ 0 and t≥0 φt (U ) = {0}. We can clearly assume that the stationary point is at the origin. Note that by reversing time, if all eigenvalues have <(λ) > 0 then 0 is repelling. Theorem. If A has an eigenvalue with <λ > 0 then 0 is unstable. Proof. If A has an eigenvalue with <λ > 0 then using JNF or otherwise decompose E = E1 ⊕ E2 into invariant subspaces E1 and E2 such that all eigenvalues of B1 = A |E1 have <λ > 0 and all eigenvalues of B2 = A |E2 have <λ ≤ 0. Now, ∃a > 0 such that all eigenvalues of B1 have <λ > a. By the lemma there is an inner product 2 on E1 such that hx, B1 xi ≥ a |x| for all x ∈ E1 . Similarly, ∀b > 0, there is an inner 2 product on E2 such that hy, B2 yi ≤ b |y| for all y ∈ E2 . Choose b ∈ (0, a). Take the direct sum inner product on E and write z ∈ E as (x, y), with x ∈ E1 and y ∈ E2 . Then v(x, y) = (B1 x, B2 y) + o(|x|).   2 2 Let C = {(x, y) : |x| ≥ |y|} and V (z) = 12 |x| − |y| . Then d V (z) = hx, xi ˙ − hy, yi ˙ dt = hx, B1 xi + o(|x| |z|) − hy, B2 yi + o(|y| |z|) 2

2

2

≥ a |x| − b |y| + o(|z| ). 2 Given α ∈ (0, a − b), ∃δ > 0 such that V˙ ≥ |x| on C ∩ U where U = {z : |z| ≤ δ}. So V increases along all orbits in C ∩ U \ {0}. They cannot cross ∂C because V decreases across ∂C. They cannot stay in C ∩ U for all t > 0 else by La Salle’s Invariance Principle they converge to a set of equilibria with strictly larger V than initially - but there are no equilibria except 0 in C ∩ U because V increases in C ∩ U \ {0}, V (0) = 0 and V (z(0)) > 0. Thus there exist points z arbitrarily close to 0 such that φt (z) leaves U for some t > 0. This contradicts stability of 0, hence 0 is unstable.

2.2.2

Equivalence of “attracting” and “asymptotically stable”

Suppose φ : R × X 7→ X is a continuous flow on a locally compact metric space X. Definition. Λ ⊂ T X is attracting for φ if ∃ compact K ⊂ X such that φt (K) ⊂ int K ∀t > 0 and t≥0 φt (K) = λ.

11

2.2. THE DAMPED PENDULUM

Definition. Λ ⊂ X is (Lyapunov) stable if ∀ neighbourhoods U of Λ, ∃ a neighbourhood V of Λ such that φt (V ) ⊂ U ∀t ≥ 0. Definition. Λ ⊂ X is asymptotically stable for φ if it is stable and ∃ a neighbourhood W of Λ such that ∀x ∈ W , d(φt (x), Λ) → 0 as t → +∞. Theorem. Λ ⊂ X is attracting iff compact, invariant and asymptotically stable. Proof of ⇒. φt is continuous, K isTcompact, so φtT (K) is compact, therefore Λ = T φ (K) is compact. Now φ φ (K) ⊂ t s t t≥0 t≥0 t≥0 φt (K) for s ≥ 0 so Λ is T forward invariant. For s ≥ 0, φ−s (Λ) = φs t≥s φt (K) = Λ, so Λ is backward invariant too. Now, let D(t) = supx∈K d(φt (x), Λ). Then φt (K) nested, so that D is nonincreasing. If D 9 0 as t → ∞ then ∃ > 0 such that ∀t > 0, ∃xt ∈ φt (K) such that ˜ = {x ∈ K : d(x, Λ) ≥ }, which is compact, so d(xt , Λ) ≥ . The xt all lie in K ˜ they have T a limit point x ˜ ∈ K. Now for all s, xt ∈ φs (K) when t ≥ s, so x ˜ ∈ φs (K) and x ˜ ∈ s≥0 φs (K). But d(˜ x, Λ) ≥ , giving a contradiction. So D(t) → 0 as t → ∞. In particular, d(φt (x), Λ) → 0 as t → ∞ for all x ∈ K. Also, for all neighbourhoods U of Λ, ∃T such that φt (K) ⊂ U for all t ≥ T — or alternatively such that D(T ) < inf x∈U / d(x, Λ), which is positive by the compactness of Λ. Let V = φT (K). It is a neighbourhood of Λ because ∀t > 0 Λ ⊂ φt (V ) ⊂ int V . Thus Λ is stable. Proof of ⇐. Let Λ = {x ∈ X : d(x, Λ) ≤ } and W be a neighbourhood of Λ. Then Λ compact gives ∃0 > 0 such that Λ ⊂ W ∀ ≤ 0 . Given  ≤ 0 , there exists a neighbourhood V of Λ such that φt (V ) ⊂ Λ 2 for all t ≥ 0 (Λ stable). Also d(φt (x, Λ)) → 0 ∀x ∈ Λ , so ∀x0 ∈ Λ ∃t0 and an open neighbourhood N of x0 such that φt0 (x) ∈ int V ∀x ∈ N (as φt0 is continuous). So φt (x) ∈ Λ 2 ∀t ≥ t0 . The neighbourhoods N form an open cover of the compact Λ , so finitely many suffice. Let T () = maxi ti . Then φt (Λ ) ⊂ Λ 2 for all t ≥ T (). So by iteration ∃Tn () such that φt (Λ ) ⊂ Λ 2n for all t ≥ Tn (). Then we define D : W 7→ R+ by D(x) = supt≥0 d(φt (x), Λ). If D(x) > 0 then ∃t1 (x) > 0 such that d(φt (x), Λ) ≤ 21 D(x) for all t ≥ t1 (x). Then for all y near enough to x the supremum of d(φt (y), Λ) is attained in [0, t1 (x)] and D is continuous at x. If D(x) = 0 then x ∈ Λ (by compactness) and ∀ > 0 there exists a neighbourhood V of Λ such that y ∈ V ⇒ d(φt (y), Λ) ≤  ∀t > 0 (Λ stable). So D(y) <  for y ∈ V and D is continuous at x. D is non-increasing along orbits and goes to zero along each orbit. Let w = ˜ = {x ∈ W : D(x) ≤ w } inf x∈W d(x, Λ) > 0 by compactness of Λ. Let K / 2 T ˜ ∈ K ˜ ∀t > 0 and ˜ ˜ w is compact, φt (K) t≥0 φt (K) = Λ, because K ⊂ Λ 2 and T ˜ so φt (Λw2) ⊂ Λ 2wn for t ≥ some Tn , so d(x, Λ) = 0 for all x ∈ t≥0 φt (K) ˜ for all t ≥ 0 because x ∈ Λ as Λ is compact, and conversely each x ∈ Λ lies in φt (K) Λ is negatively invariant. To achieve φt (K) ⊂ int K ∀t > 0 we modify D to Z ∞ ∗ D (x) = e−s D(φs (x)) ds. 0

Then D∗ is decreasing along orbits except on Λ. So defining K = {x ∈ W : D∗ (x) ≤ w 2 } we obtain Λ attracting.

12

CHAPTER 2. BASIC EXAMPLES AND IDEAS 1. An equivalent definition of Λ attracting is:T∃ a compact neighbourhood K of Λ such that φt (K) ⊂ K for all t ≥ 0 and t≥0 φt (K) = Λ. One way to prove the equivalence is to show that this second definition is equivalent to compact, invariant and asymptotically stable. The proof of stability works as before and the proof of ⇐ gives a neighbourhood K. 2. Similarly, it is equivalent to Trequire either “∃ compact K ⊂ X and T > 0 such that φT (K) ⊂ int K and t≥0 φt (K)T= Λ” or “∃ compact neighbourhood K and T > 0 such that φT (K) ⊂ K and t≥0 φt (K) = Λ”.

2.3

Damped Pendulum with torque

This has the ODE θ¨ + k θ˙ + sin θ = F , with k, F > 0. The equivalent linked system is θ˙ = p p˙ = −kp − sin θ + F. Alternative motivations for this differential equation come from AC electricity generation and the Josephson junction with steady current. Again, we investigate how H = 12 p2 − cos θ changes with time. H˙ = −kp2 + F p, which is < 0 for p outside [0, F/k] but > 0 for p in (0, F/k). H is still useful. Let E = maxp∈[0,F/k] H(θ, p) and choose any E 0 > E. Proposition. For all x0 , φt (x0 ) eventually enters H < E 0 and stays there forever after. So H acts as a “bounding function”. Proof. If H(x) > E then H˙ < 0. Let M = inf t≥0 H(φt (x)). If M > E 0 then the flows must converge to a set of equilibria in {H = M } — but there are none, as equilibria must have H˙ = 0. So all orbits eventually enter {H < E 0 }. Once in {H < E 0 } they cannot escape: suppose τ is the time of first escape from {H < E 0 } for the orbit of x. Then H(φτ (x)) = E 0 . Thus H˙ < 0 and it is not an escape but an entry — giving a contradiction. We are left with analysing the dynamics in {H < E 0 }. We seek the equilibria, which require p = 0 and sin θ = F . There are two equilibria if F < 1, one in F = 1 and none if F > 1. Performing linear stability analysis gives that one equilibrium is √ √ 2 2 a saddle and the other is a focus if k4 < 1 − F 2 and a node if k4 > 1 − F 2 . The process of annihilation of saddle and node at F = 1 is called a “saddle-node bifurcation”. A number of questions spring to mind: 1. Where do the stable and unstable manifolds of the saddle go? The unstable manifold (“separatrix”) is particularly important as it separates orbits which go two opposite ways at the saddle. A reasonable guess is that the left branch of the stable manifold goes into the sink, but what about the rest? 2. Where do the orbits go when F > 1, when there are no equilibria to converge to? We will tackle the second question first. We have an annulus A = {H < E 0 } which no orbits can escape and which contains no equilibria. Imagine flattening the annulus and regarding it as a subset of R2 .

2.3. DAMPED PENDULUM WITH TORQUE

2.3.1

13

The Poincar´e - Bendixson Theorem

Theorem (Poincar´e - Bendixson Theorem). If the forward orbit of x (o+ (x)) under a C 1 vector field on an open subset U of the plane lies in a compact subset of U for all t ≥ 0 then ω(x) is either a periodic orbit or contains an equilibrium. Proof. Given x with o+ (x) bounded then ω(x) 6= ∅ (by compactness). Suppose ω(x) contains no equilibria and take y ∈ ω(x). ω(x) is compact and invariant, so ω(y) ⊂ ω(x) and is non-empty. Take z ∈ ω(y). Then z is not an equilibrium, so there exists a C 1 arc Σ transverse to v through z. Then o+ (y) comes arbitrarily close to z as t goes to ∞, so in particular it crosses Σ arbitrarily close to z. If o+ (y) is not periodic then successive intersections with Σ are distinct (by uniqueness in backwards time). But if the first two intersections are y1 6= y2 then o+ (x) is eventually bounded away from y (by the Jordan Curve Theorem). So o+ (y) is periodic, intersecting Σ at one point only. Now ω(x) is invariant so o+ (y) ⊂ ω(x). Take a transverse section Σ0 at y. If + o (x) intersects Σ0 at y then o+ (x) is periodic and ω(x) = o+ (y). Else the successive intersections xn on Σ0 converge monotonically to y. The return times τn converge to the period T of o+ (y), so are bounded, τn < T 0 . Thus given  > 0, d(φt (xn ), φt (y)) <  for all t ∈ [0, T 0 ] and xn close enough to y. But o+ (xn ) is trapped between the piece from xn to xn+1 and o+ (y), so d(φt (xn ), o+ (y)) <  for all t ≥ 0. Thus ω(x) = o+ (y). Definition. A periodic orbit γ such that ∃x ∈ / γ with ω(x) = γ is called a limit cycle. We apply Poincar´e-Bendixson to the damped forced pendulum for F > 1 to obtain that for all x ∈ S1 × R, ω(x) is a periodic orbit. Now that we have the existance of a periodic orbit, we have further questions: 1. Is it unique? 2. What is the homotopy class of the periodic orbit(s)? 3. Is it (Are they) stable? 4. What if equilibria are present? 5. What are the possibilities when ω(x) contains an equilibrium? In general uniqueness may not hold – the ideal pendulum has uncountably many periodic orbits. Even when true it is hard to prove, but for the damped forced pendulum it is possible. The same technique will answer questions 2 and 3. The simplest case is problem 2. Consider an infinitesimal parallelipiped spanned by the columns of B moving with the flow. The volume of the parallelipiped, V = det B. Theorem. Under a vector field x˙ = v(x), dV = (div v)V. dt

14

CHAPTER 2. BASIC EXAMPLES AND IDEAS

Proof. det B(t + h) = det(B(t) + hDvB(t) + o(h)) = det B (det(I + hDv + o(h)) = det B (1 + h div v + o(h)) as required.

Theorem. div v < 0 implies no invariant sets of positive volume. Proof. Invariance implies that V (t) is constant, but div v < 0 implies that V decreases.

Proposition. There does not exist a contractible periodic orbit for the damped forced pendulum. Proof. Such an orbit bounds an invariant disk of area A > 0. But div v = −k < 0, giving a contradiction. We must also have at most one non-contractible periodic orbit, for the area between two non-contractible periodic orbits is an invariant annulus. We now have the negative divergence test (or Dulac criterion) for non-existance of periodic orbits for 2D vector fields. It is clear that div v > 0 is just as good, and so is div(gv) ≷ 0 for g > 0, as Z M= g dV is a “mass” on Ω. Ω

Theorem. div v < 0 in 2D also implies that periodic orbits are attracting. Proof. Given a periodic orbit γ such that div v < 0 in a neighbourhood of γ choose a transverse arc Σ to γ. Then, by continuity of φ, the forward orbits of points of Σ near γ(0) come back to Σ close to γ(0). In fact, by div v < 0 they come back closer by a RT div v(γ(t)) dt factor e 0 , where T is the period of γ. We prove this as follows. Take a volume A formed by letting an arc I from γ(0) on Σ flow for a time  > 0. After a time T we obtain a skewed parallelogram from A. The area A satisfies RT A(T ) ∼ e 0 div v(γ(t)) dt , A(0)

so the height must decrease as the base stays the same length – so RT |I 0 | ∼ e 0 div v(γ(t)) dt < 1. |I|

Now choose an interval J ⊂ Σ small enough so the above applies. Let U be the region swept out by φt (J). Then φt (U ) ⊂ U for all t ≥ 0. After a time T + δ, φt (U ) ⊂ U 0 , which is the analog of U , but started from T J 0 , the first return of J to R 0 div v dt Σ. U is thinner than U by a factor of e . Thus t≥0 φt (U ) = γ and γ is attracting.

2.3. DAMPED PENDULUM WITH TORQUE

15

We will come across other ways of testing for homotopy class, uniqueness and stability, but the negative divergence test is easiest if it works. For the damped forced pendulum with F > 1, for all x ∈ S1 × R, ω(x) is a periodic orbit and is the same for all x ∈ S1 × R. We can get more information about it by considering “nullclines”, curves on which θ˙ = 0 or p˙ = 0. Which direction does the periodic orbit rotate? θ˙ = 0 iff p = 0 and p˙ = 0 iff θ . We obtain this picture. p = F −sin k

16

CHAPTER 2. BASIC EXAMPLES AND IDEAS

γ ⊂ {p > 0} because p(0) < 0 implies that p increases until positive and can never go negative again. Thus θ˙ > 0 on γ. By periodicity γ spends some time in H˙ < 0 and R some in H˙ > 0, as γ H˙ dt = 0. It spends some time in p˙ > 0 and some in p˙ < 0. What about 0 < F < 1? It is important to decide where the stable and unstable manifolds of the saddle go.

+ WD must exit the annulus {H < E 0 } downwards and WL− goes at least to the p˙ = 0 nullcline. There are three possible cases for the phase portrait, determined by WU+

1. WU+ hits p = 0. 2. WU+ escapes annulus. 3. WU+ remains in annulus and converges to saddle after one revolution.

2.3. DAMPED PENDULUM WITH TORQUE

17

The first case happens with small k, because k = 0 conserves H = 12 p2 − cos θ − F θ. This is the phase portrait.

WU+ hits p = 0 and hence does so for all nearby k. The second case occurs for k large. If we rescale time by putting s = kt, then p dθ = ds k F − sin θ dp = −p + . ds k dp But k is large, so we get dθ ds = 0 and ds = −p. Both the first two cases are open, meaning that {(k, F ) ∈ (0, ∞) × (0, 1) : case 1 (case 2)} is open. But (0, ∞) × (0, 1) is connected, so case 3 occurs in between. In case 3 we get homoclinic orbits — those whose α and ω limit sets consist of equilibria but are not themselves equilibria. For all x above the homoclinic, ω(x) = homoclinic ∪ saddle. When F = 1 we have a “saddle-node” instead of the saddle and sink. It has JNF   0 0 0 −k

and the linearised system has a line of equilibria. The generic phase portrait near a saddle-node equilibrium is

• one sided stable manifold W − (1D)

18

CHAPTER 2. BASIC EXAMPLES AND IDEAS • half plane stable manifold W + (2D) • two sided strong stable manifold W ++ (1D)

Definition. A strong stable manifold is a set  x : d(φt (x), 0) ≤ d(x, 0)e−αt for α > 0 small enough. (We have the time reversed picture if the non-zero eigenvalue is positive.) For the damped pendulum with torque F = 1 we obtain 3 cases according to the behaviour of WU++ . See the next page for the pictures. For F = 1 and k large we have a homoclinic to the saddle-node.

Crossing either of the curves where there is a homoclinc to the saddle-node or saddle generates (or destroys) a periodic orbit in a “homoclinic bifurcation”. Theorem (Extension of Poincar´e-Bendixson). If v is a planar C 1 vector field and o+ (x) is bounded then ω(x) is non-empty, compact, connected and either a periodic orbit or the union of a non-empty compact set of equilibria E and a possibly empty set of trajectories whose α and ω limit sets lie in E. Proof. It is enough to consider the case where ω(x) contains both an equilibrium and a non-equilibrium. Call the non-equilibrium γ(0) and its trajectory γ. Then ω(γ) consists of equilibria, else is q ∈ ω(γ) is not an equilibrium we can take a transverse arc Σ through q and γ ∩ Σ = {q} by the same argument as before and γ is a periodic orbit. But then ω(x) = γ by the same argument as before. Similiarly α(x) consists of equilibria.

2.3. DAMPED PENDULUM WITH TORQUE

The damped forced pendulum

19

20

2.4

CHAPTER 2. BASIC EXAMPLES AND IDEAS

The van der Pol equation

This is x ¨ + (x2 − β)x˙ + x = 0,

β > 0.

√ √ Damping is positive for |x| > β and negative for |x| < β. A generalisation is the system x ¨ + f (x)x˙ + g(x) = 0. We could write it as a Rsystem using y = x, ˙ but it is more usual to introduce y = x x˙ + F (x), where F (x) = 0 f (x) dx. Then x˙ = y − F (x) y˙ = −g(x).

This reflects the original motivation: an electronic oscillator. We find the equilibria at g(x) = 0 and y = F (x). For the van der Pol equation there is one equilbrium at the origin, a repelling focus for 0 < β < 2 or repelling node for β > 2.

Every non-zero orbit moves from quadrant to quadrant clockwise. We try to find a bounding function (H such that H is decreasing R x when large and H is bounded below). We try H = 21 y 2 + G(x), where G(x) = 0 g(x) dx. Now H˙ = −g(x)F (x), so H does not provide a bounding function, though it does tell us that the origin is repelling. It will be useful to consider the change ∆H is H per half revolution. We will assume that g(0) = 0, g 0 (x) > 0, f (0) < 0, f 0 (x) ≥ C > 0 so that F (x) has precisely three zeroes at a < 0 < b.

21

2.4. THE VAN DER POL EQUATION

Let Y1 be the first intersection of the backwards orbit of (b, 0) with the y axis. For Y ≥ Y1 the forward orbit of (0, Y ) must enter x ≥ b and hit the negative y axis at y 0 . Z ∆HY Y 0 =

H˙ dt

Z =

−g(x)F (x) dt.

We will break the path into three pieces.

Z

b

−g(x)F (x) dx y − F (x)

b

g(x) dx y 1 − F (x)

∆HY B = 0

Z = 0

> 0 but decreases at Y increases. The same holds for HB 0 Y 0 .

Z ∆H

BB 0

−g(x)F (x) dt Z B =− F (x) dy

=

B0

< 0 and decreases to −∞ as Y increases. Hence ∆HY Y 0 decreases to −∞ as Y increases and starts positive at y = Y1 . It thus has a unique zero Y0 in between. If we assume further that g and F are both odd then the phase portrait is symmetric with respect to reflection in y = −x and so the unique zero corresponds to a unique periodic orbit. It attracts the orbit of all nonzero points because they must cross the y axis and then H is driven monotonically to H0 = H(0, Y0 ) at successive intersections. If we do not assume odd symmetry then we need to consider the full revolution to the positive y axis. If we let Y2 be the first intersection with the positive y axis of the backward orbit from (a, 0) then for Y > max Y1 , Y2 we can divide the orbit into five pieces and deduce that ∆Hone revolution decreases as Y increases. Also, for 0 < Y ≤ min Y1 , Y2 , ∆H > 0. Thus there must be a zero of ∆H > 0, but we can no longer be sure that it is unique. In general the question of the number of periodic orbits is hard.

2.4.1

Topics arising from the van der Pol equation

Hopf bifurcation As β decreases through zero our periodic orbit shrinks to 0 at β = 0 and is annihilated. Similarly the source at 0 turns into a sink which attracts all orbits.

22

CHAPTER 2. BASIC EXAMPLES AND IDEAS

This is called Hopf bifurcation (Andronov, Poincar´e). It is very common, for instance feedback in a PA system as gain is increased, singing wires in the wind as speed passes a critical value. Nearly conservative systems For β small it is√only interesting to consider x small since energy decreases for large amplitude (x ∼ β). Then the equation is close to the harmonic oscillator x ¨ + x = 0. We can use this toHdetermine the approximate size and shape of the periodic orbit. We compute ∆H = −g(x)F (x) dt around a periodic orbit of the harmonic oscillator with radius r. r3 cos3 t − βr cos t ∆H = −r cos t 3   r2 2 = πr β − . 4 I



 dt

To first order in β we expect the periodic orbit of the van der Pol oscillator to√be close to the periodic orbit of the harmonic oscillator for which ∆H = 0, thus r ≈ 2 β and T ≈ 2π. Relaxation Oscillators For β large, what is the periodic orbit? We rescale x by   x3 x˙ = β y − +x 3 x y˙ = − . β 3

√

3

β and y by β 2 to obtain

If β is large then x˙ is large unless y ≈ x3 − x. Then y˙ = motion along the curve until the turning points.

−x β

gives the direction of

We can get the approximate period by integrating dt over the slow phases: Z

2 3

T ≈2 − 23

dy . y˙

23

2.4. THE VAN DER POL EQUATION We use y ≈

x3 3

− x and dy = (x2 − 1)dx. Thus T ≈

β 2

(3 − 2 log 2) for large β.

Poincar´e Index Continuity of the vector field imposes strong restrictions of the possible arrangements of equilibria and limit cycles. Given a vector fields v on R2 and any simple closed curve γ on which v 6= 0 define the index of γ to be the number of times v rotates around 0 for one revolution around γ (taken anticlockwise). For instance, a small curve around an attracting focus has index +1 and a small curve around a saddle has index −1.

• Index is preserved under continuous deformations of γ in the clan of curves such that v 6= 0 on them. We can therefore defined the index of an isolated equilibrium to be the index of all small enough curves aroung it. Attracting and repelling foci/nodes, improper nodes, stars and centres have index +1, whereas saddles have index −1. The index of a small curve surrounding no equilibria is 0. • The index of a “sum” of closed curves is the sum of the indices. • Hence the index of γ is the sum of the indices of the equilibria inside γ. • Index of a periodic orbit is +1. So every periodic orbit must surround at least one equilibrium and the sum of the indices of the equilibria surrounded must be +1. Floquet multipliers of a periodic orbit Take a transverse section Σ to γ and let f : Σ 7→ Σ be the first return map. Then γ(0) is a fixed point of f . Definition. λ = f 0 (γ(0)) is the (Floquet) multiplier of γ. λ does not depend on the choice of Σ and λ > 0 for orientable surfaces. If |λ| < 1 the periodic orbit is attracting and if |λ| > 1 the periodic orbit is repelling. We say that γ is hyperbolic is |λ| = 6 1. In higher dimensions, the multipliers are the eigenvalues of Dfγ(0) .

24

CHAPTER 2. BASIC EXAMPLES AND IDEAS

Chapter 3

Bifurcations of 2D flows We have seen that the dynamics of planar vector fields is fairly simple — the ω-limit sets are of limited types. There can be interesting changes in qualitative behaviour as parameters are varied (bifurcations).

3.1

Structural stability

The first step is a non-bifurcation result. Theorem (Andronov-Pontryagin). Suppose a C 1 vector field v points inwards on the disk D2 ⊂ R2 , flow φ all its equilibria and periodic orbits are hyperbolic and there are no saddle connections, then the flows φ˜ for all C 1 -close vector fields v˜ are topologically equivalent, that is φ˜τ (t,x) (h(x)) = h(φt (x)) for some near-identity homeomorphism h : D2 7→ D2 and continuous τ : R+ × D2 7→ R+ which is an orientation-preserving homeomorphism of R+ for each x ∈ D2 and τ (t, φs (x)) = τ (t + s, x). Vector fields v such that all C 1 -close v˜ have topologically equivalent flows are called “structurally stable”. Note. The C 1 -norm is kvk1 = supx∈D2 max (|v(x)| , kDv(x)k). We’ll prove a small part of this and then investigate what happens when any of the hypotheses are not satisfied. Definition. An equilibrium x of a C 1 vector field v is non-degenerate if 0 is not an eigenvalue of Dvx . Theorem (Persistence of non-degenerate equilibria). If x0 is a non-degenerate equilibrium of C 1 vector field v0 and U is a neighbourhood of x0 then there exists a subneighbourhood V and  > 0 such that kv − v0 k1 <  on V implies that v has a unique equilibrium x ˜ ∈ V , and it is non-degenerate and depends C 1 on v. This is an application of the Implicit Function Theorem, an important result in analysis which we will use frequently. Here is a general statement. 25

26

CHAPTER 3. BIFURCATIONS OF 2D FLOWS

Theorem (Implicit Function Theorem). Let X, Y and Z be open subsets of complete normed vector spaces and F : X × Y 7→ Z be C 1 . Suppose F (x0 , y0 ) = z0 and ∂F ∂x : X 7→ Z is invertible there. Then for all small enough neighbourhoods V of x0 there exists a neighbourhood W of y0 such that y ∈ W implies that ∃!˜ x(y) ∈ V with F (˜ x(y), y) = z0 . Furthermore, ∂F is invertible at (˜ x (y), y) and the mapping  ∂F −1∂x∂F d˜ x 1 x ˜ : W 7→ V is C , with dy = − ∂x ∂y . We will not prove this here, but we will apply it to three cases. Persistence of Equilibria Given a C 1 vector field v on Rn , let F (x, v) = v(x), F : Rn × “C 1 vector fields” 7→ Rn . If we have a vector field v0 and equilibrium x0 , F (x0 , v0 ) = 0. If ∂F ∂x is invertible at (x0 , v0 ) then there is a locally unique equilibrium x ˜(v) for each v near v0 . We ∂v conclude by saying that ∂F ∂x = ∂x is invertible if no eigenvalue is 0. Return map to a C 1 transverse section for a C 1 vector field is C 1 Suppose Σ is a given C 1 transverse section to v. Suppose it is given by choosing 1 co-ordinate xn for which vn is non-zero on Σ and let x0 stand for the remaining coordinates. Then Σ has form xn = ζ(x0 ) for some C 1 ζ. We can also write Σ as {x : α(x) = 0} for C 1 α. Now v induces a C 1 flow φ : R × Rn 7→ Rn , t × x 7→ φt (x). Suppose x ¯ ∈ Σ and x) ∈ Σ for some t¯ > 0. Then we would like to deduce that ∃ C 1 t˜(x), Σ 7→ R such φt¯(¯ that f (x) = φt˜(x) (x) ∈ Σ, so the return map f is C 1 . We apply the IFT to F : (x0 , t) 7→ α(φt (x0 , ζ(x0 ))). 0 0 Now F (¯ x0 , t¯) = 0 and ∂F ∂t = dα v(φt (x , ζ(x )) 6= 0, so it is invertible. Hence there 0 0 exists a locally unique t˜(x ) such that F (x , t˜(x0 )) = 0. The return time t˜ is C 1 .

Persistence of non-degenerate periodic orbits Definition. A periodic orbit is non-degenerate if it has no Floquet multiplier +1. A periodic orbit of v corresponds to a fixed point of the return map f to some transverse section Σ. Non-degenerate means that (I − Dfx ) is invertible. It can be proved that f depends C 1 on v. The equation to solve for a fixed point of f if f (x, v) = x. We apply the IFT to F (x, v) = f (x, v) − x, F : Σ × “C 1 vector fields” 7→ Σ. Now ∂F ∂x is invertible, so we can apply the IFT to get the required result.

3.2

Saddle-node equilibria

Definition. A saddle-node is an equilibrium with an eigenvalue 0, but this does not suffice to determine the local phase portrait. In addition we require :1. 0 is a simple eigenvalue (multiplicity 1), 2. there is no other spectrum on the imaginary axis, 3. a non-degeneracy assumption on the quadratic part of the Taylor expansion (to be revealed later).

3.2. SADDLE-NODE EQUILIBRIA

27

The linearised picture has a line of equilibria, which is not typical in nonlinear systems. We will find that all sufficiently smooth nonlinear systems possess an invariant curve tangent to the 0-eigenvector such that the motion relative looks the same (topological equivalence) as for the linear system, but the dynamics on the invariant curve is in general non-trivial. This invariant curve is called a centre manifold W c .

Theorem (Centre manifold theorem). Given an equilibrium with some spectrum on the imaginary axis, let E c be the span of eigenvectors corresponding to eigenvalues on the imaginary axis and let E h be the span of eigenvectors corresponding to eigenvalues off the imaginary axis. Choose the corresponding co-ordinate system (c, h) and write the vector field as c˙ = C(c, h), h˙ = H(c, h). Then there is a locally invariant submanifold of the form h = W (c) = o(|c|) and the dynamics is topologically equivalent to ( c˙ = C(c, W (c)) . h˙ = ∂H ∂h |0 .h We will not prove this theorem, but knowing the result we can compute W c to any desired accuracy by searching for it as a Taylor series and comparing coefficients. The goal is to determine the dynamics c˙ = C(c, W (c)) on W c to sufficiently high order to determine its topological type. Example. ( x˙ = x2 + xy + y 2 = X(x, y) y˙ = −y + x2 + xy = Y (x, y) Solution. This is already in a good co-ordinate system, c = x and h = y. We look for y = W (x) = a2 x2 + a3 x3 + a4 x4 + . . . . Invariance means that the two expressions for y˙ on y = W (x) must be equal. 1. y˙ = Y (x, W (x)) = −(a2 x2 + a3 x3 + a4 x4 + . . . ) + x2 + x(a2 x2 + a3 x3 + . . . ) 2 2 2 2 2. y˙ = ∂W ∂x X(x, W (x)) = (2a2 x + 3a3 x + . . . )(x + x(a2 x + . . . ) + (a2 x + 2 . . . ) ).

Comparing coefficients yields y = x2 − x3 + o(x4 ) and the dynamics on W c is x˙ = X(x, W (x)) = x2 + x3 − 2x5 + o(x5 ). The first term suffices to determine local topological type and the full local phase portrait is as above. Example (Damped pendulum with F = 1). ( θ˙ = p p˙ = −kp − sin θ + 1 has an equilibrium at ( π2 , 0) with eigenvalue 0.

28

CHAPTER 3. BIFURCATIONS OF 2D FLOWS

Solution. It is not necessary to transform to co-ordinates in which E c and E h are axes. We just need to shift the origin to ( π2 , 0) by using φ = θ − π2 and look for an invariant manifold tangent to E c . We set p = a2 φ2 + a3 φ3 + . . . and compare coefficients. 1 2 After some fiddling we get the center manifold p = 2k φ − 2k13 φ3 + O(φ4 ) with 1 2 φ − 2k13 φ3 + O(φ4 ). Again the local picture is determined by the first motion φ˙ = 2k term. We can now give the non-degeneracy condition referred to earlier: the quadratic part of the vector field on the centre manifold is non-zero. Notes. 1. Centre manifolds are in general not unique, for instance x˙ = x2 , y˙ = −y: we can take any trajectory on the left ∪ the positive x axis. 2. But they all have the same Taylor expansion. 3. There is not necessarily any analytic centre manifold, and the Taylor expansion need not converge, or if it does the limit need not be a centre manifold. Now we want to find out what happens to a saddle-node if we change some parameters of the vector field. We consider the damped forced pendulum with F = 1+µ. We add the equation µ˙ = 0 and compute the 2D centre manifold p = W (φ, µ) of the extended system, which should be tangent to the span of the eigenvectors and generalised eigenvectors for eigenvalue 0, that is   1 0 0

µ k

and

  0 1. k 1

We write p = µk + a11 φ2 + a12 φµ + a22 µ2 + . . . and fiddle a bit to get p = 1 2 + 2k φ − k13 φµ + k15 µ2 + . . . , and the dynamics on the centre manifold is ( φ˙ = µk + µ˙ = 0

1 2 2k φ

+ o(φ2 + |µ|)

.

This is easy to analyse, we have a curve of equilibria µ = − 12 φ2 + o(φ2 ) and so have an attracting and repelling equilibrium for each µ < 0 and no equilibria for µ > 0.

We put in contracting dynamics in the p direction to obtain the following transition.

3.3. OTHER CASES OF DEGENERATE EQUILIBRIA

29

This is called a saddle-node bifurcation. We obtain an equivalent picture whenever we have a saddle-node equilibrium for µ = 0 such that the coefficient a 6= 0 in the CM flow ( φ˙ = aµ + bφ2 + o(φ2 + |µ|) . µ˙ = 0

3.3

Other cases of degenerate equilibria

If either a or b is zero we need to compute the flow on the centre manifold to higher order. The transcritical bifurcation Here, a = 0 and b 6= 0 and x˙ = bx2 +cxµ+dµ2 +O(3) with c2 > 4bd. Then we obtain two curves of equilibria, both transverse to µ = const and the dynamics looks like this. Proof. If we ignore O(3) the equation for equilibria gives a line pair through (x, µ) = (0, 0). Write it as P Q = 0 by factorising the quadratic. Then to prove that there is a curve of equilibria near P = 0, for F (P, Q) = P Q + O(3) write G(R, Q) = F (RQ, Q) = RQ2 + Q3 O(1) and then let H(P, Q) = QG2 = R + QO(1). Then ∂H ∂R is invertible and H(0, 0) = 0, so by the IFT there is a locally unique solution R = R(Q), ie P = QR(Q). The transcritical bifurcation is common where 0 is always an equilibrium. The pitchfork bifurcation In this case a = b = 0, x˙ = cxµ + dµ2 + ex3 + o(x3 + µ2 ), with c, e 6= 0. There exist two curves of equilibria, x ∼ − dc µ and µ ∼ − ec x2 . This is clear if there is no remainder, and if the remainder is present the IFT can be applied to scaled problems for x = µξ(µ) and µ = x2 ν(x). The picture depends on the sign of c and e. It is “supercritical” if ce < 0 — one attracting equilibrium gives two attracting and one repelling as cµ increases. It is “subcritical” if ce > 0, in this case one repelling equilibrium gives two repelling and one attracting as cµ increases. The pitchfork is common in systems with odd symmetry, x(−x, ˙ µ) = −x(x, ˙ µ). Then a = b = d = 0.

3.3.1

Imperfections

If a one-parameter family of vector fields vµ has a saddle-node bifurcation and then we make a small change to the family, represented by an additional parameter , then we can make a 3D centre manifold (x, µ, ). (By the IFT) The curve µ ∼ − ab x2 of equilibria for  = 0 persists to a curve µ ∼ − ab x2 + g(x, ) for some g, but all this

30

CHAPTER 3. BIFURCATIONS OF 2D FLOWS

does for small  is to move the saddle-node bifurcation by O() in (x, µ), giving a topologically equivalent bifurcation.

On a transcritical bifurcation x˙ = bx2 + cxµ + dµ2 + O(3) + f (x, µ, )

on the centre manifold.

We find generically one of two cases for  > 0 (depending on f ).

Lastly we consider the effect of imperfection on a pitchfork. x˙ = cxµ + dµ2 + ex3 +o(x3 +µ2 )+f (x, µ, ) on the centre manifold. Generically the pitchfork breaks as shown (or the other way!).

3.4

Hopf Bifurcation

Another way the hypotheses of Andronov-Pontryagin is if there is an equilibrium with a pair of eigenvalues on the imaginary axis. Suppose they are simple and there is no other spectrum on the imaginary axis. Suppose we have a one-parameter family of vector fields going through this case at µ = 0. Then the equilibrium persists and moves smoothly with µ, so we could shift the origin as a function of µ to keep the equilibrium at 0. Furthermore, simple eigenvalues of a matrix Aµ move smoothly with µ. Thus we have eigenvalues α(µ) ± ıβ(µ) with α(0) = 0 and β(0) 6= 0. Take a centre manifold and use polar co-ordinates (r, θ, µ). ˜(r) : R+ 7→ R Theorem. Suppose α0 (µ) 6= 0. Then, for r small, ∃ a smooth function µ such that for µ = µ ˜(r) the orbit of (r, θ = 0) is periodic and µ ˜(0) = 0, µ ˜0 (0) = 0. Proof. WLOG use co-ordinates so that the linear part is      x˙ α(µ) −β(µ) x = y˙ β(µ) α(µ) y and WLOG α0 (µ) > 0, β(µ) > 0. Then, switching to polars, r˙ = α(µ)r + O(r2 ) dr 2 and θ˙ = β(µ) + O(r). So dθ = α(µ) β(µ) r + O(r ). Then the return map from θ = 0 to

31

3.4. HOPF BIFURCATION

θ = 2π is well defined: r0 = ρ(r, µ). Now ∃ a periodic orbit through r > 0, θ = 0 iff ρ(r, µ) = r. It is most convenient to write this as F (µ, r) = log ρ(µ, r) − log r = 0. Since

d dθ

log r =

α β

+ O(r), F (µ, r) =

2πα(µ) β(µ) + O(r), so we can extend F to 2πα0 (0) 6= 0, so we can apply the IFT β(0)

r = 0. Now F (0, 0) = 0 and ∂F ∂µ (0, 0) = to deduce a locally unique curve of solutions µ = µ ˜(r), provided we show that F is C 1 . Now F is C 1 for r > 0 by the basic theorem of ODEs extended to time- and parameter-dependent vector fields. But what about at r = 0? To tackle this we will perform some further co-ordinate changes to simplify the differential equations. It is easiest to do this in the complex co-ordinate z = reıθ = x + ıy. The linear part is already in the form z˙ = λz where λ = α + ıβ. Suppose we have a Taylor expansion 3

z˙ = λz + a11 z 2 + a12 z z¯ + a22 z¯2 + O(|z| )

a11 , a12 , a22 ∈ C.

We seek to eliminate the quadratic terms by change of variable z = w + b11 w2 + b12 ww ¯ + b22 w ¯2

b11 , b12 , b22 ∈ C. 3

The inverse function theorem gives w = z − b11 z − b12 z z¯ − b22 z¯2 + O0 (|z| ) Then (after substituting and fiddling), we find that if we choose a11 b11 = λ a12 b12 = ¯ λ a22 b22 = α − 3ıβ 2

3

we eliminate all quadratic terms to obtain w˙ = λw + O(|w| ). Now use polar cod 2 ordinates w = reıθ to obtain dθ log r = α β + O(r ). Then F = log ρ(µ, r) − log r = ∂F 2 1 2π α(µ) β(µ) + O(r ) is C at r = 0 also with ∂r = 0. So the IFT gives a locally unique solution µ = µ ˜(r) for r small and µ ˜(0) = 0, µ ˜0 (0) = 0.

Remarks In fact α0 (µ) > 0 WLOG, and then for (µ, r) small, F (µ, r) < 0 for µ < µ ˜(r) so intersections with θ = 0 move towards 0 and for µ > µ ˜(r), F (µ, r) > 0 and intersections with θ = 0 move away from 0. If we carry out the elimination to higher order we can determine the topological type of the flows for (µ, r) small. 4 Suppose w˙ = λw + a111 w3 + a112 w2 w ¯ + a122 ww ¯ 2 + a222 w ¯ 3 + O(|w| ). We try to 3 2 eliminate all cubic terms by change of variable w = v + b111 v + b112 v v¯ + b122 v¯ v2 + 3 b222 v¯ . We find that we cannot eliminate all the cubic terms, the best we can do is if we choose a111 b111 = 2λ b112 = 0 a122 b122 = ¯ 2λ a222 b222 = ¯ 3λ − λ

32

CHAPTER 3. BIFURCATIONS OF 2D FLOWS 4

and we reduce the equation to v˙ = λv + av 2 v¯ + O(|v| ), where a = a112 . This is called a “normal form”. In polars, ( r˙ = αr + <(a) r3 + O(r4 ) , θ˙ = β + =(a) r2 + O(r3 ) so we expect a periodic orbit of radius r ∼ Now

d dθ

log r = F =

α+<(a) r 2 β+=(a) r 2

2π [log r]θ

q α(µ) − <(a) .

+ O(r3 ) and

 
2π =(a) 2 2 = r + O(r3 ). α + <(a) r 1− β β

2 0 Thus the solution curve has µ ∼ − α<(a) 0 (µ) r . WLOG α (µ) > 0. If <(a) < 0 we get a supercritical bifurcation and if <(a) > 0 we get a subcritical bifurcation.

3.5

Homoclinic Bifurcation

A third way in which Andronov-Pontryagin can fail is if there is a saddle connection. Let’s consider the simplest interesting case: a 2D vector field with a saddle and a homoclinic orbit to it. Suppose this occurs at µ = 0 in a smooth one-parameter family of vector fields vµ . The saddle is hyperbolic (and therefore persistent), so we can take it at 0 for all µ. It can also be proved that the local stable and unstable manifolds W± = {x : d(φt (x), 0) ≤  ∀t ≷ 0} respectively, for  small, vary smoothly with µ, so for any t > 0 so do φ±t and W± . In particular, if we choose a transverse section Σ to the homoclinic orbit at µ = 0 then the first intersections of φ±t (W± ) with Σ vary smoothly with µ. Choose a co-ordinate s along Σ with s+ = 0 WLOG − for all µ, oriented as shown. Assume ds dµ 6= 0 (WLOG > 0) and analyse the first return map from s > 0 to s0 on Σ. Clearly lims→0 s0 (s) = s− and s0 (s) is increasing, but 0 results will depend crucially on the slope ds ds . This is decided by the eigenvalues of the saddle as follows. We can choose co-ordinates (depending smoothly on µ) so that the linearisation at 0 is ( x˙ = −α(µ)x , where −α < 0 and β > 0 are the eigenvalues. y˙ = β(µ)y

33

3.5. HOMOCLINIC BIFURCATION Furthermore we make W± go exactly along the axes ( x˙ = −xf (x, y, µ) y˙ = yg(x, y, µ)

near 0

f (0, 0, µ) = α g(0, 0, µ) = β.

Then we compute the “transit map” from x = ξ to y = η, ξ, η small.

Z

Z Z f (x, y) x˙ x˙ dt = dy = − d(log y) x xy˙ g(x, y) α ∼ − ∆ log y. β

∆ log x =

The slope is given by

δx1 ξ

∼

α βη



y0 η

 αβ −1

. This cannot be deduced by differentiaR

tion. Instead use volume changes by a factor e δx1 ηy˙ δy0 xξ˙

R

=e

div v dt

div v dt

. Thus

∼ e(b−a)T ,

where the transit time Z T =

Z dt =

dy = y˙

Z

dy 1 1 η ∼ ∆ log y = log . yg β β y0

  α −1 αξ y0 β 1 ∼ . Caution: the ∼ is not strong So δx δy0 βη η enough to deduce behaviour for α = β. To obtain the return map to Σ we must compose 3 maps. The first and third are diffeomorphisms, as they are achieved in bounded time, so they do not change the power law of the second map. Hence we obtain:

For α > β an attracting periodic orbit is born as µ increases through 0. For α < β a repelling periodic orbit is destroyed as µ increased through 0. The case α = β requires further analysis.

34

CHAPTER 3. BIFURCATIONS OF 2D FLOWS

The period of the periodic orbit is dominated by the time spent near the saddle. For η α > β it has y0 ∼ Cµ for some C, so T ∼ β1 log Cµ → ∞ as µ & 0. For α < β it has x1 ∼ −Cµ, so Z T =

1 x1 1 ξ dx ∼ − log ∼ log → ∞ as µ % 0. x˙ α ξ α C |µ|

Let’s compute its Floquet multiplier. If α > β slope at fixed point ∼ C

0



y0 η

 αβ −1 ∼C

0



Cµ η

 αβ −1 → 0 as µ & 0.

If α < β, slope at fixed point ∼ C

0



x1 ξ

1− αβ ∼C

0



C |µ| ξ

1− αβ → ∞ as µ % 0.

We define the Lyapunov (or Floquet) exponents of a periodic orbit to be the logarithm of the Floquet multiplier divided by the period. In both the above cases the Lyapunov exponent is β − α. Note. In 2D there is only one Lyapunov exponent and we can get it using R H volume change: the Floquet multiplier is e div v dt , so the Lyapunov exponent is T1 div v dt, the average of div v around the orbit.

3.5.1

Finding homoclinic bifurcations in nearly Hamiltonian systems

In 2D, a Hamiltonian system is one ( x˙ = ∂H ∂y y˙ = − ∂H ∂x for some H : R2 7→ R. It automatically conserves H. If H has a saddle point s then it gives a saddle equilibrium for v. The level set {H = H(s)} often contains a homoclinic orbit. In a Hamiltonian system the eigenvalues −α and β always satisfy α = β, so we can’t immediately apply the previous results. But suppose we have a two-parameter family vµ, of the form  x˙ =

 Hy + fµ, (x), −Hx

f0,0 ≡ 0.

Then under suitable conditions, we will find a curve of homoclinic bifurcations in the (µ, ) plane. Choose a transverse section Σ to the homoclinic orbit of (µ, ) = (0, 0). Then for nearby (µ, ) the saddle persists and local stable/unstable manifolds move smoothly to obtain.

35

3.5. HOMOCLINIC BIFURCATION

Let us evaluate ∆H = H(s− ) − H(s+ ). We get a homoclinic orbit if ∆Hµ, = 0. Now Z s− Z s dH dH ∆H = dt + dt dt s+ dt s ! Z − Z s

s

+

=

(DH.fµ, (x) dt) s+

s

Z

s−

=

Z s! (fµ, .∇H) dt.

+ s+

s

To leading order in µ,  we can R evaluate this by integrating along the homoclinic γ of µ,  = 0. Define M (µ, ) = γ DH.fµ, (x(t)) dt. If M (µ, ) has a curve Γ of simple zeroes then use the IFT to deduce that vµ, has a curve of homoclinic bifurcations ˜ along a nearby curve Γ. Return to the DFP If you’ve forgotten(!), this has equations θ˙ = p and p˙ = −kp − sin θ + F . The case k = F = 0 is Hamiltonian, H(θ, p) = 21 p2 − cos θ. This has two homoclinics to the saddle at p = 0, θ = π (or −π) with equations p = ±2 cos θ2 . 

Z M (k, F ) =

DH γ+

Z

0 −kp + F



Z sin θ

dt =

p






γ+

0 −kp + F

 dt

−kp2 + F p dt

= γ+

Now use dt =

dθ θ

dθ p .

= Z

π

F − 2k cos

M (k, F ) = −π

θ dθ 2

= 2πF − 8k For k > 0, note that α > β. The homoclinic bifurcation generates an attracting periodic orbit.

Other cases of saddle connections: • n-D vector field with homoclinic to hyperbolic equilibrium • homoclinic to saddle-node • heteroclinic cycle • 2 homoclinics to the same saddle.

36

CHAPTER 3. BIFURCATIONS OF 2D FLOWS

The energy method can also be used to find approximate periodic orbits for nearly Hamiltonian systems. In general, Hamiltonian systems have a family of periodic orbits labelled by energy. Choose a transverse section and compute ∆H for the first return for the perturbed system. I M (µ, , E) = DH.fµ, dt, γ(E)

where γ(E) is the periodic orbit of the unperturbed system with energy E. If M has a simple zero as a function of E then vµ, has a periodic orbit near γ(E).

3.6

Saddle-node of periodic orbits

Take a periodic orbit γ with Floquet multiplier +1. Take a transverse section Σ with co-ordinate s. The return map f has f (γ(0)) = γ(0) and f 0 (γ(0))0 = +1. Suppose f 00 6= 0. If we change a parameter µ then f changes smoothly with µ and we obtain (if ∂f ∂µ |0 > 0):

As µ increases through 0 an attracting and a repelling fixed point collide and annihilate.

This is a saddle-node bifurcation of periodic orbits. We can extend this to higher dimensional vector fields by constructing a centre manifold for fixed points of the return map f : Σ 7→ Σ analogously to the centre manifold for equilibria.

Chapter 4

Introduction to dynamics in 3D We restrict attention to periodically forced 2D systems, as many of the interesting phenomena already occur here. For instance, the periodically forced van der Pol oscillator, x ¨ + (x2 − β)x˙ + x = A cos ωt, which may be rewritten as   x˙ = y − F (x) y˙ = −x + A cos φ  ˙ φ=ω

for φ ∈ S1 . Remember that F (x) =

x3 − βx. 3

Or the periodically additively forced pendulum, θ¨ + k θ˙ + g sin θ = F + A cos ωt, which is  ˙  θ = p p˙ = −kp − g sin θ + F + A cos φ.  ˙ φ=ω The parametrically forced pendulum is θ¨ + k θ˙ + (α +  sin ωt) sin θ = 0 may be written as  ˙  θ = p p˙ = −kp − (α +  sin φ) sin θ, a vector field on S1 × R × S1 .  ˙ φ=ω

4.1

Time-T map

At least conceptually, dynamics of a periodically forced 2D system, x˙ = v(x, t) = v(x, t + T ) can be reduced to iteration of a 2D map f . Define φs,t to be the map which, gives x(t) given x(s), that is x(t) = φs,t (x(s)). Then let f be φ0,T , the “time-T map”. Now φ0,nT = f ◦ · · · ◦ f = f n . So we can find x(nT ) from x(0) by iterating. Also, φmT +s,nT +t = φ0,t ◦ f n−m ◦ φ−1 0,s (0 ≤ s, t < T ), so we can get from any time to any other time by knowing f and φ0,s for s ∈ [0, T ). The problem is that in general f is not explicitly available. But we can often get sufficient information about f to determine a lot about the dynamics. 37

38

4.2

CHAPTER 4. INTRODUCTION TO DYNAMICS IN 3D

Existence of periodic orbits

The period of any periodic orbit of x˙ = v(x, φ), φ˙ = 1, φ ∈ ST1 is a multiple of T . Period T orbits correspond to fixed points of f , period nT orbits correspond to period n orbits of f , or fixed points of f n . We therefore want the fixed points of a map. It is sometimes easy. Contraction Mapping If f maps a closed set B into itself and contracts distances in B then f has a unique attracting fixed point in B. We can often verify this even without a formula for f . Example. We prove the existance of a periodic orbit for the van der Pol oscillator x ¨ + (x2 − β)x˙ + x = A cos ωt,

β < 0.

2 Solution. We consider the Riemannian metric dl2 = dx2 + (dy − |β| 2 dx) . We find 1 1 − |β|T ˙ ≤ − |β| δl for a line element. Thus δl(T ) ≤ δl(0)e 2 . In particular, after that δl 2 K − 12 |β|T , one period δl(T ) ≤ λδl(0), λ = e < 1. Let K = l(f (0)). For all R ≥ 1−λ f maps the disc of l-radius R into itself and contracts by at least λ. Thus f has a fixed point in R2 which attracts all orbits. This is a “globally attracting fixed point”.

Lefschetz Index This is the discrete-time analog of the Poincar´e index. Let γ be a simple closed curve which passes through no fixed points. Then the index of γ under f is the number of revolutions of f (x) − x around 0 for one revolution of x about γ.

If the index of γ under f is non-zero then γ surrounds at least one fixed point. The # Real positive multipliers > 1 index of a small curve about a non-degenerate fixed point is (−1) . Perturbation If the unforced case has an equilibrium then x˙ = v(x), φ˙ = 1 then this will persist for  small in the system x˙ = v (x, φ), φ˙ = 1 to give a periodic orbit. If the equilibrium has eigenvalues λj then the period-T orbit has multipliers eλj . The periodic orbit is therefore non-degenerate iff eλj 6= 1 ∀j. Harmonic balance Try Fourier series to get x(t) =

X n∈Z

2π

xn e T

ınt

.

4.3. LINEAR STABILITY

39

Energy balance For nearly Hamiltonian systems the method of energy balance can be used as before.

4.3

Linear stability of periodic orbits of T -periodic vector fields

Given a T -periodic vector field x˙ = v(x, t) and a periodic orbit γ(t) of period T , ˙ = Dv(γ(t), t)δx. Write this as x˙ = A(t)x. Let Φ(t) be the linearise about it to get δx ˙ matrix solution of Φ = A(t)Φ(t) with Φ(0) = I (the “fundamental matrix solution”). Then the solution for any x0 is x(t) = Φ(t)x0 . In particular, x(nT ) = Φ(T )n x0 , so stability is determined by the eigenvalues of Φ(T ). These are the Floquet multipliers because Φ(T ) = Df , the derivative of the return map. Φ(T ) is called the “monodromy matrix”.   a b In the 2D case, write Φ(t) = . Then the Floquet multipliers λj are the roots c d q tr2 ± of λ2 − tr λ + det = 0, that is λ± = tr 2 4 − det. If both |λ± | < 1 then the periodic orbit is attracting. If one of |λ± | > 1 then the periodic is unstable. The stability boundaries are given by tr = 1 + det, tr = H −(1 + det), det = 1 and tr2 ≤ 4 det. Note that det = e tr A(t) dt > 0.

In practice, Φ(T ) is not easy to calculate unless A(t) is constant, A(t) is piece-

40

CHAPTER 4. INTRODUCTION TO DYNAMICS IN 3D

wise constant or A(t) is diagonalisable by a constant matrix. But we can always get det Φ(T ) and we can compute tr Φ(T ) to any desired accuracy by perturbation from a known case.

4.3.1

The Mathieu Equation

This is x ¨ + k x˙ + (α +  sin t)x = 0, the linearisation about θ = 0 of the parametrically forced pendulum. The linked first order system is ( x˙ = y y˙ = −ky − (α +  sin t)x or x˙ = [C + P (t)] x. The solution for  = 0, α > 0 is   1 √ cos ωt ω sin ωt , where ω = Φ(t) = α. −ω sin ωt cos ωt For shorthand we write this as eCt . Now Φ(t) depends smoothly on  and so Φ(t) = eCt + B(t) + O(2 ), where B˙ = CB + P (t)eCt , B(0) = 0. This is “linear inhomogeneous”, so we get the solution by “variation of constants” (or more accurately “convolution with impulse response”) Z t B(t) = eC(t−s) P (s)eCs ds. 0

In particular, the (horrendous!) expression for B(2π) is   Z 2π  1 0 0 cos ω(2π − s) cos ωs ω sin ω(2π − s) − sin s 0 −ω sin ω(2π − s) cos ω(2π − s) −ω sin ωs 0   a b1 which we write as 1 (ω). Then det Φ(2π) = 1 and c1 d1

1 ω

 sin ωs ds cos ωs

tr Φ(2π) = 2 cos 2πω + (a1 + d1 ) + O(2 ). It turns out that a1 (ω) + d1 (ω) = 0 so we need to compute to higher order to see how tr Φ(2π) changes with . But we already see that the solution is linearly stable provided that ω is not within O() of a half-integer, so let us restrict attention to ω = n2 + δ with n ∈ N and δ = O(). There is a trick which will enable us to get tr Φ(2π) to O(2 ) without computing Φ(2π) to O(2 ). Note that det = ad − bc = √

r

(a−d)2 4

(a + d)2 (a − d)2 − − bc. 4 4 +bc

. Now a − d = (a1 − d1 ) + O(2 ) and   −π 0 bc = ( ω1 sin 2πω + b1 )(−ω sin 2πω + c1 ). At ω = 12 , B(2π) = and hence 0 π for ω = 21 + δ we obtain Thus tr = ±2 det 1 −

det

tr2 = 1 + π 2 2 − 4π 2 δ 2 + O(3 ) 4

4.4. NEAR-IDENTITY MAPS AND AVERAGING and thus tr = −2(1 +

41

π2 2  − 2π 2 δ 2 + O(3 )), 2

leading to instability in a wedge |δ| ≤ 2 . If we now add damping k then det Φ(T ) = e−2πk < 1. We get stability iff |tr| ≤ 1 + det. Rewrite using the trick as (1 − det)2 ≥ (a − d)2 + 4bc. For k = O() this gives instability in a region 2 ≥ k 2 + 4δ 2 , which is how you work a swing. For ω = n2 with n > 1 we find that B(2π) = 0, so we need to compute Φ(2π) to higher order. We obtain the stability diagram:

Notes. 1. Need to do nonlinear analysis to see where the instability leads, for the parametrically forced pendulum with k > 0 the instability saturates onto a period 4π orbit (period-doubling bifurcation) near threshold. 2. There are thin strips of stability for α < 0. Thus in principle one can stabilise an inverted pendulum by choosing forcing frequency and amplitude correctly. 3. The nth tongue has a width ∆α = n for k = 0, but this is a special feature of the forcing having only one Fourier component, and in general each opens linearly.

4.4

Near-identity maps and averaging

If the time-T map happens to be close to the identity, say f (x) = x + g(x), C ∞ then it can be proved that there exists an asymptotic series v(x, ) = v1 (x) + 2 v2 (x) + . . . (n) (n) of autonomous vector fields such that ∀n ∈ N, f − φ1 = O(n ), where φ1 is the time-1 map of v1 (x) + 2 v2 (x) + · · · + n vn (x). It is clear that we can start with

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CHAPTER 4. INTRODUCTION TO DYNAMICS IN 3D (1)

v1 (x) = g(x): then f is the Euler step for φ1 . Unfortunately the series often fails to converge, but this “flow approximation” is still useful. The first terms in the series can be computed explicitly for time-T maps of T periodic vector fields which are small on the scale of one period of the forcing, i.e. x˙ = v(x, t, ), period T , T  1, by the “method of averaging”. Crudely, we average over one period and write x ¯˙ = 

1 T

Z

T

v(¯ x, t, ) dt. 0

To make a proper derivation, and a procedure that can be pushed to arbitrarily higher order, introduce a co-ordinate change x = y + w(y, t, ), w(y, t + T, ) = w(y, t, ) and choose w to push the time-dependence of v to higher order:  −1   ∂w ∂w y˙ =  I +  v¯(y + w) + v˜(y + w, t, ) − , (∗) ∂y ∂t where we have split v = v¯ + v˜ into mean and zero-mean parts. The leading order time-dependence can be eliminated by choosing Z t w(y, t, ) = v˜(y, s, ) ds + w0 (y, ), 0

where w0 is chosen to give w zero mean. Then the time-dependence of the RHS of (∗) is only via the terms w, and hence is O(2 ). This procedure can be iterated, but the first step often suffices and it gives y˙ = ¯ v (y, ) + O(2 ). Example (A 1D example). x˙ = x sin2 t. Solution. This has w = − y4 sin 2t and y˙ = 2 y. Compare the exact solution x(t) = t 1 t x0 e( 2 − 4 sin 2t) with the approximate one x = y + w = x0 e 2 (1 − 4 sin 2t).

4.4.1

Application of averaging to the periodically forced van der Pol equation   3  x   x ˙ = y − α − x  3 y ˙ = −x + β cos φ   φ˙ = ω

For α, β small, dynamics is close to rotation in x, y with frequency 1. So change to rotating co-ordinates u, v for α, β, ω − 1 small.      u cos φ − ω1 sin φ x = . v − sin φ − ω1 cos φ y   u˙ Now average with respect to φ to obtain to leading order in α: v˙ (
u˙ = α2 u − σv − u4 (u2 + v 2 )
v˙ = α2 σu + v − v4 (u2 + v 2 ) − αγ where σ =

1−ω 2 αω

and γ =

β 2αω

are assumed O(1).

4.5. DIFFERENCES FROM AUTONOMOUS SYSTEMS

43

We apply chapters 2 and 3 (and extensions) to obtain:

Entrainment to the forcing frequency in regions I, II and IVb, and for some initial conditions in IVa, whereas in III and for other initial conditions in IVa the averaged motion goes to a limit cycle, implying quasiperiodic oscillation. There is hysteresis on crossing the cusp via II or IVa. The time-dependent remainder terms which are ignored by averaging cause locking of quasiperiodic motion to a ratio of the rotation frequency and fattening of the curve OS of homoclinic bifurcations into a wedge of “chaos”.

4.5

Differences from autonomous systems

Given an autonomous vector field x˙ = v(x) with an attracting periodic orbit of period T , if we add equation φ˙ = 1 on R/2πZ we obtain an attracting invariant 2-torus for

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CHAPTER 4. INTRODUCTION TO DYNAMICS IN 3D

the full system.

Define the winding ratio of the flow on the torus to be # revolutions in x − y plane . t→∞ # revolutions in φ

w = lim

˙ ˙ Then w = 2π T and the motion is conjugate to θ = w, φ = 1. Now if we perturb the system to x˙ = v(x) + v1 (x, φ) it can be proved that there is still an attracting invariant 2-torus for small  and it has a winding ratio w() near w(0). But in general the motion is not conjugate to a constant vector field: the phenomenon of “frequency locking” occurs: w() will have intervals of  on which it is rational pq and the flow on the (flattened) torus typically looks like:

i.e. some attracting and some repelling periodic orbits. Hence the periodically forced van der Pol has frequency locking regions (depending on α). Given a saddle fixed point 0 for the time-T map f we can define W ± (0) = {x : d(f n (x), 0) → 0 as t → ∞} respectively. They can be proved to be smooth curves tangent to the stable and unstable eigenvectors respectively, just as in the case of a vector field. But unlike vector field vase, in discrete-time W + and W − can cross each other. The only constraint is the backward and forward images of any intersection give another intersection, because both W ± are invariant under f . Because det Df > 0 this obliges a second orbit of intersections in between. A situation like this is almost certain to occur when an autonomous family of vector fields with a homoclinic bifurcation is forced periodically.

4.5. DIFFERENCES FROM AUTONOMOUS SYSTEMS

45

We get these portraits.

What are the consequences of transverse homoclinic orbits? If you follow W ± far enough they are bound to intersect in yet another homoclinic orbit (typically two).

Obtain a box B bounded by pieces of W ± . Now consider its forward images. After a finite number N of iterations, f N (B) ∩ B consists of 2 strips 0 and 1. Let g = f N . Then we can prove that for all sequences a = (an )n∈Z ∈ {0, 1}. There exists a point xa ∈ B such that g n (xa ) ∈ an for all n ∈ Z, which is deterministic chaos, behaviour

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CHAPTER 4. INTRODUCTION TO DYNAMICS IN 3D

as random as a sequence of coin tosses.

References ◦ P.A. Glendinning, Stability, Instability and Chaos, CUP, 1994. Fairly fun to read and probably the “best” book for the course.

◦ P.G. Drazin, Nonlinear Systems, CUP, 1992. Again a fairly fun read but not as relevant to the course.

◦ Guckenheimer & Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer, 1983. Rather above the level of this course but probably worth a bit of bedtime reading if you’re interested.

◦ Clark Robinson, Dynamical Systems, CRC Press, 1994. Feeling brave? This one certainly isn’t bedtime reading.

Related courses There is a course on Dynamical Systems in Part 2B.

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