Bifurcation Theory

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Bifurcation Theory

Ale Jan Homburg The axiomatization and algebraization of mathematics has led to the illegibility of such a large number of mathematical texts that the threat of complete loss of contact with physics and the natural sciences has been realized. (Vladimir Arnold) Contents

1 Introduction 5 1.1 The Rosenzweig-MacArthur model...... 8 1.2 Exercises...... 12

2 Elements of nonlinear analysis 17 2.1 Calculus...... 17 2.1.1 Nemytskii operators...... 19 2.2 Implicit function theorem...... 21 2.2.1 Applications of the implicit function theorem...... 23 2.3 Lyapunov-Schmidt reduction...... 24 2.4 Floquet theory...... 26 2.5 Poincar´ereturn maps...... 30 2.6 Center manifolds...... 33 2.7 Normal forms...... 36 2.8 Manifolds and transversality...... 41 2.9 Sard’s theorem...... 42 2.10 Exercises...... 46

3 Local bifurcations 51 3.1 The saddle-node bifurcation...... 51 3.1.1 One dimensional saddle-node bifurcation...... 51 3.1.2 Higher dimensional saddle-node bifurcation...... 53 3.1.3 Saddle-node bifurcation of periodic orbits...... 54 3.2 The Hopf bifurcation...... 56 3.2.1 The planar Hopf bifurcation...... 56 3.2.2 Higher dimensional Hopf bifurcation...... 64 3.3 The period-doubling bifurcation...... 66 3.4 The Bogdanov-Takens bifurcation...... 69 3.5 Exercises...... 71

3 4

4 Nonlocal bifurcations 77 4.1 Homoclinic loop...... 78 4.2 Shil’nikov homoclinic loop...... 84 4.3 Blue sky catastrophe...... 85 4.4 Homoclinic tangencies...... 86 4.5 Exercises...... 94

5 Structural stability and bifurcation 99 5.1 Hyperbolicity and Peixoto’s theorem...... 101 5.2 The Hartman-Grobman theorem...... 108 5.3 The Smale horseshoe and the Arnold cat map...... 109 5.4 Stability theorems...... 113 5.5 Global aspects of bifurcations...... 117 5.6 Exercises...... 119 Chapter 1

Introduction

The qualitative or geometric study of dynamical systems originates with Henri Poincaréwho studied differential equations appearing in problems from celestial mechanics. The study of dynamical systems has been one of the most successful fields of mathematical research with an explosive development in the last fifty years. The research field has gradually developed into a more fragmented field. There is a theoretical core aimed at the description of typical dynamics, i.e., common properties of most dynamical systems, ignoring pathological systems. This part of the research follows a natural development where dynamics of increasing complexity is studied. The focus of the research here has shifted from geometrical techniques to stochastic techniques, always in combination with analysis. Such a combination of techniques has given a special flavor to the study of dynamical systems. On the other hand, with the development of a toolbox for the study of dynamics, the body of applied research using techniques from dynamical systems has expanded dramatically. Whether in physics, engineering, biology, psychology or economics, tools from dynamical systems have been highly successful in understanding phenomena where evolution is the focus. Dynamical systems ideas also appear frequently in other areas of mathematics; there is for instance a lively interaction between number theory and combinatorics with the part of dynamical systems theory called ergodic theory. This syllabus concerns the study of changes of dynamical properties, as the rules defining the dynamical system changes. This is the research area of bifurcation theory. It has been powerful in the study of typical dynamics, and is especially powerful in applications in other sciences. My aim was to compose a syllabus on bifurcation theory that serves as a first course on it, assuming standard bachelor courses on topology, analysis and ordinary differential equations. I wanted it to address techniques, albeit sometimes in simplified settings, in a detailed manner and further to give a broad overview of results, and demonstrate its usefulness in applications and the modern study of dynamical systems. I also tried to make visible the scientific heritage of the field; the collection of techniques and results that have been guiding its development. A course on bifurcation theory does not substitute a course on dynamical systems theory. It leaves out many topics, including topological dynamical systems, differentiable chaotic dynamical systems, and ergodic theory. Various aspects and

5 6 CHAPTER 1. INTRODUCTION topics are only skimmed or left out in this course. A first course such as this one leaves out advanced topics, which are interesting from a bifurcation theory point of view. This includes equivariant dynamical systems, Hamiltonian systems, dynamics on networks, infinite dimensional systems (partial differential equations, delay differential equations) and random dynamical systems (stochastic differential equations). The term bifurcation was originally used by Poincaréto describe the splitting of equilibria in a family of differential equations. In modern use, a bifurcation of a dynamical system is a qualitative change in its dynamics produced by varying parameters.

2 Figure 0.1: The bifurcation diagram for fµ(x) = µ − x . The dashed curve stands for sources, the solid curve for sinks.

Example. Consider the differential equationx ˙ = f(µ, x) with µ, x ∈ R, f(µ, x) = µ − x2. Equilibria occur if f(µ, x) = 0 and lie on the curve µ = x2. Thus there are no equilibria if µ < 0, one equilibrium √ √ at 0 if µ = 0 and two equilibria ± µ if µ > 0. If µ = 0, then Df0(0) = 0. If µ > 0, then Dfµ( µ) = √ √ √ √ √ −2 µ < 0 and Dfµ(− µ) = 2 µ > 0. This means that µ is a stable equilibrium (a sink), and − µ is unstable (a source). We obtain the qualitative picture of Figure 0.1, in which the branches of equilibria are shown in (µ, x) space. This figure is an example of a bifurcation diagram. More in general, consider a smooth differential equationx ˙ = f(µ, x) with

f(µ, x) = µ − x2 + O(|x|3 + |µx| + |µ|2) as x, µ → 0. Equilibria are solutions of the equation f(µ, x) = 0. Our objective is to compare this with the solution of the equation µ − x2 = 0 obtained by considering only the lowest order terms in the Taylor expansion. The equation f(µ, x) = 0 can be solved near (0, 0) for µ = µ∗(x), µ∗(0) = 0. One computes Dµ∗(0) = 0, D2µ∗(0) > 0. To ﬁnd the number of solutions on either side of the curve, we ﬁrst note that

2 Dxf(µ, x) = −2x + O(|x| + |µ|) 7

as x, µ → 0. In a neighborhood of (µ, x) = (0, 0), there is a unique function x∗(µ) near µ = 0, x∗(0) = 0, ∗ ∗ such that Dxf((µ, x (µ)) = 0 since Dxxf(0, 0) 6= 0. The point x (µ) corresponds to the maximum of f(µ, x). If α(µ) = f(µ, x∗(µ)), then α(µ) < 0 implies no solutions near (0, 0) and α(µ) > 0 implies two solutions near (0, 0). It is easy to verify that α(µ) = µ + O(|µ|2) as µ → 0.

3 Figure 0.2: The surface x − λ1x − λ2 = 0 in (λ1, λ2, x) space, with projection of the folds onto the cusp

in the (λ1, λ2) plane.

2 Example. As a slightly more complicated example, suppose x ∈ R, λ = (λ1, λ2) ∈ R and consider the equation

3 4 2 2 2 f(λ, x) = x − λ1x − λ2 + O(|x| + |λ1x | + |λ1x| + |λ2x| + |λ2| + |λ1λ2|) = 0 (0.1) in a small neighborhood of (λ2, λ2, x) = (0, 0, 0). As before, consider the polynomial equation

3 x − λ1x − λ2 = 0 (0.2)

and try to compare (0.1) with (0.2). For (0.2) it is easy to determine the bifurcation curves in (λ1, λ2)- space. In fact, these curves are exactly the curves across which the number of solutions changes. That is, multiple solutions occur on these curves. Any multiple solution of (0.2) must satisfy (0.2) as well as

2 3x − λ1 = 0. (0.3)

2 3 Equations (0.2) and (0.3) uniquely deﬁne λ1, λ2 as functions of x, namely, λ1 = 3x , λ2 = 2x . This is 3 2 a parametric representation of the cusp 4λ1 − 27λ2 = 0. The basic question is whether the solutions to (0.2) have the same qualitative properties as those to

(0.1). The procedure is the same as before. The bifurcation curves solve f(λ, x) = 0 and Dxf(λ, x) = 0.

Near λ = 0, x = 0, these equations uniquely determine λ1, λ2 as function of x. One checks λ1 = 2 3 3 4 3x + O(|x| ) and λ2 = 2x + O(|x| ) as x → 0. These are the parametric representation for a cusp with 3 2 essentially the same form as 4λ1 − 27λ2 = 0. One must show that the number of solutions changes by two as this cusp is crossed. This involves discussing Dxxf(λ, x) and Dλ2 f(λ, x) along this curve. 8 CHAPTER 1. INTRODUCTION

We are often interested in the dynamics after long times and less in transients. We will concentrate on attractors, and bifurcations thereof.

Definition 0.1. A compact invariant set A of a differential equation x˙ = f(x) with flow φt is an attractor if

(i) A is Lyapunov stable: for each open neighborhood U of A there is a neighborhood V ⊂ U of A so

that φt(x) ∈ U for each t ≥ 0, x ∈ V .

(ii) A is asymptotically stable: there is an open neighborhood U of A so that φt(x) → A (i.e. the

distance between φt(x) and A goes to zero) as t → ∞.

(iii) A is minimal: there is no strict and nontrivial subset of A satisfying the ﬁrst two properties.

Other deﬁnitions have been proposed to deal with compact invariant sets that attract many points but not necessarily an open neighborhood. One could also demand the existence of a point x so that

ω(x) = A. Here, the omega-limit set of a point x is the set of limit points of φt(x) as t → ∞: y ∈ ω(x) if there are tn → ∞, n → ∞, with limn→∞ φtn (x) = y. The basin of attraction of an attractor A is the set of all points x with φt(x) → A, t → ∞. In the following we consider examples of typical appearance of ideas and results from bifurcation theory.

1.1 The Rosenzweig-MacArthur model

The discussion here of predator-prey models and in particular the Hopf bifurcation in the Rosenzweig- MacArthur model is taken from the syllabus Modeling Population Dynamics by Andr´ede Roos, which is available from his website. Volterra studied the dynamics of the following predator-prey model,

F˙ = rF − aF C, (1.1) C˙ = εaF C − µC.

In these equations F and C represent the abundance of prey (food) and predators (consumers), respectively. The parameter r represents the exponential growth rate of prey in the absence of the predator, while µ represents the death or mortality rate of the predators in the absence of prey. Encounters between prey and predators are assumed to be proportional to the product of the abundances F and C. The parameter a represents the attack rate of predators. The parameter ε represents the conversion efficiency, i.e. the efficiency with which predators convert consumed prey into offspring. Notice that only positive parameter values are meaningful. We do not include the derivation of the following facts. The system (1.1) has a trivial steady state µ r (0, 0) and an internal steady state ( εa , a ). The function −µ ln(F ) + εaF − r ln(C) + aC is invariant under the flow and has a minimum at the internal steady state, which explains oscillations in predator and prey abundance around the internal steady state. See Figure 1.1. 1.1. THE ROSENZWEIG-MACARTHUR MODEL 9

Figure 1.1: Oscillatory behavior in the Lotka-Volterra model (1.1).

A slightly more complicated and realistic version of the Lotka-Volterra predator-prey model (1.1) assumes that prey do not grow indeﬁnitely in the absence of predators, but will ultimately reach a maximum prey abundance. Hence, prey do not grow exponentially, but follow, for example, a logistic growth equation, as in the following set of diﬀerential equations, F F˙ = rF (1 − ) − aF C, K (1.2) C˙ = εaF C − µC

The system has steady states at (0, 0) and (K, 0). Moreover, it turns out that this system has an µ r µ µ asymptotically stable internal steady state εa , a 1 − εaK for K > Kc = εa . For K > Ks = µ 1 1 q r εa 2 + 2 1 + µ , the linearization at the internal steady state has complex conjugate eigenvalues with negative real part: the internal steady state is a stable focus. See Figure 1.2. Both the basic Lotka-Volterra predator-prey model and its variant with logistic prey growth assume that the total predation of prey equals aF C. This assumption implies that the feeding rate of a single predator equals aF , the product of the attack rate and the current prey abundance. Hence, according to this formulation predators never get satiated as they will eat more and more the more prey there are around: for inﬁnitely large prey abundances the predator feeding rate will also become inﬁnite. The amount of prey eaten by a single predator per unit of time is referred to as the predators functional response. The basic Lotka-Volterra model assumes that the predator functional is a linear function of the prey abundance. This form is also known as a type I functional response:

ϕ1(F ) = aF.

For larger values of the prey abundance this functional response is not very realistic, as predators get quickly satiated when prey availability is very high. A formulation of the predators functional response which accounts for the satiation of predators at high food densities is due to Holling. This particular form of the functional response is referred to as Hollings type II functional response or simply a type II 10 CHAPTER 1. INTRODUCTION

Figure 1.2: A stable focus in the model (1.2). functional response. The most mechanistic mathematical formulation of the type II functional response is aF ϕ (F ) = 2 1 + ahF in which the parameter a again denotes the rate at which a single predator searches for (i.e. attacks) prey whenever it is not currently consuming a prey item and the parameter h is the average time span a predator uses to consume a prey it has caught. The predator-prey model that accounts for both logistic prey growth and a predator type II (or satiating) functional response can now be represented by the following system of ordinary diﬀerential equations, F ahF F˙ = rF (1 − ) − C, K 1 + ahF (1.3) ahF C˙ = ε C − µC, 1 + ahF This model is also referred to as the Rosenzweig-MacArthur model after two authors that have investi- gated the dynamics of this model in great detail. The model analysis can be summarized as:

(i) The model possesses a trivial steady state

(F1,C1) = (0, 0)

which is always a saddle point.

(ii) The model possesses a prey-only steady state

(F2,C2) = (K, 0)

µ which is stable as long as K < Kc = a(ε−µh) . For larger values of K the prey-only steady state is a saddle point. 1.1. THE ROSENZWEIG-MACARTHUR MODEL 11

Figure 1.3: The bifurcation diagram for the Rosenzweig-MacArthur model.

(iii) The model possesses a internal steady state (i.e. with positive abundances of both prey and predator) µ εr(aK(ε − µh) − µ) (F ,C ) = , . 3 3 a(ε − µh) a2(ε − µh)2K

when K > Kc. This steady state becomes unstable and limit cycles occur when the carrying ε+µh capacity K exceeds the value Ks = ah(ε−µh) . Figure (1.4) illustrates the series of changes in dynamics that occur for increasing values of the carrying capacity K.

(i) For K < Kc ( top-left panel) the internal steady state is biologically irrelevant, as the predator abundance in steady state would be negative. The prey carrying capacity is below the abundance needed by the predator to sustain itself.

(ii) For Kc < K < Ks (top-right panel) and K only slightly larger than Kc the internal steady state is a stable node. From every initial condition it is approached smoothly, i.e. in a non-oscillatory fashion.

(iii) For larger K, but still K < Ks (bottom-left panel) the internal steady state turns into a stable spiral and is approached from any initial condition in an oscillatory manner.

(iv) For K > Ks (bottom-right panel) the internal steady state has become an unstable spiral. Starting from an initial condition close to the steady state, the trajectory spirals away from the steady state and approaches a stable limit cycle surrounding the steady state. This stable limit cycle arises when the steady state changes from a stable into an unstable spiral.

Acknowledgements In composing this syllabus, I made liberal use of existing texts on dynamics and bifurcations theory. Liberal is tantamount to theft. 12 CHAPTER 1. INTRODUCTION

Figure 1.4: Solution curves in the phase plane of the Rosenzweig-MacArthur predator-prey model. Two isoclines where F˙ = 0 (dashed line) and C˙ = 0 (dotted line) are drawn. The isoclines intersect in the internal steady state.

1.2 Exercises

Exercise 1.2.1. Investigate and sketch the bifurcation diagram of equilibria for the system x˙ = fµ(x), as µ varies near 0, with

3 (i) fµ(x) = µx − x ,

2 (ii) fµ(x) = µx − x ,

2 3 (iii) fµ(x) = µ x − x ,

2 3 (iv) fµ(x) = µ x + x ,

2 3 5 (v) fµ(x) = µ αx + 2µx − x , for various α.

Exercise 1.2.2. Compute the bifurcation diagram of equilibria in

x˙ = y, y˙ = µx − x3 − y.

Find a symmetry of the phase diagram and relate the bifurcation diagram to the symmetry.

Exercise 1.2.3. Consider the equation given in (0.1),

3 4 2 2 2 f(λ, x) = x − λ1x − λ2 + O(|x| + |λ1x | + |λ1x| + |λ2x| + |λ2| ) = 0. 1.2. EXERCISES 13

Solutions where both f(λ, x) = 0 and Dxf(λ, x) = 0 occur for λ along a cusp. Show that the number of solutions of (0.1) changes by two as this cusp is crossed.

Exercise 1.2.4. A Josephson junction consists of two closely spaced superconductors separated by a weak connection. Brian Josephson suggested that it should be possible for a current to pass between the two superconductors, by a quantum mechanical tunneling eﬀect. A simpliﬁed model is

φ¨ + αφ˙ + sin(φ) = I on the circle R/2πZ, with real parameters α, I. As a system,

φ˙ = y, (2.1) y˙ = I − sin(φ) − αy. on the cylinder R/2πZ × R.

(i) Analyse existence and stability of equilibria.

(ii) Prove the existence of a periodic solution for I > 1 by following these steps:

(a) Find an invariant strip S = R/2πZ × (y1, y2) with 0 < y1 < y2.

(b) Consider the ﬁrst return map P on Σ = {0} × (y1, y2); P (y0) = (y1) where (0, y1) is the ﬁrst

point on Σ from the positive orbit with initial condition (0, y0). Prove that P is the time 2π map of dy I − sin(φ) − αy = . (2.2) dφ y (c) Apply the intermediate value theorem to ﬁnd a ﬁxed point of P and thus a periodic solution of (2.1).

(iii) Prove uniqueness of the periodic solution by following these steps:

1 2 (a) Consider the function E = 2 y − cos(φ) along solutions of (2.2). Show that

Z 2π dE dφ = 0 0 dφ along periodic solutions.

dE (b) Prove dφ = I − αy. (c) Combine the two previous steps to show

Z 2π y(φ) dφ = 2πI/α 0 along periodic solutions. (d) Derive a contradiction from the assumption that two periodic solutions exist. 14 CHAPTER 1. INTRODUCTION

Exercise 1.2.5. Consider the Lorenz system

x˙ = −10x + 10y, y˙ = ρx − y − xz, 8 z˙ = − z + xy. 3 (i) The origin is a ﬁxed point. Find the two other ﬁxed points p±.

(ii) Compute the eigenvalues at the origin.

(iii) Show that the characteristic polynomial at p± is given by 41 8 160 P (λ) = λ3 + λ2 + (10 + ρ)λ + (ρ − 1). 3 3 3 (iv) Show that P (λ) has a negative real root.

(v) For ρ ≥ 14, show that P (λ) is monotonically increasing. So it has one real root λ0 and complex conjugate roots α ± i β with β 6= 0.

41 41 (vi) Find a parameter value ρ1 for which α = 0. Hint: Note that λ0 + 2α = − 3 , where 3 is the 2 41 coeﬃcient of λ . So, ﬁnd ρ = ρ1 such that λ0 = − 3 is a real root.

41 41 (vii) Show that λ0 > − 3 for ρ < ρ1 and λ0 < − 3 for ρ > ρ1.

(viii) Show that the complex eigenvalue α + i β crosses the imaginary axis as ρ increases through ρ1.

Exercise 1.2.6. The problem of the buckling of a rod was already studied by Leonhard Euler. In the problem of the buckling of an initially straight rod subjected to an axial load λ, the following boundary value problem arises: θ00 + λ sin(θ) = 0, for 0 < x < 1, with boundary conditions

θ0(0) = θ0(1) = 0.

Here, θ(x) is the angle that the tangent to the rod makes with the horizontal at the spatial position x. The variable θ is related to the vertical displacement w(x) of the rod through the relation w0 = sin(θ) with w(0) = w(1) = 0.

(i) Consider ﬁrst the linearized equation v00 + λv = 0, v0(0) = v0(1) = 0. Find its nontrivial solutions.

(ii) For the nonlinear boundary value problem, solutions (θ(x), p(x)) = (θ(x), θ0(x)) satisfy

G(λ, θ(x), p(x)) = G(λ, θ(0), p(0))

for 0 ≤ x ≤ 1, where 1 G(λ, θ, p) = p2 + λ(1 − cos(θ)). 2 Sketch the level sets of G in the (θ, p)-plane. 1.2. EXERCISES 15

(iii) Solutions must satisfy

G(λ, θ(1), 0) = G(λ, θ0, 0)

if θ0 = θ(0). Denote by s0 the minimum positive value, depending on (λ, θ0), so that the solution (θ(s), p(s)) of θ˙ = p, (2.3) p˙ = −λ sin(θ),

with (θ(0), p(0)) = (−θ0, 0), satisﬁes (θ(s0), p(s0)) = (θ0, 0). Prove that

2 Z π/2 1 s0 = √ du. p 2 2 λ 0 1 − sin(θ0/2) sin(u)

Hint: Use invariance of the function G to derive a ﬁrst order diﬀerential equation for θ(s), integrate and change variables to bring the integral to the desired form.

(iv) Write Z π/2 1 γ(θ0) = 2 du, p 2 2 0 1 − sin(θ0/2) sin(u) √ so that s0 = γ(θ0)/ λ. Prove that (λ, θ0) corresponds to a nontrivial solution of the nonlinear boundary value problem if and only if there is an integer m ≥ 1 such that

2 2 λ = m γ (θ0).

Sketch the resulting curves, for varying m, in the (λ, θ0)-plane. 16 CHAPTER 1. INTRODUCTION Chapter 2

Elements of nonlinear analysis

In this chapter we present results and techniques that are used in bifurcation analysis. This includes the implicit function theorem and its derived method of the Lyapunov-Schmidt reduction. We also discuss Sard’s theorem which is an essential tool in transversality theory.

2.1 Calculus

We collect some results from calculus on Banach spaces, mostly without proof. A detailed account can be found e.g. in [30]. Let X and Y be Banach spaces. We let L(X,Y ) be the Banach space of bounded linear operators from X to Y with the operator topology.

Let U be an open subset of X. A function f : U → Y is said to be Fr´echet diﬀerentiable at x0 if there is a bounded linear operator Df(x0): X → Y such that

|f(x0 + h) − f(x0) − Df(x0)h| = o(|h|) as |h| → 0. The operator Df(x0) is called the Fréchet derivative of f at x0. The Fréchet derivative has properties very similar to the derivative in finite dimensions. For example, if f : X → Y , g : Y → Z are Fréchet differentiable, then the chain rule holds for g ◦ f:

Dg ◦ f(x) = Dg(f(x))Df(x).

Qn If f : X → Y , Y = i=1 Yi, f = (f1, . . . , fn) with each fi being Fréchet differentiable, then f is Fréchet QN differentiable and Df(x) = (Df1(x), . . . , Dfn(x)). If X = i=1 Xi, f : X → Y , f(x) = f(x1, . . . , xN ), xi ∈ Xi, then one can define partial Fréchet derivatives Dif(x) ∈ L(Xi,Y ) with respect to xi in the usual way:

|f(x1, . . . , xi + h, . . . , xN ) − f(x1, . . . , xi, . . . , xN ) − Dfi(x)h| = o(|h|)

17 18 CHAPTER 2. ELEMENTS OF NONLINEAR ANALYSIS

as |h| → 0. If Df(x),Dif(x) exist and h = (h1, . . . , hN ) ∈ X with hi ∈ Xi, then

N X Df(x)h = Dif(x)hi. i=1 Also the following mean value theorem is valid.

Theorem 1.1. If f :[a, b] → X is Fréchetdifferentiable and there is a continuously differentiable function α :[a, b] → R such that |Df(t)| ≤ α0(t) for t ∈ [a, b], then

|f(b) − f(a)| ≤ α(b) − α(a), |f(b) − f(a)| ≤ sup |Df(t)||b − a|. t∈[a,b]

To deﬁne higher order derivatives, we need the concept of multilinear operator. An operator f : Qn X → Y is called multilinear if X = j=1 Xj and for each x = (x1, . . . , xn) ∈ X and each integer k = 1, 2, . . . , n, f(x1, . . . , xn) is linear in xk for all other variables ﬁxed. Qn Lemma 1.1. Suppose X = i=1 Xi and Y are Banach spaces. If f : X → Y is multilinear, then the following statements are equivalent:

(i) f is continuous at each point x ∈ X,

(ii) f is continuous at x = 0,

(iii) f takes bounded sets to bounded sets and there is a constant K > 0 such that

n Y |f(x1, . . . , xn)| ≤ K |xi|Xi , i=1

(iv) f is Fr´echetdiﬀerentiable.

The norm |f| of a multilinear map f from X to Y is the inﬁmum of those K satisfying (iii). The Qn set of bounded multilinear maps from X = j=1 Xj to Y will be denoted by L(X1,...,Xn,Y ). This is a Banach space with the usual rules for addition and scalar multiplication and the above norm. If n X1 = ··· = Xn = X, we let L(X,...,X,Y ) = L (X,Y ).

Lemma 1.2. There is an isomorphism between the Banach spaces L(X1,L(X2,L(X3,...,L(Xn,Y )))) and L(X1,...,Xn,Y ).

n A multilinear form f ∈ L (X,Y ) is symmetric if f(x1, . . . , xn) is invariant under any permutation σ(1, 2, . . . , n) of the integers 1, 2, . . . , n giving the indices. If U is an open set in X and f : U → Y has a 2 Fréchet derivative Df(x) for x ∈ U, then we say that f has a second Fréchet derivative D f(x0) at x0 if 2 2 Df(·): U → L(X,Y ) has a Fréchet derivative at x0. In this case, D f(x0) ∈ L(X,L(X,Y )) = L (X,Y ) 2 2 by Lemma 1.2. It is possible to show that D f(0) is a symmetric bilinear form; that is D f(x0)(h1, h2) = 2 D f(x0)(h2, h1) for all (h1, h2) ∈ X ×X. By induction one can define the Fréchet derivative of all orders N N with the Nth derivative D f(x0) ∈ L (X,Y ) and symmetric. 2.1. CALCULUS 19

If X and Y are Banach spaces and U is an open subset of X, the space CN (U, Y ), N ≥ 0, is the space of functions f : U → Y such that the jth derivative Djf(x) exists for each x ∈ U, 0 ≤ j ≤ N, and the mapping x 7→ Djf(x) of U into Lj(X,Y ) is continuous for each x ∈ U. If f ∈ CN (U, Y ) for all integers N ≥ 0, we say f ∈ C∞(U, Y ). Write xk for the k tuple (x, . . . , x). We now state Taylor’s theorem.

Theorem 1.2. If f ∈ Cn(U, Y ), then

1 f(x + h) = f(x) + Df(x)h + ... + Dn−1f(x)hn−1 + R (x, h) (1.1) (n − 1)! n

for x ∈ U, x + sh ∈ U, 0 ≤ s ≤ 1, where

Z 1 1 n−1 n n Rn(x, h) = (1 − s) D f(x + sh)h ds. (1.2) (n − 1)! 0 Also, 1 f(x + h) = f(x) + Df(x)h + ... + Dnf(x)hn + g(x, h), (1.3) n! where g(x, h) = o(|h|n) as |h| → 0.

Borel’s theorem states that any formal Taylor series occurs as the Taylor series of a smooth function. Note the contrast with real analytic functions, where convergence is required.

∞ Theorem 1.3. Given any sequence of real numbers cα, for multi-indices α, there exists a C function f : Rn → R with α D f(0) = |α|!cα.

A more general extension theorem is the Whitney extension theorem. It states that for a closed set A, if we are given a function f and candidates for its derivatives on A, then we can extend f to all of Rn so that these candidates are indeed the derivatives of f (so long as they satisfy some Taylor polynomial-like Pn conditions). For a multi-index α = (α1, . . . , αn), we denote |α| = i=1 αi.

Theorem 1.4. Given a nonnegative integer k, suppose {fα}|α|

k−|α| for each x, y ∈ A and each multi-index α with |α| ≤ k. Here Rα is uniformly o(|x−y| ) as |x−y| → 0 n k α for each α. Then there is a function g : R → R of class C so that g = f0 on A and D g = fα on A.

2.1.1 Nemytskii operators

We discuss diﬀerentiability of Nemytskii operators. 20 CHAPTER 2. ELEMENTS OF NONLINEAR ANALYSIS

Example. Let X = C0([0, 1], C) and deﬁne f : X → X by f(x)(t) = sin(x(t)), 0 ≤ t ≤ 1. One can show that f is smooth. However, with X = L2([0, 1], C), then f is not diﬀerentiable.

The function f is called a Nemytskii operator. Given a function, in the example x 7→ sin(x), an induced operator on a function space is deﬁned.

Theorem 1.5. Let f : Rn → Rm be C1. Deﬁne

0 n 0 m N : C ([0, 1], R ) → C ([0, 1], R ) by N(φ)(t) = f(φ(t)). Then N is C1.

Proof. For φ ∈ C0([0, 1], Rn), ψ ∈ C0([0, 1], Rn), deﬁne B(φ)ψ in C0([0, 1], Rm) by

B(φ)ψ(t) = Df(φ(t))ψ(t).

One checks that B is a bounded linear map. We claim that DN(φ) = B(φ). Namely, compute for t ∈ [0, 1],

(N(φ + ψ) − N(φ) − B(φ)ψ)(t) = f(φ(t) + ψ(t)) − f(φ(t)) − Df(φ(t))ψ(t) Z 1 = (Df(φ(t) + sψ(t)) − Df(φ(t)))ψ(t) ds. 0 Since φ([0, 1]) is compact, Df is uniformly continuous on φ[0, 1]. For ε > 0, there exists δ > 0 such that if x1 ∈ φ([0, 1]) and |x2 − x1| < δ, then kDf(x2) − Df(x1)k < ε. Therefore,

Z 1

|(N(φ + ψ) − N(φ) − B(φ)ψ)(t)| = (Df(φ(t) + sψ(t)) − Df(φ(t)))ψ(t) ds 0 ≤ ε|ψ(t)| ≤ ε|ψ|

Taking the supremum over t ∈ [0, 1] we get |N(φ + ψ) − N(φ) − B(φ)ψ| ≤ ε|ψ|. Since ε is arbitrary, this proves that DN(φ) = B(φ). Finally we establish that B : C0([0, 1], Rn) → L(C0([0, 1], Rn), C0([0, 1], Rm)) is continuous. Let 0 n φ1 ∈ C ([0, 1], R ). We will show that B is continuous at φ1. Note that

(B(φ2) − B(φ1))ψ(t) = (Df(φ2(t)) − Df(φ1(t)))ψ(t).

Let E be a compact neighborhood of φ1([0, 1]). Let ε > 0. As above, there is δ > 0 such that if x2, x1 ∈ E, |x2 − x1| < δ, then kDf(x2) − Df(x1)k < ε. Then, if |φ2 − φ1| < δ,

|(B(φ2) − B(φ1))ψ(t)| ≤ kDf(φ2(t)) − Df(φ1(t))k|ψ(t)| ≤ ε|ψ|.

The same argument works for the Nemytskii operator on the space B(N, Rn) of bounded sequences in Rn.

Theorem 1.6. Let f : Rn → Rm be C1. Deﬁne

n m N : B(N, R ) → B(N, R ) by N(φ)(t) = f(φ(t)). Then N is C1.

2.2 Implicit function theorem

Recall that a continuous map T : S → S on a metric space is a contraction if, for some 0 < λ < 1,

d(T (x),T (y)) ≤ λd(x, y) for all x, y ∈ S. We state the Banach contraction theorem.

Theorem 2.1. Let (S, d) be a complete metric space and T : S → S a contraction. Then there is a n unique ﬁxed point x¯ in S. Moreover, for any x0 ∈ S, T (x0) → x¯ as n → ∞.

Proof. If x = T (x) and y = T (y) are ﬁxed points, then d(x, y) = d(T (x),T (y)) ≤ λd(x, y) shows that x = y: the ﬁxed point is unique if it exists.

Take any x0 ∈ S. Then, for n ≥ 1

n+1 n n n−1 n d(T (x0),T (x0)) ≤ λd(T (x0),T (x0)) ≤ · · · ≤ λ d(T (x0), x0).

So, for m > n > 0,

m n m m−1 n+1 n d(T (x0),T (x0)) ≤ d(T (x0),T (x0)) + ··· + d(T (x0),T (x0)) ≤ λn (λm−1 + ··· + λn)d(T (x ), x ) ≤ d(T (x ), x ). 0 0 1 − λ 0 0

n The sequence {T (x0)} is therefore a Cauchy sequence. Since S is complete, it converges to a point x¯ ∈ S as n → ∞. Also, m+1 m d(T (¯x), x¯) = lim d(T (x0),T (x0)) = 0, n→∞ so that T (¯x) =x ¯.

To treat dependence of the ﬁxed points on parameters, consider a metric space (S, d), a set A, and a map T : S × A → S. We say that T is a uniform contraction if there is 0 < λ < 1 with

d(T (x, a),T (y, a) ≤ λd(x, y) for all x, y ∈ S, a ∈ A. The next theorem is the uniform contraction theorem. 22 CHAPTER 2. ELEMENTS OF NONLINEAR ANALYSIS

Theorem 2.2. Let U, V be open sets in Banach spaces X,Y , let T : U¯ ×V → U¯ be a uniform contraction on U¯. Let g(y) be the ﬁxed point of T (·, y) in U¯. If T ∈ Ck(U¯ × V ),X), 0 ≤ k < ∞, then g ∈ Ck(V,X).

Proof. If k = 0, then

Suppose k = 1. Since |T (x1, y) − T (x2, y)| ≤ λ|x1 − x2| for all x1, x2 ∈ U, y ∈ V it follows that

|DxT (x, y)| ≤ λ < 1 for (x, y) ∈ U × V . Formally diﬀerentiating g(y) = T (g(y), y), the operator Dg(y) should be a solution of the operator equation in M

M − DxT (g(y), y)M = DyT (g(y), y). (2.1)

Since DxT (g(y), y)| ≤ λ < 1, this equation has a unique solution M(y). We must show that

|g(y + η) − g(y) − M(y)η| = o(|η|) as |η| → 0. If γ = γ(η) = g(y + η) − g(y) then

γ = T (g(y) + γ, y + η) − T (g(y), y) (2.2) = DxT (g(y), y)γ + DyT (g(y), y)η + ∆ with

∆ = T (g(y) + γ, y + η) − T (g(y), y) − DxT (g(y), y)γ − DyT (g(y), y)η.

Since T is C1, for any 0 < ε < 1 − λ, there is δ > 0 such that |∆(γ, η)| < ε(|γ| + |η|) if |γ|, |η| < δ. Since γ = γ(η) is continuous in η, γ(η) → 0 as η → 0, we may further restrict δ so that |∆(γ(η), η)| < ε(|γ(η)| + |η|) if |η| < δ. From (2.2), 1 |γ(η)| ≤ (|D T (g(y), y)| + ε)|η| =: k|η| 1 − λ − ε y if |η| < δ. Therefore, |γ(η) − M(y)η| < ε(1 + k)(1 − λ)−1|η| for any 0 < ε < 1 − Λ, |η| < δ. This implies that Dg(y) exists and is equal to M(y). Suppose the result holds for k − 1 and k > 1. Thus, if T is Ck, then g is Ck−1 at least and the fact that Dg(y) satisﬁes (2.1) implies that g is Ck.

The implicit function theorem is a corollary of the uniform contraction theorem.

Theorem 2.3. Suppose X,Y,Z are Banach spaces, U ⊂ X,V ⊂ Y open and F : U × V → Z a Ck map, k ≥ 1. Assume F (x0, y0) = 0 for some (x0, y0) ∈ U × V and DxF (x0, y0) has bounded inverse. Then k there is a neighborhood U1 × V1 ⊂ U × V of (x0, y0) and a C function f : V1 → U1, f(y0) = x0 so that

F (x, y) = 0 for (x, y) ∈ U1 × V1 if and only if x = f(y). 2.2. IMPLICIT FUNCTION THEOREM 23

−1 Proof. If L = (DxF (x0, y0)) , G(x, y) = x − LF (x, y), then the ﬁxed points of G are solutions of

F = 0. The function G has the same smoothness properties as F , G(x0, y0) = x0, DxG(x0, y0) = 0.

Therefore, |DxG(x, y)| ≤ θ < 1 for all x, y in a neighborhood U1 × V1 of (x0, y0). Is is easy to choose

this neighborhood so that G : U1 × V1 → U1. The result now follows from the uniform contraction theorem.

2.2.1 Applications of the implicit function theorem

The following example discusses ﬁnding equilibria of diﬀerential equations using the implicit function theorem.

n m Example. Consider a family of diﬀerential equationsx ˙ = fµ(x), x ∈ R , µ ∈ R . Let x0 be an

equilibrium for µ = µ0. Suppose Dfµ0 (x0) has no eigenvalue 0. Then there exist a neighborhood U of

x0 and ε > 0 so that for any µ with |µ − µ0| < ε, fµ has a unique equilibrium in U. This statement is

proved by the implicit fuction theorem. Indeed, the equation fµ(x) = 0 can be solved near (µ0, x0) for

x =µ ¯ withx ¯(µ0) = x0 since Dfµ0 (x0) is invertible. An application of the implicit function theorem on Banach spaces yields the following similar state- n ment. Let x0 be an equilibrium of a diﬀerential equationx ˙ = f(x) on R . Suppose Df(x0) has no n n eigenvalue 0. Then there exist a neighborhood U of x0 and ε > 0 so that any g : R → R with

sup n |g(x) − f(x)|, |Dg(x) − Df(x)| < ε has a unique equilibrium in U. To obtain this statement, ∈R consider the equation F (g, x) = g(x) = 0 for F : Rn × C1(Rn, Rn) → Rn and note that F is continuously diﬀerentiable and DxF (f, x0) = Df(x0) is invertible. The implicit function can thus be applied to yield

the statement, as above for the family fµ.

We use the implicit function theorem to prove the existence and uniqueness theorem for ordinary diﬀerential equations. We include parameter dependence. n Suppose Λ is a Banach space, Ω ⊂ R × R , G ⊂ Λ are open sets, f :Ω × G is continuous, Dxf(t, x, λ) is continuous for (t, x) ∈ Ω, λ ∈ Λ, and consider the initial value problem

x˙ = f(t, x, λ), (2.3) x(σ) = ξ,

where (σ, ξ) ∈ Ω.

Theorem 2.4. Under the above hypotheses, there is a solution x(σ, ξ, λ)(t) of (2.3) which is unique and continuous in (σ, ξ, λ, t) in its domain of deﬁnition. If f ∈ Ck(Ω × G, Rn), then x(σ, ξ, λ)(t) is Ck in (σ, ξ, λ, t) in its domain of deﬁnition.

Proof. If x is a solution of (2.3) on [σ − α, σ + α], t − σ = ατ, z(τ) = x(ατ + σ) − ξ, then z must satisfy

dz(τ) − αf(ατ + σ, ξ + z, λ) = 0, dτ z(0) = 0, 24 CHAPTER 2. ELEMENTS OF NONLINEAR ANALYSIS and conversely. Let X = {φ ∈ C1([−1, 1], Rn); φ(0) = 0}, Y = C0([−1, 1], Rn) and let F : R × R × Λ × X → Y be given by dφ(τ) F (α, σ, ξ, λ, φ)(τ) = − αf(ατ + σ, ξ + φ, λ), dτ τ ∈ [−1, 1]. Then F (0, σ0, ξ0, λ0, 0) = 0 and DφF (0, σ0, ξ0, λ0, 0)ψ = dψ/dτ for any (σ0, ξ0, λ0). If R τ ψ ∈ X, dψ/dτ = y ∈ Y , then ψ(τ) = 0 y(s)ds and ψ is continuous and linear in y. The implicit function theorem implies the existence of a unique solution z∗(α, σ, ξ, λ, τ) of F (α, σ, ξ, λ, τ) = 0 in a neighborhood of (0, σ0, ξ0, λ0, 0) and this function has the same smoothness properties as f. This proves the theorem locally. The global theorem is obtained by the process of continuation.

2.3 Lyapunov-Schmidt reduction

Let us start with a motivating example.

Example. Consider the equation y2 − λ − f(λ, y, z) = 0, (3.1) Bz − g(λ, y, z) = 0, where λ ∈ R, y ∈ R, z ∈ Rn−1, B a nonsingular matrix, f = O(|y|3 + |yz| + |z|2), g = O(|y|2 + |z|2) ∗ as |y|, |z| → 0, |λ| < λ0. For λ, y, z small, the second equation has a unique solution z = z (λ, y), z∗(λ, y) = O(|y|2) as y → 0. Putting this into the ﬁrst equation gives

y2 − λ − f(λ, y, z∗(λ, y)) = 0

∗ 3 and f(λ, y, z (λ, y)) = O(|y| ) as |y| → 0, |λ| < λ0. This equation has the form of the equation looked at in the introduction and, thus, λ arises as a bifurcation point.

In spite of the simplicity of this example, its method can be systematically generalized to equations with multidimensional state space coordinates and parameters. For simplicity we develop the theory for equations on Euclidean spaces only, but it applies likewise to equations on Banach spaces. Let A : Rn → Rn be a linear map and let N : Rn → Rn be a map with N(0) = 0 and DN(0) = 0. The problem is to determine the solutions of the equation

Ax − N(x) = 0 (3.2) for x ∈ X. The null space of A is denoted by N (A), the range by R(A). Let U : Rn → N (A) be a projection to the kernel of A. Let E : Rn → R(A) be a projection to the range of A. Lemma 3.1. There is a linear map K : R(A) → R(A), called the right inverse of A, such that AK = I on R(A), KA = I − U on Rn and (3.2) is equivalent to the equation z − KEN(y + z) = 0, (3.3) (I − E)N(y + z) = 0, (3.4) where y ∈ N (A), z ∈ (I − U)Rn and x = y + z. 2.3. LYAPUNOV-SCHMIDT REDUCTION 25

Proof. The map A is one-to-one from (I − U)Rn onto R(A). Thus, the existence of a right inverse is clear. Equation (3.2) is equivalent to

E(A − N)(y + z) = 0, (I − E)(A − N)(y + z) = 0.

The second equation gives (I − E)N(y + z) = (I − E)A(y + z). Since (I − E)A = 0, this yields (3.4). The ﬁrst equation gives EA(y + z) = EN(y + z). As EA = A and KA(y + z) = (I − U)(y + z) = z, this yields (3.3).

If it is possible to ﬁnd a solution z∗(y) for (3.3), then (3.2) is equivalent to

(I − E)N(y + z∗(y)) = 0.

In bifurcation theory this reduction method often arises in the following context. For parameters λ ∈ Λ, assume that for N : Rn × Λ → Rn we have

N(0, 0) = 0,DxN(0, 0) = 0.

Consider the equation

Ax − N(x, λ) = 0 (3.5) for (x, λ) near (0, 0). This equation is equivalent to

z − KEN(y + z, λ) = 0, (3.6) (I − E)N(y + z, λ) = 0 (3.7) for x = y + z with y ∈ N (A), z ∈ (I − U)Rn. Apply the implicit function theorem to (3.6). This provides a neighborhood V ⊂ N (A) × Λ of (0, 0) and a function z∗ : V → (I − U)Rn, z∗(0, 0) = 0, such that z∗(y, λ) satisﬁes (3.6) and is the only solution in a neighborhood of 0. Thus (3.5) has a solution near (x, λ) = (0, 0) if and only if (y, λ) ∈ V satisfy the reduced bifurcation equations

(I − E)N(y + z∗(y, λ), λ) = 0.

This method to obtain reduced bifurcation equations is called Lyapunov-Schmidt reduction.

Example. Let f : R2 × R → R2 be given as

f(x, y, λ) = (λ2 + x − x2 + y2, λ + x2 − xy).

The function f(·, λ) can be viewed as a one parameter family of vector ﬁelds on R2, generating the diﬀerential equations 2 2 2 x˙ = f1(x, y, λ) = λ + x − x + y , (3.8) 2 y˙ = f2(x, y, λ) = λ + x − xy. 26 CHAPTER 2. ELEMENTS OF NONLINEAR ANALYSIS

Goal is to ﬁnd the equilibria that occur for λ close to 0. Observe that f(0, 0, 0) = (0, 0), that Dx,yf(0, 0, 0) is not invertible and has one dimensional kernel spanned by (0, 1) and one dimensional range spanned by (1, 0). Note that Dxf1(0, 0, 0) is invertible. By the implicit function theorem, there is an open neighborhood U of (0, 0) in R2, an open neighborhood V of 0 in R, and a map X = X(y, λ) from U to V so that X(y, λ) is the unique solution in V of the equation

f1(X(y, λ), y, λ) = 0.

Now (X(y, λ), y, λ) is an equilibrium of f if and only if g(y, λ) = 0, where

g(y, λ) = f2(X(y, λ), y, λ).

A calculation shows that

Dλg(0, 0) = Dλf2(0, 0, 0) = 1. By the implicit function theorem, there exists a unique branch of solutions Λ = Λ(y) near y = 0 so that g(y, Λ(y)) = 0. One can compute the second order Taylor expansion at 0 of the functions (y, λ) 7→ X(y, λ), y 7→ Λ(y) and y 7→ X(y, Λ(y)). The Taylor expansions can be used to determine the stability of the equilibrium solutions (X(y, Λ(y)), y).

2.4 Floquet theory

Consider

x˙ = f(x) (4.1) in Rn. Let γ be a periodic solution of (4.1) of minimal period T ; γ(t) = γ(t + T ) with T > 0 minimal with this property. Linearizing the diﬀerential equation about γ we obtain the nonautonomous equation

ξ˙ = Df(γ(t))ξ, (4.2) called the ﬁrst variation equation. This leads us to the study of time periodic linear systems, the topic of Floquet theory. Consider the diﬀerential equation

u˙(t) = A(t)u(t) (4.3) with A(t) = A(t + T ) a periodic n × n matrix.

Theorem 4.1. The solutions of the system (4.3) form an n-dimensional vector space. Moreover, there exists a matrix-valued solution Π(t, t0) such that the solution satisfying the initial condition x(t0) = x0 is given by Π(t, t0)x0

Proof. This follows from the existence and uniqueness theorem for diﬀerential equations and linearity of (4.3). 2.4. FLOQUET THEORY 27

The principal matrix solution Π(t, t0) is a solution of

Π(˙ t, t0) = A(t)Π(t, t0), (4.4) Π(t0, t0) = I.

Lemma 4.1. Suppose A(t) is periodic with period T . Then the principal matrix solution satisﬁes

Π(t + T, t0 + T ) = Π(t, t0).

d Proof. By dt Π(t + T, t0 + T ) = A(t + T )Π(t + T, t0 + T ) = A(t)Π(t + T, t0 + T ) and Π(t0 + T, t0 + T ) = I we see that Π(t + T, t0 + T ) solves (4.4). Hence it is equal to Π(t, t0) by uniqueness.

We recall Liouville’s theorem.

Lemma 4.2. R t tr A(s) ds det Π(t, t0) = e t0

d Proof. From dt Π(t, t0) = A(t)Π(t, t0). A direct calculation shows d det Π(t, t ) = tr(A(t)) det Π(t, t ). (4.5) dt 0 0 We drive this equation only for the two dimensional case, to illustrate the argument. Write ! a(t) b(t) A(t) = . c(t) d(t)

Write also ! α(t) β(t) Π(t, t0) = . γ(t) δ(t)

Then (suppressing the argument t in the coeﬃcients of Π(t, t0) and A(t)), d d det Π(t, t ) = (αδ − βγ) dt 0 dt = α0δ + αδ0 − β0γ − βγ0 = (aα + bγ)δ + α(cβ + dδ) − (aβ + bδ)γ − β(cα + dγ) = (a + d)(αδ − βγ)

= tr(A(t)) det Π(t, t0).

The lemma follows by solving (4.5) with initial condition det Π(t0, t0) = 1.

We state Floquet’s theorem.

Theorem 4.2. Suppose A(t) is periodic of period T . Then the principal matrix solution has the form

(t−t0)Q(t0) Π(t, t0) = P (t, t0)e ,

where P (·, t0) is T -periodic and P (t0, t0) = I. 28 CHAPTER 2. ELEMENTS OF NONLINEAR ANALYSIS

Proof. Deﬁne the monodromy matrix

M(t0) = Π(t0 + T, t0)

as the principal matrix solution when moving on by one period. By Lemma 4.1, M(t0 + T ) = M(t0). By Liouville’s formula t +T R 0 tr A(s) ds R T tr A(s) ds det M(t0) = e t0 = e 0

is independent of t0 and positive. The logarithm of M(t0) therefore exists (see Lemma 4.3), say M(t0) =

TQ(t0) e with Q(t0 + T ) = Q(t0).

(t−t0)Q(t0) Write Π(t, t0) = P (t, t0)e . A straightforward computation shows

−1 −(t−t0)Q(t0) P (t + T, t0) = Π(t + T, t0)M(t0) e

−(t−t0)Q(t0) = Π(t + T, t0 + T )e

−(t−t0)Q(t0) = Π(t, t0)e

= P (t, t0).

Q P∞ 1 n Recall that the exponential of a n×n matrix Q is deﬁned by e = n=0 n! Q . The inverse operation, the logarithm, exists for an invertible matrix.

Lemma 4.3. If P is a n × n matrix with det P 6= 0, then there is a matrix Q with

P = eQ.

If P is real and all real eigenvalues are positive, then Q is real. In particular, if P is real, there is a real logarithm for P 2.

Proof. It is no restriction to assume that P is in Jordan normal form and to consider the case of only one Jordan block P = αI + N. Taking the formula

∞ X (−1)j+1 ln(1 + x) = xj, j j=1 put n−1 X (−1)j+1 Q = ln(α)I + N j. αjj j=1 Then eQ = P . To prove the statements on real logarithms, we only need to consider Jordan blocks with complex eigenvalues. Take the real Jordan normal form and note that from ! ! Re α Im α cos(φ) sin(φ) R = = r , −Im α Re α − sin(φ) cos(φ) 2.4. FLOQUET THEORY 29

α = reiφ, the logarithm is given by ! 0 −φ ln R = ln(r)I + . φ 0

For a real Jordan block RI + N = R(I + R−1N), the logarithm is given by

n X (−1)j+1 ln(RI + N) = ln(R) I + R−jN j. j j=1

The eigenvalues ρj of M(t0) are known as Floquet multipliers (also characteristic multipliers) and the eigenvalues γj of Q(t0) are known as Floquet exponents (also characteristic exponents). They are

T γj related via ρj = e .

Lemma 4.4. A periodic linear system is asymptotically stable if all Floquet multipliers satisfy |ρj| < 1

(or if all Floquet exponents satisfy Re γj < 0).

Example. We look at an example of a periodic linear system, Hill’s equation

x¨(t) + q(t)x(t) = 0, q(t + T ) = q(t). (4.6)

The associated system is x˙ = y, (4.7) y˙ = −qx. The principal matrix solution is given by ! c(t, t0) s(t, t0) Π(t, t0) = , c˙(t, t0)s ˙(t, t0)

where c(t, t0) is the solution of (4.6) with c(t0, t0) = 1, c˙(0, t0) = 0 and s(t0, t0) is the solution of (4.6)

with s(t0, t0) = 0, s˙(t0, t0) = 1. Liouville’s formula shows

det Π(t, t0) = 1.

and hence the characteristic equation for the monodromy matrix ! c(t0 + T, t0) s(t0 + T, t0) M(t0) = c˙(t0 + T, t0)s ˙(t0 + T, t0)

is given by ρ2 − 2∆ρ + 1 = 0,

where 1 1 ∆ = tr (M(t )) = (c(t + T, t ) +s ˙(t + T, t )). 2 0 2 0 0 0 0 30 CHAPTER 2. ELEMENTS OF NONLINEAR ANALYSIS

Figure 4.1: Stability diagram for the Mathieu equation (4.8).

A linear equation is called stable if all solutions are bounded. We obtain that Hill’s equation is stable if |∆| < 1 and unstable if |∆| > 1. A speciﬁc form of Hill’s equation is the Mathieu equation

x¨(t) = −ω2(1 + ε cos(t))x(t). (4.8)

A numerically computed stability diagram is depicted in Figure 4.1. The shaded regions are the ones where |∆| = |∆(ω, ε)| > 1, that is, where the equation is unstable. Observe that these unstable regions emerge from the points 2ω ∈ N where ∆(ω, 0) = cos(2πω) = ±1.

2.5 Poincar´ereturn maps

We continue studying smooth diﬀerential equationsx ˙ = f(x) on Rn. Recall that a set Σ ⊂ Rn is called a submanifold of codimension one (i.e., its dimension is n − 1), if it can be written as

Σ = {x ∈ U ; S(x) = 0}

where U ⊂ Rn is open, S ∈ Ck(U, R), and DS(x) 6= 0 for all x ∈ Σ. The submanifold Σ is said to be transversal to the vector ﬁeld f if DS(x)f(x) 6= 0 for x ∈ Σ.

Theorem 5.1. Suppose x ∈ Rn and let Σ be a submanifold of codimension one, transversal to f such that φ(T ; x) ∈ Σ, where φ denotes the ﬂow of f. Then there exists a neighborhood V of x and τ ∈ Ck(V, R) such that τ(x) = T and φ(τ(y), y) ∈ Σ for all y ∈ V .

Proof. Consider the equation S(φ(t, y)) = 0 for (t, y) near (T, x). Since ∂ S(φ(t, y)) = DS(φ(t, y))f(φ(t, y)) 6= 0 ∂t 2.5. POINCARE´ RETURN MAPS 31

for (t, y) in a neighborhood of (T, x) by transversality. So by the implicit function theorem, there exists a neighborhood V of x and function τ ∈ Ck(V, R) such that for all y ∈ V we have S(τ(φ(y), y)) = 0, that is, φ(τ(y), y) ∈ Σ.

The same argument works for the transition map between two diﬀerent transversals. This yields the ﬂow box theorem.

n Theorem 5.2. Suppose x ∈ R and let Σ1, Σ2 be two submanifolds of codimension one, transversal to f such that x ∈ Σ1, φ(T ; x) ∈ Σ2, where φ denotes the ﬂow of f. Then there exists a neighborhood V of x and τ ∈ Ck(V, R) such that τ(x) = T and

φ(τ(y), y) ∈ Σ1 for all y ∈ V .

If x is periodic and T is its period, then P :Σ → Σ given by

P (y) = φ(τ(y), y)

is called a Poincar´emap. Every ﬁxed or periodic point of P corresponds to a periodic orbit.

Assume now that γ is a periodic orbit with period T . Write x0 = γ(0). The following lemma connects the ﬁrst variation equation ξ˙(t) = Df(γ(t))ξ(t) and the principal matrix solution thereof to

the derivative of the ﬂow. Connecting to the notation of the previous section, we take t0 = 0 in the

following, and remove reference to t0 from the notation. So the principal matrix solution is the solution Π(t) of Π(˙ t) = Df(γ(t))Π(t) with Π(0) = id.

Lemma 5.1. The principal matrix solution (for t0 = 0) of the ﬁrst variation equation is given by

Π(t) = Dxφ(t, x0).

Moreover, f(φ(t, x0)) is a solution of the ﬁrst variation equation;

f(φ(t, x0)) = Π(t)f(x0). (5.1)

Proof. Abbreviate J(t, x) = Dxφ(t, x). Then J(0, x) = I and by interchanging t and x derivatives it

follows that J˙(t, x) = Df(φ(t, x))J(t, x). Hence J(t, x0) is the principal matrix solution of the ﬁrst variation equation.

To derive (5.1), we will demonstrate that f(φ(t, x0)) is the solution u(t) of d u(t) = Df(φ(t, x ))u(t) dt 0

with u(0) = f(x0). Therefore it equals Dxφ(t, x0)f(x0), which is the solution of this diﬀerential equation with the same initial condition. Calculate d d f(φ(t, x)) = Df(φ(t, x)) φ(t, x) = Df(φ(t, x))f(φ(t, x)). dt dt 32 CHAPTER 2. ELEMENTS OF NONLINEAR ANALYSIS

Since the ﬁrst variation equation along the periodic solution γ is periodic, the principal matrix solution is of the form Π(t) = R(t)etQ

TQ for a T -periodic matrix R, and the monodromy matrix M = e = Dxφ(T, x0) has eigenvalues indepen-

dent of the point in the orbit chosen. Note that one of the eigenvalues is one, since Mf(x0) = f(x0).

Theorem 5.3. The eigenvalues of the derivative the Poincar´emap DP at x0 plus the single value 1 coincide with the eigenvalues of the monodromy matrix M. In particular, the eigenvalues are independent

of the base point x0 and the transversal section Σ.

Proof. After a linear transformation it is no restriction to assume f(x0) = (0,..., 0, 1). Write x = (y, z) ∈ Rn−1 × R. Then Σ is locally the graph of a function s : Rn−1 → R. and we can take y as local coordinates for the Poincar´emap. Since

d ∂ φ(τ(x), x) = f(x0)Dτ(x0) + φ(T, x0) dx ∂x x=x0

we infer DP (x0)j,k = Mj,k for 1 ≤ j, k ≤ n − 1 by Lemma 5.1. Moreover, Mf(x0) = f(x0) and thus ! DP (x ) 0 M = 0 m 1 for some m, from which the claim is obvious.

Note that DP (x0) and M have equals determinants. Since the determinant of M does not vanish,

P is a local diﬀeomorphism at x0. The periodic orbit γ is an asymptotically stable periodic orbit of f if

and only if x0 ∈ γ is an asymptotically stable ﬁxed point of P . By Liouville’s formula we have

R T tr (A(t)) dt R T div (f(φ(t,x ))) dt det M = e 0 = e 0 0 .

In two dimensions there is only one eigenvalue which is equal to the determinant and hence we obtain

Theorem 5.4. Suppose f is a planar vector ﬁeld. Then a periodic point x0 is asymptotically stable if Z T div (f(φ(t, x0))) dt < 0 0 and unstable if the integral is positive.

Figure 5.1 shows the numerically computed attractor for the stroboscope map, with (x(t), x˙(t)) as coordinates and snapshots taken with periods 2π, of the Duﬃng equation

x¨(t) + 0.05x ˙(t) + x(t)3 = 7.5 cos(t).

This nonautonomous system corresponds to an autonomous system on R2 × T, x˙ = y, y˙ = −0.05y − x3 + 7.5 cos(θ), (5.2) θ˙ = 1. 2.6. CENTER MANIFOLDS 33

Figure 5.1: The Ueda attractor, studied by Yoshisuke Ueda.

The stroboscope map is the Poincaréreturn map on Σ = {(x, y, θ); θ = 0}. Diffeomorphisms arise as Poincaréreturn maps of differential equations. This goes by a construction called the suspension of a diffeomorphism, treated in the following theorem. See [36] for the proof.

Theorem 5.5. Let f : M → M be a Cr diffeomorphism on a compact manifold M. Then there exists a manifold M˜ , a Cr−1 vector field on M˜ admitting a cross section Σ and a Cr diffeomorphism h : M → Σ that conjugates f and the Poincaréreturn map on Σ.

2.6 Center manifolds

Let Cr(Rn, Rm) denote the Banach space of all Cr functions whose derivatives up to order r are uniformly n r n m bounded on R and | · |r denotes the norm on C (R , R ). Let ε > 0 and

r n m Cε = {w ∈ (R , R ); |w|r < ε}.

Consider the system x˙ = Ax + u(x, y), (6.1) y˙ = By + v(x, y),

n m r n+m n r n+m m where x ∈ R , y ∈ R , u ∈ Cε(R , R ), v ∈ Cε(R , R ), r ≥ 1, ε > 0 and A and B are constant matrices.

Theorem 6.1. Assume that all eigenvalues of A have zero real parts and all eigenvalues of B have

nonzero real parts. If r ≥ 1 and α > 0 are ﬁxed, then there exists ε0 = ε0(r, α, A, B) > 0 such that, for r every 0 < ε < ε0, there exists a C function

1 n+m n r n+m n 1 n+m m r n+m m r n m h : Cε(R , R ) ∩ Cα(R , R ) × Cε(R , R ) ∩ Cα(R , R ) → C (R , R )

such that 34 CHAPTER 2. ELEMENTS OF NONLINEAR ANALYSIS

(i) h(0, 0) = 0,

(ii) the set n Mc = {(x, y); y = h(u, v)(x), x ∈ R }

is a Cr manifold in Rn+m and is invariant under the ﬂow of (6.1).

(iii) Mc contains exactly those solutions (x(t), y(t)) of (6.1) for which

sup |y(t)| < ∞. t∈R

Moreover, Mc is unique with respect to properties (i), (ii), (iii).

The manifold Mc is called the center manifold. The following implication discusses local center manifolds. A manifold is locally invariant if the vector ﬁeld is tangent to it.

Theorem 6.2. Consider the system x˙ = Ax +u ˜(x, y), (6.2) y˙ = By +v ˜(x, y), where x ∈ Rn, y ∈ Rm and all eigenvalues of A have zero real parts and all eigenvalues of B have nonzero real parts. Suppose that u˜, v˜ are Cr with

u˜(x, y), v˜(x, y) = o(|x| + |y|) as |x| + |y| → 0. If r ≥ 1 is given, then there exists δ = δ(r) > 0 and a Cr function y = h(x) such that

(i) h(0) = 0, Dh(0) = 0,

(ii) the Cr manifold

M˜ c = {(x, y); y = h(x)) locally invariant with respect to (6.2).

Proof. Let χ : Rn → [0, 1] be C∞ with ( 1, |x| ≤ 1, χ(x) = 0, |x| ≥ 2

For ε > 0, deﬁne

u(x, y) =u ˜(xχ(x/ε), yχ(y/ε)), v(x, y) =v ˜(xχ(x/ε), yχ(y/ε)).

If we choose ε suﬃciently small, then the functions u and v satisfy the conditions of Theorem 6.1.

The manifold M˜ c is the local center manifold. The following example demonstrates how an expansion of the center manifold is obtained and how it is used to investigate stability. 2.6. CENTER MANIFOLDS 35

Example. Consider the planar system x˙ = xy + x3, (6.3) y˙ = −y − 2x2. The origin is the only equilibrium. Is it stable or unstable? The linearized vector ﬁeld at the origin is ! 0 0 and has eigenvalues 0 and −1. We seek a one dimensional center manifold y = V (x) = 0 −1 1 2 3 2 wx + O(|x| ). Then 1 y˙ = wxx˙ + O(|x|2)x ˙ = ( w2 + w)x4 + O(|x|5). 2 On the other hand, 1 y˙ = −y − 2x2 = (− w − 2)x2 + O(|x|3). 2 Comparing coeﬃcients yields w = −4. Thus the center manifold has an expansion

V (x) = −2x2 + O(|x|3) and the restriction of (6.3) is given by

x˙ = xV (x) + x3 = −x3 + O(|x|4).

It follows that the origin is stable, since it is stable on the center manifold and stable in the transverse direction. Note that putting y = O(|x|2) in the equation forx ˙ in (6.3) yieldsx ˙ = O(|x|3), which does not provide suﬃcient information to conclude about stability.

The fact that the extensions u and v ofu ˜ anv ˜ are not unique suggests that the local center manifold may not be unique. The following example shows that this is the case.

Example. Consider the system x˙ = −x3, (6.4) y˙ = −y with x, y ∈ R. Let c1, c2 be real numbers and let

2  −c e−1/(2x ), x < 0  1 h(x; c1, c2) = 0, x = 0  −1/(2x2)  c2e , x > 0.

For any c1, c2, the curve y = h(x; c1, c2) is an invariant manifold of (6.4) and is a center manifold because

h(0, c1, c2) = 0, Dh(0; c1, c2) = 0.

1 2 We also note that the right hand sides of (6.4) do not belong to Cε(R , R) since the ﬁrst derivative of −x3 goes to inﬁnity as |x| → ∞.

The next examples provide examples concerning limitations in regularity of center manifolds. 36 CHAPTER 2. ELEMENTS OF NONLINEAR ANALYSIS

Example. Consider the system x˙ = −x3, (6.5) y˙ = −y + x2 Suppose this system has a local center manifold y = h(x) where h is analytic at x = 0. Then

2 3 h(x) = a2x + a3x + ··· .

It is not diﬃcult to see that

a2n+1 = 0, n = 1, 2,...

a2 = 1,

an+2 = nan n = 2, 4,...

This implies that (6.5) has no analytic center manifolds.

Example. Consider the system x˙ = −εx − x3, y˙ = −y + x2, (6.6) ε˙ = 0. Suppose there is a C∞ center manifold y = h(x, ε) for |x| < δ, |ε| < δ. Choose n > 1/δ. Since h(x, 1/(2n)) ∞ is C in x, there are constants a1, a2, . . . , a2n such that

2n X i 2n+1 h(x, 1/(2n)) = aix + O(x ) i=1 for x suﬃciently small. One can show that ai = 0 if i is odd and, if n > 1,

(1 − (2i)/(2n))a2i = (2i − 2)a2i−2, i = 2, . . . , n

∞ and a2 6= 0. It is not diﬃcult to show that this contradicts the hypothesis that h is C .

2.7 Normal forms

We shall try to find coordinate transformations which simplify the analytic expression of a vector field. The resulting simplified vector fields are called normal forms. Recall that any n × n matrix can be transformed to a simple form, the Jordan normal form, by a linear coordinate transformation.

Theorem 7.1. Let A be a complex n × n matrix. Then there exists a linear transformation U that transforms A to a block matrix   J1 0 0 −1  .  U AU =  0 .. 0    0 0 Jm 2.7. NORMAL FORMS 37

with each block of the form   α 1 0 0 0    0 α 1 0 0     ..  Ji = αI + N =  0 0 α . 0     .   0 0 0 .. 1    0 0 0 0 α Here N is a nilpotent matrix with ones above the diagonal and zeros elsewhere.

For real matrices there is a real Jordan normal form. Here, the block matrices for real eigenvalues are the same as for the Jordan normal form. Complex conjugate eigenvalues give rise to block matrices   RI 0 0 0    0 RI 0 0     ..   0 0 R . 0     .   0 0 0 .. I    0 0 0 0 R ! ! cos(φ) − sin(φ) 1 0 with R = and I = . sin(φ) cos(φ) 0 1 A linear map x 7→ Lx induces a map ad L on the linear space Hk of vector fields whose coefficients are homogeneous polynomials of degree k. The map ad L is defined by

ad L(Y )(x) = [Y,L](x) = DL(x)Y (x) − DY (x)Lx.

In component form, we have n X ∂Li ∂Yi [Y,L] = Y − L . i ∂x j ∂x j j=1 j j

Here L = (L1,...,Ln) and x = (x1, . . . , xn).

Theorem 7.2. Let x˙ = f(x) be a Cr diﬀerential equation with f(0) = 0 and Df(0) = L. Choose a complement Gk for ad L(Hk) in Hk, so that Hk = ad L(Hk) ⊕ Gk. Then there is an analytic change of coordinates in a neighborhood of the origin which transforms the system x˙ = f(x) to

(2) (r) y˙ = g(y) = Ly + g (y) + ··· + g (y) + Rr(y)

(k) r with g ∈ Gk for 2 ≤ k ≤ r and Rr(y) = o(|y| ) as |y| → 0.

Proof. We use induction and assume thatx ˙ = f(x) has been transformed so that the terms of degree smaller than s lie in the complementary subspace Gi, 2 ≤ i < s. We introduce a coordinate transformation of the form x = h(y) = y + P (y), where P is a homogeneous polynomial of degree s whose coeﬃcients are to be determined. Substitution gives the equation

(id + DP (y))y ˙ = Ly + f (2)(y) + ··· + f (s)(y) + LP (y) + o(|y|s) (7.1) 38 CHAPTER 2. ELEMENTS OF NONLINEAR ANALYSIS

as |y| → 0. The terms of degree smaller than s are unchanged by this transformation, while the new terms of order s are

f (s)(y) + LP (y) − DP (y)L = f s(y) + ad L(P )(y). (7.2)

(s) A suitable choice of P will make f (y) + ad L(P )(y) lie in Gs, as desired.

Applying this procedure, we prove Poincar´e’stheorem on formal linearization.

Theorem 7.3. Let x˙ = f(x) be a smooth diﬀerential equation on Rn with f(0) = 0 and Df(0) = A. If the eigenvalues λi of A are nonresonant, i.e.

n X λs 6= miλi i=1

Pn ∞ for mi ≥ 0, i=1 mi ≥ 2, then x˙ = f(x) can be reduced to x˙ = Ax + r (x) by a smooth change of coordinates. Here r∞ is a ﬂat function, Dir∞(0) = 0 for all i ≥ 0.

Proof. Assume that A is diagonal. We claim that ad L acting on Hk is diagonal, has eigenvectors

m m1 mn x es = x1 ··· xn es, 1 ≤ s ≤ n, with eigenvalues λs − hm, λi. Here es is the sth standard basis vector and λ = (λ1, . . . , λn). Namely, calculate

n X mi ad L(Y ) = λ xm − xmλ x = (λ − hm, λi)xme . s x i i s s i=1 i

If all eigenvalues of ad L are different from zero, then it is invertible. This holds by assumption. So we can successively kill terms of order 2, 3,... in the Taylor expansion of f. This defines a sequence of substitutions, which for all substitutions together gives an infinite Taylor expansion of a coordinate change. Although the Taylor series may not converge, by Borel’s theorem we obtain a smooth coordinate transformation as in the statement of the theorem. If A is not diagonal, but has Jordan blocks, then also Ad L has Jordan blocks. The eigenvalues are

however the same as in the diagonal case. Therefore, ad L is invertible on Hk. The theorem therefore holds in the nondiagonal case as well.

Example. We illustrate the normal form procedure for a planar system which has an equilibrium with ! 0 −1 eigenvalues ±i. In the appropriate linear coordinate system, DL is given by . We compute 1 0 the action of ad L on H2 in the following manner. For each basis vector Yi of H2 we have

ad L(Yi) = DLYi − DYiL (7.3) 1 1 ! 1 ! ∂Yi ∂Yi ! ! 0 −1 Yi ∂x ∂y −y = − ∂Y 2 ∂Y 2 . (7.4) 1 0 Y 2 i i x i ∂x ∂y 2.7. NORMAL FORMS 39

Thus, for example, ! ! x2 2xy ad L = , (7.5) 0 x2 ! ! 0 −xy ad L = , (7.6) xy y2 − x2

and so on. We use ﬁrst order partial derivatives ∂/∂x, ∂/∂y to represent vector ﬁelds, so that for example ! 2xy 2xy(∂/∂x) + x2(∂/∂y) represents . In terms of the basis x2

∂ ∂ ∂ ∂ ∂ ∂ {x2 , xy , y2 , x2 , xy , y2 } ∂x ∂x ∂x ∂y ∂y ∂y

of H2, ad L on H2 has the following matrix:   0 −1 0 −1 0 0    2 0 −2 0 −1 0     0 1 0 0 0 −1       1 0 0 0 −1 0     0 1 0 2 0 −2    0 0 1 0 1 0

This matrix is nonsingular (its determinant is −9), so all quadratic terms of a vector ﬁeld with linear part −y(∂/∂x) + x(∂/∂y) can be removed by coordinate transformation. Calculations show that, in terms of the basis ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ {x3 , x2y , xy2 , y3 , x3 , x2y , xy2 , y3 } ∂x ∂x ∂x ∂x ∂y ∂y ∂y ∂y of H3, ad L on H3 has the following matrix:   0 −1 0 0 −1 0 0 0    3 0 −2 0 0 −1 0 0     0 2 0 −3 0 0 −1 0       0 0 1 0 0 0 0 −1     1 0 0 0 0 −1 0 0       0 1 0 0 3 0 −2 0     0 0 1 0 0 2 0 −3    0 0 0 1 0 0 1 0

Inspection shows that ad L(H3) is six dimensional and

{x(x2 + y2)(∂/∂x) + y(x2 + y2)(∂/∂y), −y(x2 + y2)(∂/∂x) + x(x2 + y2)(∂/∂y)}

is a basis for a complement G3 (columns 1,2,3,4,5 and 8 are linearly independent and the vectors (3, 0, 1, 0, 0, 1, 0, 3) and (0, 1, 0, 3, −3, 0, −1, 0) are left eigenvectors for zero for this matrix). 40 CHAPTER 2. ELEMENTS OF NONLINEAR ANALYSIS

Expressed in terms of of diﬀerential equations, we have shown that the normal form theorem gives a coordinate transformation which transforms

x˙ = −y + o(|x|, |y|), (7.7) y˙ = x + o(|x|, |y|), (7.8) into the system

u˙ = −v + (au − bv)(u2 + v2) + h.o.t., (7.9) v˙ = u + (av + bu)(u2 + v2) + h.o.t., (7.10) for suitable constants a and b. Here h.o.t. stands for higher order terms, here terms of order four and higher.

Without proof we state two stronger theorems on C1 and C∞ linearization, due to Genrikh Belitskii and Shlomo Sternberg, respectively.

Theorem 7.4. Assume f : Rn → Rn is a C2 vector ﬁeld with a hyperbolic equilibrium at the origin. Assume the eigenvalues λ1, . . . , λn satisfy the conditions

Re (λi) 6= Re (λj) + Re (λk)

1 n when Re (λj) < 0 < Re (λk). Then there is a C diffeomorphism h on R that conjugates the flow of f and the flow of Df(0) on a neighborhood of 0.

Theorem 7.5. Assume f : Rn → Rn is a C∞ vector ﬁeld with a hyperbolic equilibrium at the origin. Assume the eigenvalues λ1, . . . , λn satisfy the conditions

λi 6= k1λ1 + ··· + knλn

∞ n for all i and k1, . . . , kn ≥ 0 with k1 + ··· + kn ≥ 2. Then there is a C diffeomorphism h on R that conjugates the flow of f and the flow of Df(0) on a neighborhood of 0.

For the study of bifurcations that involve periodic orbits, parameter-dependent versions and higher degrees of diﬀerentiability are needed. The next theorem, a parameter-dependent version of Sternberg’s linearization result, gives conditions under which

2 u˙ = f(u, µ) = fu(p, µ)(u − p) + O(ku − pk ) (7.11) can be transformed into

v˙ = fu(p, µ)v (7.12) for u close to an equilibrium p. 2.8. MANIFOLDS AND TRANSVERSALITY 41

Theorem 7.6. Assume that f(u, µ) is C∞ in (u, µ). Fix any ` ≥ 1, then there exist numbers N =

N(`, fu(0, 0)) ≥ 1 and > 0 with the following property: if n X νi 6= Njνj (7.13) j=1

Pn ` for each i and all natural numbers Nj with 2 ≤ j=1 Nj ≤ N, then we can C -linearize (7.11) in B(p) ` for |µ| < , i.e. there is a coordinate transformation of class C in u ∈ B(p) and |µ| < that transforms (7.11) into (7.12).

2.8 Manifolds and transversality

Let X,Z be Banach spaces, U ⊂ X open and f ∈ C1(U, Z). A point x ∈ U is called a regular point for f if Df(x): X → Z is surjective. A point x ∈ U is called a critical point for f if x is not a regular point. A point y ∈ Z is called a regular value for f if each point in f −1(y) is a regular point. Otherwise it is called a critical value. If X = Rn, Z = Rk, then x is a critical point for f if and only if rank Df(x) < k. A set M ⊂ Rn is said to be a Cr-manifold of dimension m if, for each x ∈ M, there is an open neighborhood U of x and a map F ∈ Cr(U, Rn−m) which is regular at each point of U and M ∩ U = {x ∈ U ; F (x) = 0}. We call n − m the codimension of M.

Theorem 8.1. A Cr-manifold of dimension m in Rn is locally the graph of a Cr map. More precisely, n m n−m m n−m one can express R as the product R × R so that, if x = (u, v), u ∈ R , v ∈ R , and x0 = r n−m (u0, v0) ∈ M, then there is a neighborhood U of x0, a neighborhood V of u0 and f ∈ C (V, R ) such that M ∩ U = {(u, v); v = f(u), u ∈ V }.

m A manifold M is thus obtained by gluing together pieces of R ; there are open sets Ui ⊂ M, open m m sets Vi ⊂ R and maps ψi : Ui → R such that ψi identiﬁes Ui with Vi. The open sets Ui with the maps ψi are called charts. A collection of charts covering the manifold is an atlas. A submanifold N ⊂ M is simply a manifold N that is contained in M. We will also use the concept of an injectively immersed manifold in a manifold M; this is the image of an injective map N → M from a manifold N into M for which the derivative is injective at every point. An injectively immersed manifold can accumulate onto itself. n Suppose M is a manifold of dimension m in R . For x0 ∈ M and a neighborhood U of x0, if

M ∩ U = {x ∈ U ; F (x) = 0} and x0 is a regular point of F , then the tangent space Tx0 M is deﬁned by

n Tx0 M = {x ∈ R ; DF (x0)x = 0}.

A vector ﬁeld on M is a function X that assigns a vector X(x) ∈ TxM for each x ∈ M. For a vector m ﬁeld X on M, there will be functions Yi : Vi → R such that

−1 −1 X(ψi (y)) = Dψi (y)Yi(y).

m A vector field X is on M is thus replaced by vector fields Yi on R . A vector field X on M is of class k k k+1 C if the Yi are C functions and the manifold M is C . 42 CHAPTER 2. ELEMENTS OF NONLINEAR ANALYSIS

A vector field X on M gives rise to a flow φt : M → M built from solutions of the differential equation ˙ φt(x) = X(φt(x)).

Using charts we see that a differential equation on M is locally the same as a differential equation on Rm. We therefore have local existence and uniqueness. Global existence and uniqueness on a compact manifold is obtained by continuation. For a compact manifold M, consider the space Ck(M) of Ck vector fields on M, endowed with the k i i uniform C topology: Xn → X if D Xn → D X uniformly on M for 0 ≤ i ≤ k. For finite k one can put a norm compatible with this topology on Ck(M). Namely, cover M with a finite collection of open

sets Vi, such that each Vi is contained in the domain of a local chart Ui with coordinates hi that map Vi onto Dm(0), the unit disc in Rm. Then, for f ∈ Ck(M), deﬁne

j −1 kfkk = max max { sup kD f ◦ hi (u)k}. i 0≤j≤k u∈Dn(0)

This deﬁnes a norm on Ck(M).

k Lemma 8.1. k · kk is a complete norm on C (M).

For k = ∞, we have a complete metric d∞, compatible with the topology, given by ∞ X kX − Y kj d (X,Y ) = 2−j . ∞ 1 + kX − Y k j=1 j

Consider the space of Ck maps between compact manifolds M ⊂ Rn and N ⊂ Rp. As N is a compact subset of Rp, we get Ck(M,N) as a closed subset of Ck(M, Rp). Complete metric spaces are Baire spaces; i.e. every set that contains a countable intersection of open and dense sets is dense. A subset containing a countable intersection of open and dense sets is residual. A property shared by all members of a residual set is called generic. Two manifolds M,N in Rn are said to be transverse if either M ∩ N = ∅ or, for every p ∈ M ∩ N, n TpM + TpN = R . Note that the sum is not necessarily a direct sum. In this case, M ∩ N is a manifold of codimension equal to the sum of the codimensions of M and N in Rn. To see this, let p ∈ M ∩ N and suppose that near p, M = F −1(0) for F : Rn → Rn−dim(M) and N = G−1(0) for G : Rn → Rn−dim(N). Consider the function H : Rn × Rn → Rn × Rn−dim(M) × Rn−dim(N),

H(x, y) = (y − x, F (x),G(y)).

The transversality condition gives that p is a regular point for H, so that M ∩ N is a manifold.

2.9 Sard’s theorem

Consider a C1 map f : M → N from a smooth manifold M of dimension n to a smooth manifold N of dimension p. Recall that a point x is critical if and only if the diﬀerential at x,

Df(x): TxM → Tf(x)N, 2.9. SARD’S THEOREM 43

has rank less than p, i.e. is not surjective. We state and prove Sard’s theorem, ﬁrst a baby version and then a general version.

Theorem 9.1. Let f : U → R, where U ⊂ R, be a smooth function. If

W = {x ∈ U | f 0(x) = 0}

is the set of critical points of f, then f(W ) has Lebesgue measure zero.

Proof. Let I be an interval in U of length a > 0. There exists K > 0 so that for all x ∈ W ∩ I and y ∈ I,

|f(x) − f(y)| ≤ K|x − y|2. (9.1)

Subdivide I into k intervals Ji, 1 ≤ i ≤ k, of length a/k. Then (9.1) gives

|f(x) − f(y) ≤ K(a/k)2

2 if x ∈ W ∩ Ji, y ∈ Ji. Thus f(W ∩ I) is contained in a union of intervals of length at most Kk(a/k) . Letting k go to inﬁnity shows that f(W ∩ I) has Lebesgue measure zero. Thus also f(W ) has Lebesgue measure zero.

Theorem 9.2. Let f : U → Rp be Cr, where U ⊂ Rn is open and

r > max (0, n − p).

If W ⊂ U is the set of critical points of f, then f(W ) has Lebesgue measure zero.

Corollary 9.1. If f : U → Rp is Cr, r > max (0, n − p), then the set of regular values is dense in Rp.

Proof. If i ≥ 1 and k Wi = {x ∈ U ; D f(x) = 0 for all 1 ≤ k ≤ i},

then W ⊃ W1 ⊃ W2 ⊃ · · · . Since

f(W ) = f(W − W1) ∪ f(W1 − W2) ∪ · · · ∪ f(Wr−1 − Wr) ∪ f(Wr),

we will complete the proof provided the following three claims are true.

(i) f(W − W1) has measure zero.

(ii) f(Wi − Wi+1) has measure zero for 1 ≤ i ≤ r.

(iii) f(Wr) has measure zero.

Proof of (i). Since W = W1 when p = 1, we may assume that p ≥ 2. Letx ¯ ∈ W − W1. Then we may assume ∂f (¯x) 6= 0. ∂x1 44 CHAPTER 2. ELEMENTS OF NONLINEAR ANALYSIS

Consider the map h(x) = (f1(x), x2, . . . , xn). Since Dh(¯x) is nonsingular, h is locally a diﬀeomorphism. Let g = f ◦ h−1. Note that h is deﬁned locally in an open neighborhood V ofx ¯ and the set of critical values of g is f(V ∩ W ). Hence, if W 0 is the set of critical points of g, then g(W 0) = f(V ∩ W ). Moreover ! x1 g(x1, x2, . . . , xn) = . g˜(x2, . . . , xn)

If n = 1, then for some c ∈ Rp−1, ! x1 g(x1) = . c It is trivial that g(W 0) has measure zero if n = 1. If n = 2, then ! x1 g(x1, x2) = . g˜(x2)

Hence, (x1, x2) is a critical point of g if and only if x2 is a critical point ofg ˜. Since (i) is true for n = 1, 0 the intersection of g(W ) and {x1 = constant} has 1-dimensional measure zero. By Fubini’s theorem, g(W 0) has measure zero. An induction will show that g(W 0) has measure zero for any n ≥ 1. Since W

is compact, ﬁnitely many such V ’s will cover W . Thus, f(W − W1) has measure zero.

Proof of (ii). Letx ¯ ∈ Wi − Wi+1. By deﬁnition, all ith partial derivatives vanish but not all (i + 1)th partial derivatives vanish. Let α be an appropriate ith partial derivative of f with (without loss of generality) ∂α (¯x) 6= 0. ∂x1 Let  α(x)     x2  h(x) =  .   .   .  xn

−1 and g = f ◦ h . Again, h is deﬁned locally in an open neighborhood V ofx ¯. By deﬁnition, α(Wi) = 0, p−1 p−1 so h(Wi ∩ V ) ⊂ {0} × R . Letg ˜ be the restriction of g to ({0} × R ). So

g˜(h(Wi ∩ V )) = f(Wi ∩ V ).

Since each point in h(Wi ∩ V ) is a critical point ofg ˜,g ˜(h(Wi ∩ V )) has measure zero by induction on n.

Since Wi is compact, f(Wi − Wi+1) has measure zero.

Proof of (iii). Let In ⊂ U be a cube of length a > 0. By Taylor’s theorem, assuming for simplicity that f is Cr+1 (a similar reasoning applies if f is Cr),

|f(x) − f(y)| ≤ K|x − y|r+1 2.9. SARD’S THEOREM 45

n for all x, y ∈ I , x ∈ Wr. Here K > 0 is a constant independent of x and y. Let k ≥ 1 be an integer. n n Subdivide the cube I into k cubes, J1,...,Jkn each of which has length a/k. By the above estimate, for each i = 1, . . . , kn, √ |f(x) − f(y)| ≤ 2(K na)r+1k−r−1

r+1 for x ∈ Ji ∪ Wr, y ∈ Ji. This says that f(Ji ∩ Wr) lies in a cube of length K(a/k) . Hence, f(Wr) lies √ in the union of kn cubes whose volume is (2(K na)r+1)pkn−p(r+1). If k is suﬃciently large, then the above number will tend to zero since r > max (0, n − p).

The next theorem on regular values of functions depending on a parameter follows from Sard’s theorem.

Theorem 9.3. Let f : Rm × Rn → Rp be Cr, r > max (m − p, 0). If b ∈ Rp is a regular value of f, then the set {λ ∈ Rn ; b is a critical value of f(·, λ)} has measure zero.

Proof. Since b is a regular value of f, S = f −1(b) is a Cr manifold of dimension m + n − p. Consider m n n r 1 r the projection π : R × R → R . Since S is C , the restriction π = π|S is C . By Corollary 9.1, almost every λ ∈ Rn is a regular value of π1. Let λ¯ be a regular value of π1. Thus, the linear map 1 ¯ ¯ Dπ (¯x, λ) is surjective for every (¯x, λ) ∈ S. Since the tangent space T(¯x,λ¯)S consists of precisely the elements (y, µ) ∈ Rm × Rn satisfying

¯ ¯ Dxf(¯x, λ)y + Dλf(¯x, λ)µ = 0,

1 ¯ n m ¯ the surjectivity of Dπ (¯x, λ) implies that, for every µ ∈ R , there exists y ∈ R with Dxf(¯x, λ)y + ¯ Dλf(¯x, λ)µ = 0. Hence,

¯ ¯ ¯ p range Dxf(¯x, λ) = range Dxf(¯x, λ) + range Dλf(¯x, λ) = R .

This implies that b is a regular value for f(·, λ¯) and completes the proof.

Using charts we also get the following formulation of Sard’s theorem, for simplicity for smooth maps.

Theorem 9.4. Let f : M → N be a smooth map between manifolds M,N. Then the set of regular values is dense.

The diﬀerentiability conditions in Theorem 9.2 are necessary: Whitney [48] constructed counterex- amples such as an example, illustrated in Figure 9.1, of a C1 function f : R2 → R with an interval of critical values. 46 CHAPTER 2. ELEMENTS OF NONLINEAR ANALYSIS

Figure 9.1: The graph of a C1 function with an interval of critical values. The derivative vanishes along the nondecreasing continuous curve.

2.10 Exercises

Exercise 2.10.1. Suppose f : Rn → R and g : Rn → Rk are smooth and Dg(p) has full rank. Let M = −1 g (c), where c = g(p). Prove that f|M has a critical point at p if and only if there are λ1, . . . , λk ∈ R with k X Df(p) = λiDgi(p), i=1 where g = (g1, . . . , gk). This is the Lagrange multiplier theorem.

Exercise 2.10.2. Use Lyapunov-Schmidt reduction to ﬁnd an expression, or approximation, of the set of equilibria, as a function of the parameter λ, of the planar vector ﬁeld

f(x, y, λ) = (λ + 2x + y − x2, 2x + (1 + λ)y − xy) near the equilibrium (x, y) = (0, 0) at λ = 0.

Exercise 2.10.3. Let f : Rm → Rm with f(0) = 0. Consider the following decompositions of Rm,

m R = ker Df(0) ⊕ C, (10.1) m R = range Df(0) ⊕ C,˜ where C and C˜ are complementary to ker Df(0) and range Df(0), respectively, in Rm.

(i) Show that by choosing appropriate projections, to range Df(0) and C˜, f can be written as the system

f1 : ker Df(0) × C → range Df(0), (10.2) f2 : ker Df(0) × C → C.˜ 2.10. EXERCISES 47

(ii) Writing f1(x, y) and f2(x, y) for (x, y) ∈ ker Df(0) × C, show that Dyf1(0, 0) is invertible and that

Dxf2(0, 0) = 0.

Exercise 2.10.4. Prove Borel’s theorem, Theorem 1.3. Hint: Let ϕ be a smooth bump function that is equal to 1 on {|x| ≤ 1} and 0 outside {|x| < 2}. Write ϕδ(x) = ϕ(x/δ). Prove that for any ε > 0, there k is δ > 0 so that ϕδ(x)x has derivatives up to order |k|−1 (using multi-index notation: k = (k1, . . . , kn),

k k1 kn x = x1 ··· xn , |k| = k1 + ··· + kn) that are uniformly smaller than ε. Multiply terms in a given formal power series by suitable bump functions.

Exercise 2.10.5. Let M ⊂ Rn be a compact m-dimensional manifold. Suppose n > 2m + 1. Show that there is a projection from M to a (2m + 1)-dimensional subspace of Rn, so that the image is a manifold in R2m+1, by following the steps below.

n n ⊥ (i) Show that it suﬃces, for n > 2m+1, to ﬁnd a nonzero vector a ∈ R and a projection πa : R → a , ⊥ ⊥ where a is the codimension one subspace perpendicular to a, so that πa(M) is a manifold in a .

(ii) Deﬁne g : M × M × R → Rn and h : TM → Rn by

g(x, y, t) = t(x − y), h((p, v)) = v,

⊥ where (p, v) represents the vector v ∈ TpM. Show that πa(M) is a manifold in a if a is not in the images of g and h.

(iii) Apply Sard’s theorem, Theorem 9.4.

Exercise 2.10.6. Let M,N be two submanifolds of Rn. Show that for any ε > 0, there exists a ∈ Rn, |a| < ε, so that M+a is transverse to N. Hint: Consider g : M×N → Rn deﬁned by g(x, y) = x−y. Show that for a regular value a of g, M +a is transverse to N (note that Dg(x, y)T(x,y)(M ×N) = TxM +TyN). Apply Sard’s theorem, Theorem 9.4.

Exercise 2.10.7. What are necessary and suﬃcient conditions, in terms of Floquet multipliers, for a periodic orbit to be a hyperbolic set?

Exercise 2.10.8. Consider, for α, γ1, γ2 ∈ R, β > 0, and n ∈ N, the following linear 2π/β-periodic system ! ! x˙ x = A(t) , (10.3) y˙ y with n+1 n ! α + γ1 cos (βt) β − γ1 sin(βt) cos (βt) A(t) = n n+1 −β − γ2 sin (βt) cos(βt) α + γ2 sin (βt)

Let λi,n = λi,n(α, β, γ1, γ2), i = 1, 2, denote the two Floquet exponents associated to (10.3). 48 CHAPTER 2. ELEMENTS OF NONLINEAR ANALYSIS

(i) Show that ! ! x(t) sin(βt) = eαt y(t) cos(βt)

is a solution of (10.3) for all α, γ1, γ2 ∈ R, β > 0, n ∈ N. What does this tell you about λ1,n(α, β, γ1, γ2)?

(ii) Take n = 0. Determine λi,0, i = 1, 2, and show that the solution (˜x, y˜) = (0, 0) of (10.3) is stable

if α < 0 and unstable if α > 0, independent of β, γ1, γ2.

(iii) For which parameter combinations is (0, 0) stable if n = 1? And if n = 3?

(iv) Show that (0, 0) is stable for α < 0 and unstable for α > 0 if n is even.

(v) Take n = 1 and γ1 = γ2 = γ. Deﬁne Λi(t) ∈ C, i = 1, 2, as the two eigenvalues of A(t). Determine Λi(t), i = 1, 2 (and note that they turn out to be constant, i.e. not time dependent).

(vi) Keep n = 1 and γ1 = γ2 = γ. For which parameters is Re (Λi) < 0, i = 1, 2, and (0, 0) unstable?

Exercise 2.10.9. Find the Floquet multipliers of the following systems:

(i) x˙ = sin(2πt)x, y˙ = (−1 + cos(2πt))y,

(ii) x˙ = cos(2πt)x, y˙ = sin(2πt)x + (−1 + cos(2πt))y.

Exercise 2.10.10. Consider the periodic linear system x˙ = A(t)x with

3 2 3 ! −1 + 2 cos (t) −1 + 2 cos(t) sin(t) A(t) = 3 3 2 . 1 + 2 sin(t) cos(t) −1 + 2 sin (t)

(i) Compute the eigenvalues of A(t) to establish that they have negative real parts. ! cos(t) (ii) Verify that et/2 is a solution of the periodic linear system. Note that this solution is sin(t) unbounded as t → ∞.

(iii) What are the Floquet multipliers of the system?

Exercise 2.10.11. Consider for α ∈ R the two dimensional system

x˙ = αxy − x3 + y2, (10.4) y˙ = −y + x2 + xy.

Determine the center manifold W c((0, 0)) up to and including terms of order three. Determine the approximate ﬂow on W c((0, 0)) near (0, 0). Determine the stability of (0, 0) for all α ∈ R.

Exercise 2.10.12. Find series approximations for the reduced system on the center manifold near the origin, suﬃciently far to determine the stability of the reduced system, in the following cases: 2.10. EXERCISES 49

(i) x˙ = αx2 − y2, (10.5) y˙ = −y + x2 + xy. First suppose α 6= 0, and then let α = 0.

(ii) x˙ = −y + xz, y˙ = x + yz, (10.6) z˙ = −z − (x2 + y2) + z.

Exercise 2.10.13. Consider for β ∈ R the two dimensional system x˙ = −x3, (10.7) y˙ = −y + x2 + βx4. (i) Take β = −2. Determine the stable manifold W s((0, 0)) and the center manifold(s) explicitly by solving the appropriate equations. Is W c((0, 0)) uniquely determined? Is it analytic? If so, give an expression of W c((0, 0)) in terms of a power series.

(ii) Sketch, for β still equal to −2, the phase portrait, including the manifolds W s((0, 0)) and W c(0, 0)).

(iii) Consider the general case β ∈ R. What can you say about W c((0, 0))? Is it unique? Is it analytic? Can you give an explicit expression, or a power series expansion? Exercise 2.10.14. Consider the three dimensional system x˙ = −y + xz − x4, y˙ = x + yz + xyz, (10.8) z˙ = −z − x2 − y2 + z2 + sin(x3). Determine the stability of the critical point (0, 0, 0). Hint: determine the ﬂow on the center manifold and use polar coordinates. Exercise 2.10.15. Show that the system

x˙ = y + O(|(x, y)|2), y˙ = O(|(x, y)|2), can be transformed to

u˙ = v + au2 + O(|(u, v)|3) v˙ = bu2 + O(|(u, v)|3) or to

x˙ = v + O(|(u, v)|3) y˙ = au2 + buv + O(|(u, v)|3) 50 CHAPTER 2. ELEMENTS OF NONLINEAR ANALYSIS Chapter 3

Local bifurcations

In this chapter we review bifurcations of equilibria and periodic orbits of diﬀerential equations. We treat the codimension-one bifurcations of equilibria, these are the saddle-node bifurcation and the Hopf bifurcation. We also discuss the saddle-node bifurcation and the period-doubling bifurcation of periodic orbits. In the ﬁnal section we consider the Bogdanov-Takens bifurcation; a codimension-two bifurcation of equilibria.

3.1 The saddle-node bifurcation

In the saddle-node bifurcation a saddle equilibrium and a node (stable equilibrium) collide and disappear as a parameter changes. Figure 1.1 illustrates the bifurcation in a simple model.

Figure 1.1: The saddle-node bifurcation in the system x˙ = β + x2, y˙ = −y.

3.1.1 One dimensional saddle-node bifurcation

We consider ﬁrst the one-dimensional case. Suppose the system

x˙ = f(x, λ),

51 52 CHAPTER 3. LOCAL BIFURCATIONS

x ∈ R, λ ∈ R has an equilibrium x = 0 at λ = 0 with Dxf(0, 0) = 0. Expanding f(x, λ) in a Taylor series with respect to x at x = 0, we can write

2 3 f(x, λ) = f0(λ) + f1(λ)x + f2(λ)x + O(|x| )

with f0(0) = 0 and f1(0) = 0. Introduce a coordinate ξ = x + δ where δ = δ(λ). A computation shows

˙ 2 3 2 2 3 ξ = (f0(λ) − f1(λ)δ + f2(λ)δ + O(|δ| )) + (f1(λ) − 2f2(λ)δ + O(|δ| ))ξ + (f2(λ) + O(|δ|))ξ + O(|ξ| ).

Assume that 1 f (0) = D2f(0, 0) 6= 0. 2 2 x Then it follows from the implicit function theorem that there is a smooth function δ(λ) that annihilates the linear term in the diﬀerential equation for ξ for all suﬃciently small λ. It also follows that f 0 (0) δ(λ) = 1 λ + O(λ2). 2f2(0) The equation for ξ now contains no linear terms:

˙ 0 2 2 3 ξ = f1(0)λ + O(λ ) + (f2(0) + O(|λ|))ξ + O(|ξ| ).

Consider as a new parameter µ = µ(λ) the constant term above:

0 2 µ = f1(0)λ + φ(λ)λ

0 0 where φ is a smooth function. Note that µ(0) = 0 and µ (0) = f0(0) = Dλf(0, 0). If we assume that

Dλf(0, 0) 6= 0,

then the implicit function theorem implies the existence of an inverse function λ(µ) with λ(0) = 0. The diﬀerential equation becomes ξ˙ = µ + λ(µ)ξ2 + O(|ξ|3),

where λ(µ) is a smooth function with λ(0) = f2(0) 6= 0. Let a rescaling be given by η = |λ(µ)|ξ and β = |λ(µ)|µ. Then we get

η˙ = β + sη2 + O(|η|3)

with s = sign a(0). We have proved the following theorem.

Theorem 1.1. Suppose that a one-dimensional system

x˙ = f(x, λ),

x ∈ R, λ ∈ R, has at λ = 0 an equilibrium at x = 0, with Dxf(0, 0) = 0. Assume that the following conditions are satisﬁed: 3.1. THE SADDLE-NODE BIFURCATION 53

2 (i) Dxf(0, 0) 6= 0,

(ii) Dλf(0, 0) 6= 0.

Then there is a smooth coordinate change η = h(x, λ) and a smooth parameter change β = g(λ), transforming the system into η˙ = β ± η2 + O(|η|)3.

3.1.2 Higher dimensional saddle-node bifurcation

Let

x˙ = f(x, µ) (1.1)

be an ordinary diﬀerential equation in Rn depending on a parameter µ ∈ R.

Theorem 1.2. Assume that (1.1) for µ = µ0 has an equilibrium p for which

(i) Dxf(p, µ0) has a single eigenvalue 0 with right eigenvector v and left eigenvector w. Further,

Dxf(p, µ0) has k eigenvalues with negative real part and n − k − 1 eigenvalues with positive real part.

(ii) hw, Dµf(p, µ0)i= 6 0.

(iii) hw, Dxxf(p, µ0)(v, v)i= 6 0.

n n Then there is a smooth curve of equilibria in R × R, passing through (p, µ0) and tangent to R × {µ0} at (p, µ0).

n Proof. By a translation in state space R and parameter space, we may assume p = 0 and µ0 = 0. From T hw, Dxf(0, 0)i = h(Dxf(0, 0)) w, ui = 0, we see that w is perpendicular to the image Im Dxf(0, 0). The second condition thus gives that

n Im Dxf(0, 0) ⊕ Dµf(0, 0) = R .

Likewise, the third assumption gives that

Dxxf(0, 0)(v, v) 6∈ Im Dxf(0, 0).

Write the equation f(x, µ) as

Dxf(0, 0)x − N(x, µ) = 0 (1.2)

where N(x, µ) = f(x, µ) − Dxf(x, µ). We will apply the Lyapunov-Schmidt reduction to (1.2). In the

notation from Section 2.3, A = Dxf(0, 0) and let U be the projection onto N (A) with UR(A) = 0. Then I − U is the projection onto R(A) with (I − U)N (A) = 0. Let E be the perpendicular projection onto R(A), i.e. it projects Rw onto 0. 54 CHAPTER 3. LOCAL BIFURCATIONS

The Lyapunov-Schmidt reduction applied to (1.2) yields a reduced bifurcation equation

(I − E)N(y + z∗(y, µ), µ) = 0, (1.3)

where z∗ solves z − KEN(y + z, µ) = 0. (1.4)

Here x = y + z with y ∈ N (A) and z ∈ R(A). ∗ ∗ ∗ From (1.4) we obtain Dyz (0, 0) − KEDxN(0, 0)(I + Dyz (0, 0)) = 0, so that Dyz (0, 0) = 0. Now

∗ ∗ Dµ((I − E)N(y + z (y, µ), µ))|y=0,µ=0 = (I − E)(DxN(0, 0)Dµz (0, 0) + DµN(0, 0))

= (I − E)DµN(0, 0)

= hw, DµN(0, 0)i

which is nonzero by assumption. Further, writing y = ηv with η ∈ R,

∗ ∗ ∗ DηN(ηv + z (ηv, µ), µ) = DxN(ηv + z (ηv, µ), µ)(v + Dyz (ηv, µ)v)

∗ so that, diﬀerentiating again and using DxN(0, 0) = 0, Dyz (0, 0) = 0,

∗ Dηη((I − E)N(ηv + z (ηv, µ), µ))|η=0,µ=0 = (I − E)DxxN(0, 0)(v, v)

= hw, DxxN(0, 0)(v, v)i

which is nonzero by assumption. It follows that the reduced bifurcation equation satisﬁes the assumptions of the saddle-node bifurcation in one dimension. This concludes the proof.

3.1.3 Saddle-node bifurcation of periodic orbits

A periodic orbit can undergo a saddle-node bifurcation if a Floquet multiplier becomes 1. For a Poincar´e return map on a local cross section, this translates to the existence of a ﬁxed point with eigenvalue 1 for the linearization. We state the bifurcation results in one dimension.

Theorem 1.3. Let I,J ⊂ R be open intervals and f : I × J → R be a C2 map such that

∂ (i) f(x0, µ0) = x0 and ∂x f(x0, µ0) = 1 for some x0 ∈ I, µ0 ∈ J.

∂2 (ii) ζ = ∂x2 f(x0, µ0) < 0,

∂ (iii) η = ∂µ f(x0, µ0) > 0.

2 Then there are ε, δ > 0 and a C function α :(x0 − ε, x0 + ε) → (µ0 − δ, µ0 + δ) such that

0 00 (i) α(x0) = µ0, α (x0) = 0, α (x0) = −ηζ > 0.

−1 (ii) Each x ∈ (x0 − ε, x0 + ε) is a ﬁxed point of f(·, α(·)), i.e., f(x, α(x)) = x, and α (µ) is exactly

the ﬁxed point set of f(·, µ) in (x0 − ε, x0 + ε) for µ ∈ (µ0 − δ, µ0 + δ). 3.1. THE SADDLE-NODE BIFURCATION 55

(iii) For each µ ∈ (µ0, µ0 +δ), there are exactly two ﬁxed points x1(µ) < x2(µ) of f(·, µ) in (x0 −ε, x0 +ε) with ∂ ∂ (x (µ), µ) > 1, 0 < (x (µ), µ) < 1, ∂x 1 ∂x 2

α(xi(µ) = µ, i = 1, 2.

(iv) f(·, µ) does not have ﬁxed points in (x0 − ε, x0 + ε) for each µ ∈ (µ0 − δ, µ0).

Remark 1.1. Similar statements hold for other signs, positive or negative, of η and ζ.

Proof. Consider the function g(x, µ) = f(x, µ) − x.

∂ ∂ Observe that ∂µ g(x0, µ0) = ∂x f(x0, µ0) > 0. Therefore, by the implicit function theorem, there are 2 ε, δ > 0 and a C function α :(x0 − ε, x0 + ε) → J such that g(x, α(x)) = 0 for each x ∈ (x0 − ε, x0 + ε). and there are no other zeros of g in (x0 − ε, x0 + ε) × (µ0 − δ, µ0 + δ). A direct calculation shows that 00 α satisfies the first statement. Since α (x0) > 0, the third and fourth statement are satisfied for ε and δ sufficiently small.

Example. Consider x˙ = µ(1 + cos(2πt)) + x2 − cos(2πt)2 − 2π sin(2πt), (1.5)

where µ is a real parameter. When µ = 0, the equation has a periodic solution cos(2πt). Subtract this from the variable and set z(t) = x(t) − cos(2πt). Then

z˙ = µ(1 + cos(2πt)) + 2 cos(2πt)z + z2

We perform a number a transformations. First we remove the linear term from the equation. Note that z˙ = 2 cos(2πt)z is solved by z(t) = esin(2πt)/πz(0). Now let z(t) = esin(2πt)/πu(t). Then

u˙ = µ(1 + cos(2πt))e− sin(2πt)/π + esin(2πt)/πu2.

Next we change the coeﬃcient of the quadratic term to a constant, by a transformation of the form 2 ˙ sin(2πt)/π u(t) = w(t) + β(t)w(t) . Plugging into the equation, we see that we can take β = e − c0 for R 1 sin(2πt)/π c0 = 0 e dt > 0. Then

2 3 w˙ = µq(t) − 2µq(t)β(t)w + (c0 + O(µ))w + O(w ), (1.6)

where q(t) = (1 + cos(2πt))esin(2πt)/π. Let x 7→ P (x, µ) be the time 2π map. To ﬁnd a periodic solution, we look for zeros of h(x, µ) = P (x, µ) − x. When µ = 0, (1.6) has w = 0 as periodic solution, which corresponds to cos(2πt) for (1.5). This periodic solution is nonhyperbolic, so that P (0, 0) = 0 and

DxP (0, 0) = 1. So h(0, 0) = 0 and Dxh(0, 0) = 0. ∂ Let φ(t, x, µ) be a solution ofx ˙ = f(x, t, µ). Then ∂µ φ(t, x, µ) is a solution of ∂ ∂ v˙ = f(φ(t, x, µ), t, µ)v + (φ(t, x, µ), t, µ) ∂x ∂µ 56 CHAPTER 3. LOCAL BIFURCATIONS

with v(0) = 1. Applying this to (1.6) shows that Z 1 − sin(2πt)/π Dµh(0, 0) = (1 + 2 cos(2πt))e dt > 0. 0

3.2 The Hopf bifurcation

The Hopf bifurcation is named after Eberhard Hopf, who wrote a basic paper [22] on it, after earlier work by Henri Poincar´eand by Aleksandr Andronov, who treated the bifurcation in the plane. The Hopf bifurcation explains the creation of oscillatory behavior when an equilibrium changes stability. The book [34] by Jerrold Marsden and Marjorie McCracken oﬀers an extensive treatment of the bifurcation, while Yuri Kuznetsov [29] discusses computational aspects.

Example. Consider the system x˙ = µx + y − xf(r), (2.1) y˙ = −x + µy − yf(r), where r = px2 + y2, f(r) and f 0(r) are continuous and f(0) = 0, f(r) > 0 for r > 0. We claim that

(i) for µ ≤ 0 the origin is a stable spiral, its basin of attraction being all of R2, (ii) for µ > 0 the origin is unstable and there is a stable limit cycle whose radius increases from zero as µ increases from zero. Its basin of attraction is R2 \{0}. This easily follows from the polar form of the equations, r˙ = r(µ − f(r)), (2.2) θ˙ = −1.

3.2.1 The planar Hopf bifurcation

We will study the Hopf bifurcation in the plane. Consider a family of diﬀerential equationsx ˙ = f(λ, x), 2 λ ∈ R, x ∈ R ,. Assume that at λ = λ0 there is an equilibrium u0 for which Dxf(λ0, u0) has complex conjugate eigenvalues ±βi. By the implicit function theorem, there exists a curve u(λ) of equilibria for

λ near λ0, u(λ0) = u0. Denote the eigenvalues of Dxf at u(λ) by α(λ) ± β(λ)i.

A parameter dependent coordinate change y = x − u(λ), and a parameter change µ = λ − λ0, transforms the family of diﬀerential equations to one for which the origin is the equilibrium, for parameter values near 0. By a linear coordinate transformation, the linearization Df(λ, 0) is put into normal form ! α −β Dxf(0, 0) = β α

for λ near 0, where α, β are smooth functions of λ with α(0) = 0, β(0) 6= 0. Note that Dxf(λ, 0) has eigenvalues α ± βi, which are on the imaginary axis for λ = 0. 3.2. THE HOPF BIFURCATION 57

Figure 2.1: The left panel shows the bifurcation diagram for the supercritical Hopf bifurcation: as the parameter increases, the equilibrium becomes unstable and a stable limit cycle appears. The right panel shows the subcritical Hopf bifurcation: as the parameter increases, the equilibrium becomes unstable and unstable periodic orbit disappears.

Theorem 2.1. Consider a family of diﬀerential equations u˙ = f(u, λ), λ ∈ R, u ∈ R2, with the following properties.

(i) for λ near 0, there is an equilibrium at the origin with ! α −β Duf(0, λ) = β α

for smooth functions α = α(λ), β = β(λ) with α(0) = 0, β(0) 6= 0. Note that Duf(0, 0) has complex conjugate eigenvalues ±βi,

(ii) α0(0) > 0.

(iii) with ! αx − βy + g(x, y, λ) f(x, y, λ) = , (2.3) βx + αy + h(x, y, λ) the number (the ﬁrst Lyapunov value) −1 a = ((g + g + h + h )β + g (g + g ) 16β xxx xyy xxy yyy xy xx xx

−hxy(hxx + hyy) − gxxhxx + gyyhyy)

computed at (x, y, λ) = (0, 0, 0), is nonzero.

Then the bifurcation is as follows, depending on the sign of a:

(i) If a > 0, there is a supercritical Hopf bifurcation: for λ ≤ 0 the origin is stable and attracts all nearby orbits, while for λ > 0 the origin is unstable and there is a stable periodic orbit that attracts all nearby orbits except for the origin. 58 CHAPTER 3. LOCAL BIFURCATIONS

(ii) If a < 0, there is a subcritical Hopf bifurcation: for λ < 0 there is an unstable periodic orbit; the stable equilibrium at the origin attracts only points inside the periodic orbit, for λ ≥ 0 the origin is unstable and no other orbits stay close to the origin.

Proof. Since the linearization at the origin is in Jordan normal form, the diﬀerential equations are of form x˙ = α(λ)x − β(λ)y + g(x, y, λ), (2.4) y˙ = β(λ)x + α(λ)y + h(x, y, λ), where the functions g and h together with their ﬁrst order derivatives with respect to (x, y) vanish at the origin. Note α(0) = 0, β(0) 6= 0, and by assumption, α0(0) 6= 0. For the analysis it is convenient to use polar coordinates. By a change to polar coordinates we obtain the family

r˙ = α(λ)r + p(r, θ, λ), (2.5) θ˙ = β(λ) + q(r, θ, λ), where

p(r, θ, λ) = g(r cos(θ), r sin(θ), λ) cos(θ) + h(r cos(θ), r sin(θ), λ) sin(θ), 1 q(r, θ, λ) = (h(r cos(θ), r sin(θ), λ) cos(θ) − g(r cos(θ), r sin(θ), λ) sin(θ)) . r Since g and h and their first order derivatives vanish at the origin, the system is smooth. The equilibrium at the origin has been blown up to a circle {0} × T in the phase space R × T. We study the Poincaré first return map on R × {0}. This is made easier by applying a coordinate change that brings the system into the normal form given in Lemma 2.1 below. With this lemma we have

3 4 r˙ = αr − Re (c1)r + O(r ), ˙ 2 3 θ = β − Im (c1)r + O(r ).

A time reparametrization brings this to

r˙ = (α/β)r − ar3 + O(r4), θ˙ = 1,

2 with a = Re (c1) − (α/β )Im (c1) a smooth function of the parameter. For λ = 0, a = Re (c1). Note that the higher order terms are functions of r, θ and the parameter λ. Going through the details of the normal form computations shows that a is given by the formula in the statement of the Hopf bifurcation theorem. To ﬁnd the periodic orbit, we rescale r to make the coeﬃcient of the third order term equal to ±1.

Considering one case where c1 > 0, we obtain

ρ˙ = (α/β)ρ − ρ3 + O(|ρ|4). 3.2. THE HOPF BIFURCATION 59

This deﬁnes ρ as function of θ. Write the solution that starts at ρ0 in Taylor series ρ(θ, ρ0) = u1(θ)ρ0 + 2 3 4 3 4 u2(θ)ρ0 + u3(θ)ρ0 + O(|ρ0| ). Thenρ ˙ = (α/β)ρ − ρ + O(|ρ| ) yields

2 3 3 4 0 0 2 0 3 4 (α/β)u1ρ0 + (α/β)u2ρ0 + ((α/β)u3 − u1)ρ0 + O(|ρ0|) = u1ρ0 + u2ρ0 + u3ρ0 + O(|ρ0| ), so that

0 u1 = (α/β)u1, 0 u2 = (α/β)u2, 0 3 u3 = (α/β)u3 − u1.

Solving with u1(0) = 1, u2(0) = 0, u3(0) = 0, we get

1 − e2(α/β)θ u (θ) = e(α/β)θ, u (θ) = 0, u (θ) = e(α/β)θ . 1 2 3 2(α/β)

The return map ρ0 7→ ρ(2π, ρ0) therefore has the form

2πα/β 2πα/β 3 4 ρ(2π, ρ0) = e ρ0 − e (2π + O(α))ρ0 + O(|ρ0| ).

This map is easily analyzed: taking the displacement ρ(2π, ρ0) − ρ0 and dividing the expression for it by

ρ0, we ﬁnd a saddle-node bifurcation.

We discuss a normal form result for the Hopf bifurcation in the plane.

Lemma 2.1. A smooth parameter dependent coordinate change brings (2.3) into normal form, which in polar coordinates has an expression

3 4 r˙ = αr − Re (c1)r + O(r ), ˙ 2 3 θ = β − Im (c1)r + O(r ).

Here c1 is a smooth function of the parameter λ.

Proof. Take the diﬀerential equation (2.3) with the linear part in real Jordan normal form. Consider a complex coordinate z = x + i y. Observe that

z˙ = (α + iβ)z + r(z, z,¯ λ) (2.6) with r(z, z,¯ λ) = O(|z|2) as z → 0. We start with a coordinate transformation that removes quadratic terms. Expand (2.6) as g g z˙ = νz + 20 z2 + g zz¯ + 02 z¯2 + O(|z|3), (2.7) 2 11 2 with ν = α + iβ. Consider a coordinate w given by h h z = w + 20 w2 + h ww¯ + 02 w¯2 2 11 2 60 CHAPTER 3. LOCAL BIFURCATIONS

The inverse change of variable is given by h h w = z − 20 z2 − h zz¯ − 02 z¯2 + O(|z|3). 2 11 2 Therefore

w˙ =z ˙ − h20zz˙ − h11(z ˙z¯ + zz¯˙) − h02z¯z¯˙ + ··· g g = νz + ( 20 − νh )z2 + (g − νh − νh¯ )zz¯ + ( 02 − νh¯ )¯z2 + ··· 2 20 11 11 11 2 02 1 1 = νw + (g − νh )w2 + (g − νh¯ )ww¯ + (g − (2¯ν − ν)h )w ¯2 + O(|w|3). 2 20 20 11 11 2 02 02 Setting g g g h = 20 , h = 11 , h = 02 20 ν 11 ν¯ 02 2¯ν − ν kills the quadratic terms in (2.7) (close to the bifurcation value λ = 0). The next step is removing as many third order terms as possible, by the same technique. That is, start with an expansion g g g g z˙ = νz + 30 z3 + 21 z2z¯ + 12 zz¯2 + 03 z¯3 + O(|z|4). (2.8) 6 2 2 6 We claim that the introduction of a coordinate h h h h z = w + 30 w3 + 21 w2w¯ + 12 ww¯2 + 03 w¯3 6 2 2 6 transforms the equation into the desired form

2 4 w¯ = νw + c1w w¯ + O(|w| ).

The inverse transformation is h h h h w = z − 30 z3 − 21 z2z¯ − 12 zz¯2 − 03 z¯3 + O(|z|4). 6 2 2 6 Therefore, h h h h w˙ =z ˙ − 30 z2z˙ − 21 (2zz¯z˙ + z2z¯˙) − 12 (z ˙z¯2 + 2zz¯z¯˙) − 03 z¯2z¯˙ + ··· 2 2 2 2 g h g h g h g h = νz + ( 30 − ν 30 )z3 + ( 21 − νh − ν¯ 21 )z2z¯ + ( 12 − ν 12 − νh¯ )zz¯2 + ( 03 − ν¯ 03 )¯z3 + ··· 6 2 2 21 2 2 2 12 6 2 1 1 1 = νw + (g − 2νh )w3 + (g − (ν +ν ¯)h )w2w¯ + (g − 2¯νh )ww¯2+ 6 30 30 2 21 21 2 12 12 1 (g + (ν − 3¯ν)h )w ¯3 + O(|w|4). 6 03 03 Setting g g g h = 30 , h = 12 , h = 03 30 2ν 12 2¯ν 03 3¯ν − ν 2 annihilates all cubic terms except the w w¯ term. Here, setting h21 = g21/(ν +ν ¯) does not work, since the denominator vanishes for λ = 0. We set

h21 = 0. 3.2. THE HOPF BIFURCATION 61

The above transformations bring the diﬀerential equation into the form

2 4 z˙ = (α + i β)z − c1z z¯ + O(|z| ), as z → 0. In polar coordinates we have

3 4 r˙ = αr − Re (c1)r + O(r ), ˙ 2 3 θ = β − Im (c1)r + O(r ).

Remark 2.1. Starting with the expression (2.7), following the calculations shows that g g (ν +ν ¯) |g |2 |g |2 g c = 20 11 + 11 + 02 + 21 . 1 2|ν|2 ν 2(2ν − ν¯) 2 Higher dimensional versions of the Hopf bifurcation theorem are obtained from center manifold re- ductions. To verify the conditions one has to derive asymptotic expansions of the diﬀerential equation on the center manifold. Start with a diﬀerential equation

x˙ = αx − βy + g(x, y, z, λ), y˙ = βx + αy + h(x, y, z, λ), (2.9) z˙ = Bz + i(x, y, z, λ), for (x, y, z) ∈ R × R × Rp, B a p × p matrix depending on λ, α and β depending on λ with α(0) = 0, and g, h, i higher order terms. Then there is a local center manifold z = V (x, y, λ), tangent to {x, y} = 0 at the origin. Computing for λ = 0, local invariance means

BV (x, y, 0) + i(x, y, V (x, y, 0), 0) = DxV (x, y, 0)(−βy + g(x, y, V (x, y, 0), 0))+

DyV (x, y, 0)(βx + h(x, y, V (x, y, 0), 0)).

Example. Motivated by studies of Alan Turing [47] on models of biological cells that are inert in itself but pulse when in interaction via diffusion, Stephen Smale studied simple models of differential equations exhibiting this phenomenon. In his study, included in [34], a single cell is modeled by a vector field

x˙ = R(x)

4 on P = {x = (x1, x2, x3, x4) ∈ R ; xi ≥ 0}, with the following property. The diﬀerential equation x˙ = R(x) on P is globally asymptotically stable and possesses a unique equilibriumx ¯ that attracts every solution in P . Two interacting cells are modeled by diﬀerential equations

x˙1 = R(x1) + µ(x2 − x1), (2.10) x˙2 = R(x2) + µ(x1 − x2), 62 CHAPTER 3. LOCAL BIFURCATIONS

on P × P , where   µ1 0 0 0    0 µ2 0 0  µ =   .  0 0 µ 0   3  0 0 0 µ4 For the choice of R is Smale’s work, there exists µ, so that (2.10) is a global oscillator. That is, it has a nontrivial periodic solution that attracts all solutions in P × P except for points on the two stable dimensional manifold of the equilibrium (¯x, x¯). The example is related to the Hopf bifurcation as one can let µ move from zero to the values in Smale’s result to create the oscillator. Observe that the statements provide global descriptions of the dynamics. Turing’s work led to the study of pattern formation in reaction-diﬀusion equations such as ∂ u = d 4 u + f(u, v), ∂t u ∂ v = d 4 v + g(u, v), ∂t v for functions (x, y, t) 7→ u(x, y, t), v(x, y, t). These model density functions for chemicals that diﬀuse and

interact. Different constants du, dv give rise to different diffusion speeds of the chemicals.

Assume an equilibrium at (u0, v0): f(u0, v0) = g(u0, v0) = 0. Turing instability is the effect that without diffusion the origin is a stable equilibrium for the ordinary differential equations, but diffusion

causes instability. To study linear instability we linearize the equations about (u0, v0) to obtain ∂ u = du 4 u + au + bv, ∂t (2.11) ∂ v = d 4 v + cu + dv, ∂t v where Df(u0, v0) = a b and Dg(u0, v0) = c d . An asymptotically stable equilibrium for the system without diﬀusion occurs if ad − bc > 0 and a + d < 0 (negative trace and positive determinant of a 2 × 2 matrix indicates eigenvalues with negative real parts). To investigate stability for the system with diﬀusion, plug in terms u˜ =ue ˆ i (k1x+k2y)+σt, v˜ =ve ˆ i (k1x+k2y)+σt. from a Fourier series expansion, where k = (k1, k2) is a wave-vector. Substituting into (2.11) leads to the dispersion relation for σ,

2 2 2 2 2 σ + σ(du|k| + dv|k| − a − d) + (du|k| − a)(dvk − d) − bc = 0.

For instability, one of the roots of this equation must have positive real part. One checks that this is the case if 2 2 2 h(k ) = (du|k| − a)(dv|k| − d) − bc < 0. For any wave vector k for which this holds, the corresponding mode will grow. Now h(x) is minimal at 1 a d xmin = + 2 du dv 3.2. THE HOPF BIFURCATION 63

and has minimal value 1 2 h(xmin) = (ddu + adv) + ad − bc. 4dudv a d 2 So an instability occurs if + > 0, so that xmin > 0 (and the corresponding |k| is positive) and dv dv h(xmin) < 0. This local bifurcation, in this setting, is called a Turing bifurcation.

Example. Van der Pol’s oscillator is given by the second order diﬀerential equation

x¨ − (2λ − x2)x ˙ + x = 0

with a real parameter λ. The van der Pol oscillator was originally proposed by the electrical engineer Balthasar van der Pol. It models an oscillator with nonlinear damping, of diﬀerent signs for small and large |y|. The van der Pol equations model nonsinusoidal oscillations that van der Pol called relaxation oscillations. It is equivalent to the planar system

x˙ = y, (2.12) y˙ = −x + 2λy − x2y √ The eigenvalues of the linearization about the equilibrium at the origin are λ±i 1 − λ2. When λ crosses zero, the origin changes stability. We analyze the bifurcation directly. Going to polar coordinates (r, θ) and using the angle θ to parameterize orbits, we obtain

dr 2λr sin2(θ) − r3 cos2(θ) sin2(θ) = . (2.13) dθ 1 + 2λ sin(θ) cos(θ) − r2 cos3(θ) sin(θ)

Expanding this equation in a Taylor series for small r and λ, we obtain dr 1 = (λ − λ cos(2θ) + O(λ2))r − (1 − cos(4θ) + O(λ))r3 + O(r4). (2.14) dθ 8 We now apply transformations to simplify the equation. Note that the linear equation dr = (λ − λ cos(2θ))r (2.15) dθ is solved by λθ − λ sin(2θ) r(θ) = e e 2 r(0).

Change to a new variable ρ given by − λ sin(2θ) r = ρe 2 .

Then the linear equation becomesρ ˙ = λρ and (2.14) becomes

dρ 1 = (λ + O(λ2))ρ − (1 − cos(4θ) + O(λ))ρ3 + O(ρ4). dθ 8

3 da 1 As for the cubic terms, let η = ρ + a(θ)ρ with dθ = 8 cos(4θ). This gives 1 η˙ = (λ + O(λ2))η − ( + O(λ))η3 + O(η4). 8 64 CHAPTER 3. LOCAL BIFURCATIONS

Note that the η2 term is not present. The Poincar´ereturn map, or time 2π map, is the map 1 P (η ) = η + (λ + O(λ2))η − ( + O(λ))η3 + O(η4) 0 0 0 8 0 0

for small η0, λ. Fixed points, other than η0 = 0, are solutions to 1 λ + O(λ2) − ( + O(λ))η2 + O(η3) = 0. 8 0 0 √ √ 1 2 3 This gives a bifurcation curve λ = 8 η0 + O(η0) or η0 = 8 λ + O(λ) for η0, λ > 0 small.

3.2.2 Higher dimensional Hopf bifurcation

Higher dimensional versions of the Hopf bifurcation theorem are obtained from center manifold reduc- tions. To verify the conditions one has to derive asymptotic expansions of the diﬀerential equation on the center manifold. Start with a diﬀerential equation x˙ = αx − βy + g(x, y, z, λ), y˙ = βx + αy + h(x, y, z, λ), (2.16) z˙ = Bz + i(x, y, z, λ),

for (x, y, z) ∈ R × R × Rp, B a p × p matrix depending on λ, α and β depending on λ with α(0) = 0, and g, h, i higher order terms. Then there is a local center manifold z = V (x, y, λ), tangent to {x, y} = 0 at the origin. Computing for λ = 0, local invariance means

BV (x, y, 0) + i(x, y, V (x, y, 0), 0) = DxV (x, y, 0)(−βy + g(x, y, V (x, y, 0), 0))+

DyV (x, y, 0)(βx + h(x, y, V (x, y, 0), 0)).

This formula allows to compute the Taylor expansion up to second order for V , plugging this into the equations forx ˙ andy ˙ allows to calculate the first Lyapunov coefficient and thus to check the conditions of the Hopf bifurcation theorem. See [29] for an explicit formula for the first Lyapunov exponent in terms of the original differential equation. It is possible to find the periodic solutions by solving an equation in a space of periodic functions. We include this line of reasoning below, first for the planar case. Consider a smooth differential equation

x˙ = A(α)x + f(α, x) (2.17)

2 for x ∈ R and α ∈ R. Here A(α) is a 2 × 2 matrix. We assume that f(0, 0) = 0, Dxf(0, 0) = 0 and that the eigenvalues of A(0) are iν with ν > 0. Further, assume that the eigenvalues of A(α) are µ(α) ± iν(α) with Dµ(0) 6= 0.

Theorem 2.2. For α and a parameter a near 0, there are smooth functions α∗(a) with α∗(0) = 0, ω∗(a) with ω∗(0) = 2π, and x∗ ∗ (a), so that x∗(a)(t) is ω∗(a)-periodic and a solution to (2.17). One has

∗ x1(a)(t) = a cos(ω(a)t) + o(|a|), (2.18) ∗ x2(a)(t) = −a sin(ω(a)t) + o(|a|) (2.19) as a → 0. Any small periodic solution of (2.17) is of the form x∗(a) for some small a. 3.2. THE HOPF BIFURCATION 65

Proof. Use polar coordinates: with x1 = r cos(ϕ) and x2 = r sin(ϕ), (2.17) becomes

ϕ˙ = β(α) + Φ(α, ϕ, r), (2.20) r˙ = αr + R(α, ϕ, r) (2.21) with Φ(α, ϕ, 0) = 0, R(α, ϕ, 0) = 0, DrR(α, ϕ, 0) = 0 and these functions are 2π-periodic in ϕ. Elimi- nating t one obtains

dr α = r + R∗(α, ϕ, r) (2.22) dϕ β(α)

∗ ∗ ∗ with R (α, ϕ, 0) = 0, DrR (α, ϕ, 0) = 0 and R is 2π-periodic in ϕ. The 2π-periodic solutions of (2.22) near r = 0 yield periodic solutions of (2.17) near x = 0 and conversely. We seek solutions r(θ) of (2.22) with r(2π) = r(0) = a. Note that such a solution is 2π-periodic. Apply the variation of constants formula to (2.22). Then r(2π) = r(0) = a yields an equation of the form

(1 − e−2π/β(α))a = h(α, a) (2.23) for a function h with h(α, 0) = 0. Dividing by a gives

1 − e−2π/β(α) − g(α, a) = 0 (2.24) for a function g with g(α, 0) = 0 and Dαg(0, 0) 6= 0. Apply the implicit function theorem to determine α = α∗(a) satisfying (2.24) and α∗(0) = 0. One obtains ω∗(a) from the equation for ϕ in (2.20).

In Theorem 2.2 we used the implicit function theorem to discuss the Hopf bifurcation in Rn for n = 2. The same ideas yield a similar theorem for arbitrary n. In fact, suppose

x˙ = A(α)x + f(α, x) (2.25)

n is a smooth diﬀerential equation where x ∈ R , α ∈ R, A(α) is an n×n matrix, f(α, 0) = 0, Dxf(α, 0) = 0 and ! B(α) 0 A(α) = (2.26) 0 C(α), ! α β(α) B(α) = (2.27) −β(α) α, with β(0) = 1 and (eC(α)2π − I)−1 exists.

Theorem 2.3. Assume the above hypotheses. For α and a parameter a near 0, there are smooth functions α∗(a) with α∗(0) = 0, ω∗(a) with ω∗(0) = 2π, and x∗ ∗ (a), so that x∗(a)(t) is ω∗(a)-periodic 66 CHAPTER 3. LOCAL BIFURCATIONS and a solution to (2.25). One has   a cos(ω(a)t)    −a sin(ω(a)t)    ∗  0  x (a)(t) =   + o(|a|) (2.28)  .   .    0 (2.29) as a → 0. Any small periodic solution of (2.17) is of the form x∗(a) for some small a.

Proof. Let x = (r cos ϕ, −r sin ϕ, rx˜) in (2.25) and eliminate t to obtain

dr α = r + R(α, ϕ, r, x˜), dϕ β(α) dx˜ 1 = C(α)˜x + X˜(α, ϕ, r, x˜), dϕ β(α) where the functions R, X˜ are 2π-periodic in ϕ, vanishing at r = 0 and Dr,x˜R, Dr,x˜X˜ also vanish at r = 0. The proof of Theorem 2.2 can now be repeated almost verbatim to obtain the conclusion stated in the theorem.

The discrete time analogue of the Hopf bifurcation, the unfolding of a diffeomorphism with a fixed point and a linearization with complex conjugate eigenvalues on the unit circle, is called the Neimark- Sacker bifurcation after independent work on it by Juri Neimark and Robert Sacker. In its unfolding an invariant circle appears. In terms of bifurcations of periodic orbits of differential equations, an invariant torus appears in a bifurcation where complex conjugate Floquet multipliers move through the unit circle.

3.3 The period-doubling bifurcation

A periodic solution of a family of differential equations can loose stability if a Floquet multiplier moves through −1 and becomes larger than 1 in modulus, while the moduli of the other Floquet multipliers remain inside the unit disc. Because the critical Floquet multiplier is negative, the center manifold near the periodic solution is a Möbiusband. As we shall see, periodic orbits of roughly double the period can occur in an unfolding. Thus a stable periodic orbit looses stability and becomes of saddle type, while the stability is inherited by a periodic orbit of a doubled period. The analysis can be done by studying the Poinca´rereturn map on a local cross section, transforming the bifurcation problem to one for diffeomorphisms. We treat the bifurcation for one dimensional maps, corresponding to flows restricted to a center manifold.

3 Theorem 3.1. Let I,J be open intervals and f : I × J → R be a C map such that f(x0, µ0) = x0 and Dxf(x0, µ0) = −1 for some x0 ∈ I, µ0 ∈ J. Assume 3.3. THE PERIOD-DOUBLING BIFURCATION 67

Figure 3.1: The period-doubling bifurcation.

∂2 1 ∂ ∂2 (i) η = ∂µ∂x f(x0, µ0) + 2 ∂µ f(x0, µ0) ∂x2 f(x0, µ0) < 0,

2 ∂3 ∂2 (ii) ζ = −2 ∂x3 f(x0, µ0) − 3 ∂x2 f(x0, µ0) < 0.

3 Then there are ε, δ > 0 and C functions ξ :(µ0 −δ, µ0 +δ) → R with ξ(µ0) = x0 and α :(x0 −ε, x0 +ε) → 0 00 R with α(x0) = µ0, α (x0) = 0 and α (x0) = −2η/ζ > 0, such that

(i) f(ξ(µ), µ) = ξ(µ) and ξ(µ) is the only ﬁxed point in (x0 − ε,0 +ε) for µ ∈ (µ0 − δ, µ0 + δ).

(ii) ξ(µ) is an attracting ﬁxed point of f(·, µ) for µ ∈ (µ0 − δ, µ0) and is a repelling ﬁxed point for

µ ∈ (µ0, µ0 + δ).

(iii) For each µ ∈ (µ0, µ0 + δ), the map f(·, µ0) has, in addition to the repelling ﬁxed point ξ(µ), exactly

two period-two points x1(µ), x2(µ) in the interval (x0 − ε, x0 + ε). Moreover, α(xi(µ)) = µ and

xi(µ) → x0 as µ → µ0 for i = 1, 2.

(iv) For µ ∈ (µ0 − δ, µ0], there are no period-two points in (x0 − ε, x0 + ε).

Remark 3.1. Similar statements hold for other signs, positive or negative, of η and ζ.

Proof. Since ∂ f(x , µ ) = −1, ∂x 0 0 and thus diﬀerent from 1, we can apply the implicit function theorem to obtain a diﬀerentiable function

ξ with f(ξ(µ), µ) = ξ(µ), for µ close to µ0 and with ξ(µ0) = x0. This proves the ﬁrst statement. Diﬀerentiating f(ξ(µ), µ) = ξ(µ) with respect to µ gives

d ∂ ∂ f(ξ(µ), µ) = f(ξ(µ), µ) + f(ξ(µ), µ)ξ0(µ) = ξ0(µ) dµ ∂µ ∂x and hence ∂ f(ξ(µ), µ) 0 ∂µ ξ (µ) = ∂ . 1 − ∂x f(ξ(µ), µ) 68 CHAPTER 3. LOCAL BIFURCATIONS

0 1 ∂ Thus ξ (µ0) = 2 ∂µ f(x0, µ0). Since

2 2 d ∂ ∂ ∂ 0 f(ξ(µ), µ) = f(x0, µ0) + 2 f(x0, µ0)ξ (µ0) dµ µ=µ0 ∂x ∂µ∂x ∂x we get d ∂ ∂2 1 ∂ ∂2 f(ξ(µ), µ) = f(x0, µ0) + f(x0, µ0) 2 f(x0, µ0) dµ µ=µ0 ∂x ∂µ∂x 2 ∂µ ∂x and the second assumption yields the second statement. For the third and fourth statements consider the change of variables y = x − ξ(µ) and the function

g(y, µ) = f(f(y + ξ(µ), µ), µ) − ξ(µ).

Observe that the ﬁxed points of f(f(·, µ), µ) correspond to solutions of g(y, µ) = y. Moreover, ∂ g(0, µ) = 0, g(0, µ ) = 1. ∂y 0 A direct computation shows that ∂2 g(0, µ ) = 0, ∂y2 0

so that the graph of the second iterate of f(·, µ0) is tangent to the diagonal at (x0, µ0) with vanishing second derivative. A direct calculation also shows that, by the third assumption, the third derivative does not vanish: ∂3 ∂3 ∂2 2 g(0, µ ) = −2 f(x , µ ) − 3 f(x , µ ) < 0. ∂y3 0 ∂x3 0 0 ∂x2 0 0 Therefore, 1 g(y, µ ) = y + ζy3 + o(y3) (3.1) 0 6

for y → 0. Since ξ(µ) is a fixed point of f(·, µ), we have that g(0, µ) = 0 in an interval around µ0. Therefore there is a differentiable function h such that g(y, µ) = yh(y, µ). To find the period-two points of f(·, µ) different from ξ(µ) we must solve h(y, µ) = 1. From (3.1) we obtain 1 h(y, µ ) = 1 + ζy2 + o(y2) 0 6 as y → 0. So ∂ ∂2 1 h(0, µ ) = 0, h(0, µ ) = 0, (0, µ ) = ζ. 0 ∂y 0 ∂y2 0 3 On the other hand,

∂ 1 ∂ ∂2 h(0, µ0) = lim g(y, µ0) = g(0, µ0) = ∂µ y→0 y ∂µ ∂µ∂y d ∂ d ∂ 2 f(f(ξ(µ), µ), µ)| = f(ξ(µ), µ) | = −2η dµ ∂x µ=µ0 dµ ∂x µ=µ0

By the implicit function theorem, since η 6= 0, there is ε > 0 and a diﬀerentiable function β :(−ε, ε) → R such that h(y, β(y)) = 1 for |y| < ε and β(0) = µ0. Diﬀerentiating h(y, β(y)) with respect to y, we obtain 3.4. THE BOGDANOV-TAKENS BIFURCATION 69

that β0(0) = 0. The second diﬀerentiation yields β00(0) = ζ/(6η) > 0. Therefore β(y) > 0 for y 6= 0 and

the new period-two orbit appears only for µ > µ0.

Note that since g(·, µ) has three ﬁxed points near x0 for µ close to µ0, and the middle one, ξ(µ), is unstable, the other two must be stable. In fact, a direct calculation shows that ∂ ∂ 1 ∂2 ∂3 1 g(y, β(y)) = g(0, µ ) + g(0, µ )y + g(0, µ )y2 + o(y2) = 1 + ζy2 + o(y2) ∂y ∂y 0 2 ∂y2 0 ∂y3 0 6 for y → 0. Since ζ < 0, the period-two orbit is stable.

3.4 The Bogdanov-Takens bifurcation

In the previous sections local bifurcations of codimension one were treated. The saddle-node bifurcation of an equilibrium involved an eigenvalue 0, the Hopf bifurcation involved two complex conjugate eigenvalues. For periodic orbits, there are the saddle-node bifurcation with a Floquet multiplier 1, the period-doubling bifurcation with Floquet multiplier −1 and the Hopf bifurcation with complex conjugate Floquet multipliers on the unit circle. Local bifurcations of higher codimension arise if nondegeneracy conditions are violated (for instance, the relevant second order derivative for the saddle-node does vanish; the codimension two bifurcation is the cusp), or if the nonhyperbolic part is higher dimensional. We will consider, as a single example, the unfolding of a diﬀerential equationu ˙ = f(u) in the plane with an equilibrium at the origin f(0, 0) = 0 and a double zero eigenvalue for the linearization: ! 0 1 Df(0) = . 0 0

Note that Df(0, 0) is nilpotent, but not semi-simple. This is called the Bogdanov-Takens bifurcation after the mathematicians Rifkat Bogdanov and Floris Takens who studied it independently. The analysis starts with the derivation of a normal form. Under certain genericity assumptions, there exist smooth parameter dependent coordinate change and a smooth reparametrization, so that the diﬀerential equations are smoothly equivalent to

x˙ = y, (4.1) 2 3 y˙ = β1 + β2x + x ± xy + O(k(x, y)k ). for (x, y) small, with parameters β1, β2. Take for deﬁniteness the case with a minus sign in front of the xy term in the equation fory ˙. The descriptions in the following are illustrated in Figure 4.1. The next steps are the analysis of the truncated normal form, without the O(k(x, y)k3) term, and the analysis with inclusion of these higher order terms. The analysis of the truncated normal form is already nontrivial. Finding bifurcations of 2 equilibria is however straightforward. Equilibria are located on the x-axis and satisfy β1 + β2x + x = 0. A saddle-node bifurcation occurs along the curves

2 T− = {4β1 − β2 = 0, β2 < 0} 70 CHAPTER 3. LOCAL BIFURCATIONS

Figure 4.1: The bifurcation diagram of the Bogdanov-Takens bifurcation. and 2 T+ = {4β1 − β2 = 0, β2 > 0}.

To the left of the curve, for smaller β1 values, there are two equilibria ! −β ∓ pβ2 − 4β E = 2 2 1 , 0 ± 2

The vertical axis β1 = 0 is a line of parameters for which the linearization about E− has a pair of eigenvalues whose sum is zero. If moreover β2 < 0, a Hopf bifurcation occurs. That is, a Hopf bifurcation occurs along

H = {β1 = 0, β2 < 0}.

Now make a round trip near the Bogdanov-Takens point (β1, β2) = (0, 0), starting in region 1 where there are no equilibria and hence no limit cycles. Entering into region 2 through T− brings a saddle and a node. The node turns into a focus and looses stability across the Hopf bifurcation curve H.A stable periodic orbit exists in region 3 . If we continue clockwise and return to region 1 , the periodic orbit has disappeared. So there must be another bifurcation. It turns out there is a curve

6 P = {β = − β2 + o(β2), β < 0} 1 25 2 2 2 of homoclinic orbits to E+ that completes the bifurcation diagram. For the normal form that is not truncated, ﬁnding curves of saddle-node and Hopf bifurcations involves applying the implicit function theorem. Checking nondegeneracy conditions is straightforward. The analysis of the homoclinic bifurcation is not diﬀerent. In addition one has to prove that no other periodic orbits exist and there are no other bifurcations of periodic orbits. 3.5. EXERCISES 71

3.5 Exercises

Exercise 3.5.1. Formulate and prove a theorem based on the implicit function theorem that can be used to show that a small perturbation of a family of diﬀerential equations with a saddle-node bifurcation has a nearby saddle-node bifurcation. A possible formulation can have the following ﬂavor: suppose x˙ = f(x, µ) with µ ∈ R unfolds a saddle-node bifurcation. Consider a family x˙ = f(x, µ, ν) with f(x, µ, 0) = f(x, µ). Prove that x˙ = f(x, µ, ν) unfolds a saddle node for each small value of ν.

Exercise 3.5.2. Suppose that f : R3 → R2 is given by

f(x, y, λ) = (λ − x2 + xy, −2y + x2 + y2)

and u = (x, y). Show that the diﬀerential equation generated by f satisﬁes the conditions of theorem 1.2 and hence has a saddle-node bifurcation. Draw the phase portrait near (u, λ) = (0, 0) for λ < 0, λ = 0, λ > 0.

Exercise 3.5.3. For the following equations, determine bifurcations and stability of critical elements. Sketch the bifurcation diagram.

(i) y¨ +y ˙ + y(2y2 + λ(λ − 2)) = 0,

(ii) y¨ +y ˙ + eλy − 1 − (y − 1)2.

Exercise 3.5.4. Determine the bifurcation diagram for the phase portrait of the diﬀerential equation

xx¨ + ax˙ 2 = b,

where a and b are real parameters.

Exercise 3.5.5. This exercise considers discrete time dynamical systems, obtained by iterating maps, and center manifolds in this context. Compute the reduced system on the center manifold to third order for the following maps and describe their bifurcations.

2 1 2 (i) f(x, y) = (x + xy − y , 2 y + x ).

3 1 2 (ii) f(x, y) = (x + x + αxy, 2 y − x ) for various α.

1 1 3 (iii) f(x, y) = (y, − 2 x + 2 y − y ).

Hint: write down the equation that the center manifold satisﬁes, and use power series methods.

Exercise 3.5.6. Find center manifolds to second order and reduced systems to third order for the following problems:

(i) x˙ = x2y + αz2, y˙ = −y + x2 + zy, z˙ = z − y2 + xy.

1 2 (ii) (x, y, z) 7→ (−x + yz, − 2 y + x , 2z − xy). 72 CHAPTER 3. LOCAL BIFURCATIONS

Exercise 3.5.7. Treat remaining signs in Theorem 3.1, see Remark 3.1.

Exercise 3.5.8. The Brusselator is a diﬀerential equation in the plane that models autocatalytic reaction

x˙ = 1 − (b + 1)x + ax2y, (5.1) y˙ = bx − ax2y.

with parameters a, b > 0. The variables x, y represent concentrates of two reactants. Do a local bifurcation analysis of equilibria.

Exercise 3.5.9. Consider the two systems

r˙ = λr ± r3, (5.2) θ˙ = 1 + ar2,

written in polar coordinates. Show that the minus sign has a supercritical Hopf bifurcation and the plus sign a subcritical Hopf bifurcation.

Exercise 3.5.10. Consider the predator-prey model

y x˙ = x b − x − , 1 + x (5.3) x y˙ = y − ay , 1 + x where x, y ≥ 0 represent the populations and a, b > 0 are parameters.

(i) Show that a positive ﬁxed point (x∗, y∗), x∗ > 0, y∗ > 0, exists for all a, b > 0.

(ii) Show that a Hopf bifurcation occurs at the positive ﬁxed point if

4(b − 2) a = b2(b + 2)

and b > 2.

Exercise 3.5.11. Garrett Odell considered the predator-prey system

x˙ = x(x(1 − x) − y), (5.4) y˙ = y(x − a),

where x ≥ 0 represents prey population, y ≥ 0 represents predator population, and a ≥ 0 is a control parameter.

(i) Find and classify the ﬁxed points.

(ii) Sketch the phase portrait for a > 1 and show that the predators go extinct.

(iii) Show that a Hopf bifurcation occurs at a = 1/2. 3.5. EXERCISES 73

(iv) Sketch all topologically diﬀerent phase portraits for 0 < a < 1.

Exercise 3.5.12. The second order equation

y¨ +y ˙3 − 2λy˙ + y = 0 is known as Rayleigh’s equation. Convert it into a ﬁrst order system and investigate the Hopf bifurcation at λ = 0. Determine the stability type of the periodic orbit.

Exercise 3.5.13. Show that the equation

x˙ = λx − y + xy2, (5.5) y˙ = x + λy + y3 has a subcritical Hopf bifurcation. Hint: Change to polar coordinates and compute, explicitly, the Poincar´emap deﬁned on the positive x-axis. Recall that Bernoulli’s equation z˙ = a(t)z + b(t)zn+1 is transformed to a linear equation by the change of variables w = z−n.

Exercise 3.5.14. Richard Fitzhugh proposed the following system as a model of nerve impulse trans- mission: x˙ = −x(x − a)(x − 1) − y + qH(t), (5.6) y˙ = b(x − γy), where 0 < a < 1, γ > 0, b > 0 and q are constants, H(t) is the Heaviside function H(t) = 0 for t < 0 and H(t) = 1 for t ≥ 0. In this model, q represents the stimulus, x represents the response and y represents a recovery variable. Suppose that 3 γ ≤ a2 − a + 1 and show that there is a critical value q0 of q such that a Hopf bifurcation occurs at q0. Discuss the stability of the resulting periodic orbit.

Exercise 3.5.15. Consider the diﬀerential equation in polar coordinates, given by

r˙ = r(−2λ2 + 3λr2 − r4), (5.7) θ˙ = 1.

Study periodic solutions for λ near zero.

Exercise 3.5.16. Consider the variational van der Pol equation

u˙ = u − σv − u(u2 + v2), (5.8) v˙ = σu + v − v(u2 + v2) − γ, on the plane, depending on real parameters σ, γ.

(i) Find the locus of saddle-node bifurcations in the (σ, γ)-plane. 74 CHAPTER 3. LOCAL BIFURCATIONS

Figure 5.1: Exercise 3.5.18 treats the Hamiltonian saddle-node bifurcation for planar Hamiltonian systems. Here the bifurcation diagram is shown, with the parameter on the horizontal axis.

(ii) At which points is the saddle-node bifurcation not of codimension one? (I.e. at which points do conditions (i) and (iii) of Theorem 1.2 fail? This involves tedious calculations). q (iii) Analyze the pitchfork bifurcation which takes place at (σ, λ) = ( √1 , 8 ). 3 27 (iv) Find the Hopf bifurcations.

Exercise 3.5.17. Show that the system

x¨ + µx˙ + νx + x2x˙ + x3 undergoes Hopf bifurcations on the lines B1 = {µ = 0, ν > 0} and B2 = {µ = ν, ν < 0}. Show that the former is supercritical and occurs at the ﬁxed point (x, x˙) = (0, 0) while the latter is subcritical and √ occurs simultaneously at the ﬁxed points (x, x˙) = (± −ν, 0).

Exercise 3.5.18. Suppose that u˙ = f(u, λ) is a planar Hamiltonian diﬀerential equation with parameter λ, i.e. with u = (q, p), ∂H ∂H f(q, p, λ) = − , ∂p ∂q for a Hamiltonian function H(q, p, λ).

(i) Prove that if f(u0, λ0) = 0 and the linearization Duf(u0, λ0) has a zero eigenvalue, then this eigenvalue has algebraic multiplicity two. In particular, a planar Hamiltonian system can not have a saddle-node bifurcation.

(ii) Suppose that f(u0, λ0) = 0 and Duf(u0, λ0) has a zero eigenvalue with geometric multiplicity one.

Let v span the kernel of Df(u0, λ0) and let w be a left eigenvector for the eigenvalue 0. Prove the following result: Suppose that

hw, Dλf(u0, λ0)vi= 6 0, hw, Duuf(u0, λ0)(v, v)i= 6 0. 3.5. EXERCISES 75

Then

2 (a) There exists s0 > 0 and a smooth curve γ ∈ R × R so that γ(0) = (u0, λ0) and f(γ(s)) = 0, |s| < s0. 2 (b) The curve γ has a quadratic tangency to R × {λ0} at (u0, λ0). (c) The stability of the equilibrium on γ changes at s = 0.

(iii) Reformulate the hypotheses of the bifurcation result in terms of the Hamiltonian, so that there is no mention of the vectors v, w.

(iv) Discuss this bifurcation for the following system:

q˙ = p, (5.9) p˙ = sin(q) − αq − β.

Exercise 3.5.19. Consider the planar Hamiltonian system ∂ q˙ = H(q, p), ∂p ∂ p˙ = − H(q, p), ∂q where H : R2 → R is a smooth function. Assume H has a nondegenerate critical point at (q, p) = (0, 0), and that the linearized vector ﬁeld about (0, 0) has a complex imaginary pair of eigenvalues on the imaginary axis. Now consider the family of vector ﬁelds depending on a real parameter λ, ∂ ∂ q˙ = H(q, p) + λ H(q, p), ∂p ∂q ∂ ∂ p˙ = − H(q, p) + λ H(q, p). ∂q ∂p (i) Find the sheet of periodic orbits near (0, 0, 0) in (λ, q, p) space by applying Theorem 2.2.

(ii) Formulate and prove a corresponding theorem for a sheet of periodic orbits in Hamiltonian systems in Rn, by applying Theorem 2.3. Exercise 3.5.20. Consider a family of diﬀerential equations

x˙ = fλ(x) (5.10) on Rn, depending on a real parameter λ. Let Γ be a ﬁnite group with a linear action on Rn (each γ ∈ Γ is given by a matrix) and assume that (5.10) is Γ-equivariant:

fλ(γx) = γfλ(x) for γ ∈ Γ.

(i) Let ϕt denote the ﬂow generated by (5.10). Show that γϕt(x) is a solution for each γ ∈ Γ. 76 CHAPTER 3. LOCAL BIFURCATIONS

n For a point x ∈ R , the subgroup of Γ that ﬁxes x is called the isotropy subgroup Σx of x:

Σx = {γ ∈ Γ; γx = x}.

Associated to a subgroup Σ of Γ, its ﬁxed point subspace Fix Σ is deﬁned by

n Fix Σ = {x ∈ R ; γx = x for all γ ∈ Σ}.

(ii) Show that a ﬁxed point subspace is invariant under the ﬂow ϕt.

Now assume that in (5.10), f0(0) = 0 and Df0(0) is singular. Also assume that Fix Γ = {0}.

(iii) Suppose Σ is an isotropy subgroup with dim Fix Σ = 1. Write Fix Σ = Rv. Assume Df0(0)v = 0 ∂ and ∂λ Df0(0)v 6= 0. Show the existence of a smooth branch x = sv, λ = λ(s), with s near zero and 2 λ(0) = 0, so that fλ(s)(sv) = 0. Hint: Write fλ(sv) = h(s, λ)v for h : R → R. Since Fix Γ = {0}, the origin is an equilibrium for all λ, and hence, h(0, λ) = 0. So h(s, λ) = sk(s, λ). Apply the implicit function theorem.

Exercise 3.5.21. This exercise is a follow-up to the previous exercise, and provides an example. Con- sider the planar diﬀerential equation

x˙ = λx + A(x2 + y2)x + Bx3, y˙ = λy + A(x2 + y2)y + By3, where λ is a parameter and A, B are constants satisfying

A + B 6= 0, 2A + B 6= 0,B 6= 0.

(i) Find the equilibria and their eigenvalues. Sketch the bifurcation diagram by sketching phase diagrams for λ = 0 and λ 6= 0.

(ii) Show that the diﬀerential equation is D4-equivariant, for the representation of D4 (the symmetry group of the square) generated by

γ1(x, y) = (x, −y),

γ2(x, y) = (−y, x).

(iii) Find the ﬁxed point spaces of the isotropy subgroups, and relate this to the computed bifurcation diagram.

2 Exercise 3.5.22. Consider the quadratic family fa(x) = a − x . Show that a period doubling bifurcation occurs at a = 3/4.

2 Exercise 3.5.23. Consider the family fa,b(x, y) = (a − x − by, x). Show that this is a diﬀeomorphism for b 6= 0. Discuss period doubling bifurcations for a near 3/4 and small values of b. Chapter 4

Nonlocal bifurcations

The previous chapter focused on bifurcations of equilibria and periodic orbits of diﬀerential equations. This chapter continues with bifurcations of connecting orbits between equilibria and periodic orbits. The Korteweg-de Vries equation (or KdV equation for short) is a partial diﬀerential equation,

∂ ∂3 ∂ u(x, t) + u(x, t) + 6u(x, t) u(x, t) = 0, (0.1) ∂t ∂x3 ∂x for u : R2 → R, modeling waves on shallow water surfaces. Its history starts with experiments of John Scott Russell in 1834, followed by theoretical investigations of, among others, John Strutt (Lord Rayleigh) and Joseph Boussinesq in 1871 and, finally, Diederik Korteweg and Gustav de Vries in 1895. It is well known to admit solitons, or solitairy waves: moving solutions with a fixed shape, u(x, t) = φ(x − ct), with limξ→±∞ φ(ξ) = 0. To find such solitons, plug u(x, t) = φ(x − ct) into the KdV equation to obtain the ordinary differential equation dφ d3φ dφ −c + + 6φ = 0, dξ dξ3 dξ or by integrating, d2φ − cφ + + 6φ2 = 0 (0.2) dξ2 (without integration constant as we want φ to converge to zero at infinity). This equation has a solution that converges to 0 as ξ → ±∞, which yields a soliton 1 1√ u(x, t) = − c sech2 ( c(x − ct)) 2 2

2 for (0.1). This formula involves the hyperbolic secant sech (x) = ex+e−x . Writing (0.2) as a ﬁrst order system,

dφ = ψ, dξ (0.3) dψ = cφ − 6φ2, dξ

77 78 CHAPTER 4. NONLOCAL BIFURCATIONS the soliton is given by a homoclinic orbit; a nontrivial solution that converges to the equilibrium at (0, 0) for ξ → ±∞. Note that these homoclinic orbits exist for all values of c 6= 0; the qualitative picture does not change. Other perturbations of (0.3) will destroy homoclinic orbits and thus lead to bifurcations. The following two sections will study two diﬀerent instances of homoclinic bifurcations.

4.1 Homoclinic loop

Deﬁnition 1.1. Given an equilibrium p for a diﬀerential equation, a nontrivial orbit h(t) with

lim h(t) = lim h(t) = p t→∞ t→−∞ is called a homoclinic orbit.

The goal of homoclinic bifurcation theory is to investigate the recurrent dynamics near a homoclinic orbit h(t). In other words, we are interested in finding all orbits that stay in a fixed tubular neighborhood of a given homoclinic orbit for all times. A natural way for approaching this problem is to use Poincaréfirst-return maps. Denote by Σ a local cross section placed at h(0). Starting with an initial condition u0 in Σ, we then follow u(t) until it hits Σ again, say at time t = T , and define the first-return map P via P (u0) := u(T ) ∈ Σ. The main issue is that a solution that starts near h(0) may not return to a neighborhood of h(0); see Figure 1.1. Thus, in general, the domain of the Poincarémap P does not contain an open neighborhood of h(0) in Σ. Furthermore, any solution that does return will spend a very long time near the equilibrium, thus spoiling most finite-time error estimates for P from standard variation-of-constant formulas. For computational purposes, it is useful to consider two cross sections Σin and Σout and write the first-return map P as a composition of transition maps between the cross sections.

Figure 1.1: The domain of the ﬁrst-return map on a cross section does not contain an open neighborhood of h(0) (one often has wedge shaped domains as shown here).

We will treat homoclinic bifurcations in the plane, avoiding additional complications that arise in higher dimensions. 4.1. HOMOCLINIC LOOP 79

Figure 1.2: The phase diagrams of a nondegenerate homoclinic orbit h(t) to an equilibrium with eigenvalues λ, µ satisfying λ + µ < 0 is shown. A unique attracting single-round periodic orbit bifurcates to one side of µ = 0 (here for µ < 0), and its period goes to inﬁnity as µ tends to zero.

Theorem 1.1. Let

x˙ = fa(x) (1.1)

be one family of vector ﬁelds on the plane R2, depending on a real parameter a ∈ R, with the following assumptions.

(i) For a = 0, there is hyperbolic equilibrium p. The linearization Df0(p) has real eigenvalues λ0, µ0

with µ0 < 0 < λ0 and

µ0 + λ0 < 0.

(ii) At a = 0, there is a homoclinic orbit h0(t) with

lim h0(t) = lim h(t) = p. t→∞ t→−∞

(iii) M 6= 0, where ∞ Z R t ∂ − div f0(h0(s)) ds M = e 0 f0(h0(t)) × fa(h0(t))|a=0 dt. −∞ ∂a

Then there exists δ > 0, ε so that for a ∈ (a0, a0 + δ), there is a stable limit cycle ζa(t) of (1.1) of period

T (a) such that T (a) goes to inﬁnity as a → a0 and ζa approaches h as a → a0. Furthermore, if U denotes the ε neighborhood of h, then ζa is the unique limit cycle in U for a ∈ (a0, a0 + δ). There is no

limit cycle in U for a ∈ (a0 − δ, a0].

Remark 1.1. The integral M is called a Melnikov integral after Viktor Melnikov. The condition M 6= 0 is a generic unfolding condition: M measures, at ﬁrst order in a, the splitting of stable and unstable

manifolds of pa. The condition is equivalent to transversality of stable and unstable manifolds of pa in s u the combined state and parameter space: ∪aW (pa) × {a} intersects ∪aW (pa) × {a} transversally along

h0 × {0}.

Proof. Take local coordinates x, y near the equilibrium in which the local stable and unstable coordinates s u are linear: Wloc(0) ⊂ {x = 0}, Wloc ⊂ {y = 0}. We now have the following expression for the diﬀerential equations: x˙ = µx + g(x, y), (1.2) y˙ = λy + h(x, y). 80 CHAPTER 4. NONLOCAL BIFURCATIONS

Here g and h are smooth functions with

g(x, y) = O(|x||y| + |x|2), h(x, y) = O(|y||x| + |y|2) as (x, y) → 0. Since 1 + h(x, y)/(λy) 6= 0 for y small, we can divide the equations by 1 + h(x, y)/(λy) (this amounts to a time reparametrization). We end up with equations

x˙ = µx + j(x, y), (1.3) y˙ = λy with j(x, y) = O(|x||y| + |x|2).

For s > 0 small, let Σin = {x = s, |y| < s} and Σout = {|x| < s, y = s} are local cross sections, transverse to the flow. Without loss, h intersects Σin and Σout. By a rescaling we get s = 1. Define the local transition map Ploc : D ⊂ Σin → Σout by letting P (x) be the first point φt(x, 1) ∈ Σout. Here

D = (0, s) is the domain of Ploc. Lemma 1.2 gives asymptotic expansions for Ploc.

The global transition map Pglob :Σout → Σin is a local diﬀeomorphism by the implicit function 2 theorem. We can write Pglob(x, 1) = a0 + a1x + O(x ) as x → 0, where a0, a1 are smooth functions of the parameter. The composition P = Ploc ◦ Pglob thus has an expansion

−µ/λ −µ/λ P (1, y) = (a0 + a1y + O(y ), 1), where a0, a1, µ, λ are smooth functions of the parameter and where one can diﬀerentiate this formula to obtain estimates for derivatives. The theorem now follows easily from a graphical analysis.

Lemma 1.1. In smooth local coordinates (x, y) near p, and after a multiplication by a smooth positive function, the vector ﬁeld f is given by the set of diﬀerential equations

x˙ = λsx + f s(x, y), y˙ = λuy, where f s(x, y) satisﬁes

f s(x, y) = O(|x|2), as kxk → 0.

Proof. We may take smooth coordinates x = (x, y) near p, so that

W s(p) = {y = 0}, (1.4) W u(p) = {x = 0}, (1.5)

In such coordinates we have

s x˙ s = µx + f (x, y), u x˙ u = λy + f (x, y), 4.1. HOMOCLINIC LOOP 81

for quadratic and higher order terms f s, f u with f s(x) = O(|x|k(x, y)k) as x → 0 and f u(x) = O(|y|k(x, y)k) as y → 0. Dividing the vector ﬁeld by the positive function λu + f u(x, y)/y gives an expression with f u = 0 and still f s(x, y) = O(|x|kxk). s s ∂2 s Write f (x, y) = kxy + g (x, y) with ∂x∂y g (x, y) = 0 in (x, y) = (0, 0). A direct computation shows that a polynomial coordinate change removes the term kxy from the diﬀerential equation for x. Indeed, let u = x + axy and v = y. Thenv ˙ = λv and

u˙ = µu + (k + aµ + aλ)uv + ... where the dots stand for terms O(|u|2) and terms of order three and higher. As µ + λ 6= 0, we can choose a such that k + aµ + aλ = 0. This does the trick. We may now assume f s(x, y) = O(|x|2) + O(|x|k(x, y)k2). To ﬁnish the proof, we must remove terms O(xy) from the equation for x. Consider a coordinate change u = x + p(y)x for a function x 7→ p(x) that vanishes at x = 0, and v = y. Write the diﬀerential equations in the new coordinates as

y˙s = µu + P (u, v)u andv ˙ = λv. Note that P (0, v) =p ˙ + h.o.t., where h.o.t. stands for higher order terms in p, v. We look for a function p so that P (0, v) vanishes. Considering p as variable, this yields diﬀerential equations

p˙ = h.o.t. v˙ = λv

We can obtain p by constructing the one dimensional unstable manifold for this system of diﬀerential equations.

in out Lemma 1.2. Let Ψ:Σ → Σ be the first return map given by the flow φt of f. Write β = µ/λ. In coordinates near p as in Lemma 1.1, There exists ω > 0, φ > 0, so that as y → 0, y 7→ Ψ(y) satisfies

k d β β+ω−k Ψ(y) − φy ≤ Ck+l|y| , dyk for positive constants Ck+l.

Remark 1.2. One can smoothly linearize the vector ﬁeld restricted to the local stable manifold. This brings f s(x, y) = O(|x|2|y|) as (x, y) → 0. In this case one has φ = 1.

Remark 1.3. If the diﬀerential equation is smoothly linearized near the equilibrium, then Ψ(y) = yβ.

Proof. Let Σin ⊂ {x = δ},Σout ⊂ {y = δ} be cross-sections close to p. By a rescaling we may assume δ = 1. Note that in the normal form

s x˙ s = µx + f (x, y),

x˙ u = λy, 82 CHAPTER 4. NONLOCAL BIFURCATIONS

this leaves us with f s(x, y) = δO(|x|2). It suffices to consider the first hit map Sin :Σin → Σout as the first hit map Sout :Σout → Σin is a local diffeomorphism by the Flow Box Theorem, Theorem 5.2. We will show that Sin satisfies

k in β β+ω−k D in S (y) − φy ≤ Ck+l|y| , xu

for positive constants Ck+l. Write down the variation of constants formula for an orbit (x(t), y(t)) starting in a point (1, y) ∈ Σin. out 1 λt λ(t−τ) Observe that this orbit reaches Σ at time τ = − λ ln y. Thus y(t) = e y = e and Z t µt µ(t−v) s λ(v−τ) xs(t) = e + e f (x(v), e )dv. (1.6) 0 We will mimic Picard iteration on a weighted Banach space to construct the orbit and at the same time obtain asymptotics for it. Let

B = {x : [0, τ] → Es | sup |x(t)|eµt < ∞}. t∈[0,τ] Equipped with the norm kxk = sup |x(t)|e−µt, t∈[0,τ] B is a Banach space. Write Γ : B → B for the map given by the right hand side of (1.6). We claim that there is a ball U ⊂ B so that Γ(U) ⊂ U. Compute Z t −λ t µt µ(t−v) s λ (v−τ) kΓxk = sup e s e + e f (x(v), e u )dv t∈[0,τ] 0 Z t −µv s λ (v−τ) = sup 1 + e f (x(v), e u )dv t∈[0,τ] 0 Z τ ≤ 1 + Cδ e−µv|x(v)|2dv, 0 for some C > 0, using Lemma 1.1. It follows that, for δ small enough, there exists R > 0 so that for kxk ≤ R, also kΓxk ≤ R. We see that the orbit x satisﬁes a bound

|x(t)| ≤ Ceµt. (1.7)

Writing f s(x, y) = gs(x) + gu(x, y) with gu(x, 0) = 0, compute Z t Z t µt µ(t−v) s µ(t−v) u λ(v−τ) x − e − e g (x(v)) dv = e g (x(v), e ) dv 0 0 Z t ≤ eµ(t−v)Cδe2µveλ(v−τ) dv. 0 This implies the lemma with Z ∞ φ = 1 + eµvgs(x(v)) dv. 0 Bounds for derivatives are obtained similarly by diﬀerentiating the variation of constants formula and estimating as before. 4.1. HOMOCLINIC LOOP 83

Denote the homoclinic orbit for a = 0 by h0(t). Take a line piece S through h0(0), perpendicular s s s u to the homoclinic orbit. For a 6= 0, consider the solutions ha(t) in W (pa) with ha(0) ∈ S and ha(t) in u u s u W (pa) with ha(0) ∈ S. We consider ha(t) for t ≥ 0 and ha(t) for t ≤ 0.

u s Lemma 1.3. With d = ha(0)−ha(0) measuring the signed distance between stable and unstable manifolds of pa, measured inside S, we have d = Ma + O(a2), with ∞ Z R t ∂ − div f0(h0(s)) ds M = e 0 f0(h0(t)) × fa(h0(t))|a=0 dt. −∞ ∂a

Proof. Denote the homoclinic orbit for a = 0 by h0(t). Take a line piece S through h0(0), perpendicular s s s u to the homoclinic orbit. For a 6= 0, consider the solutions ha(t) in W (pa) with ha(0) ∈ S and ha(t) in u u s u W (pa) with ha(0) ∈ S. We consider ha(t) for t ≥ 0 and ha(t) for t ≤ 0. Since the stable and unstable manifolds are smooth and depend smoothly on a, we may develop

s s 2 ha(t) = h0(t) + ah1(t) + O(a ), t ≥ 0, u u 2 ha(t) = h0(t) + ah1 (t) + O(a ), t ≤ 0.

˙s s Note that ha(t) = fa(ha(t)) gives ∂f h˙s (t) = f (h (t)) + aDf (h (t))hs(t) + a a (h (t))| + O(a2), a 0 0 0 0 1 ∂a 0 a=0 s so that h1 satisfies ∂f h˙s(t) = Df (h (t))hs(t) + a (h (t))| 1 0 0 1 ∂a 0 a=0 u The solution h1 satisfies a similar differential equation. Write s s u u ∆ (t) = f0(h0(t)) × h1(t), ∆ (t) = f0(h0(t)) × h1 (t). Then, with ∆(t) = ∆u(t) − ∆s(t), we have

d = a∆(0) + O(a2).

Compute

˙s ˙ s ˙s ∆ (t) = Df0(h0(t))h0(t) × h1(t) + f0(h0(t)) × h1(t) ∂f = Df(h (t))f (h (t)) × hs(t) + f (h (t)) × (Df (h (t))hs(t) + a (h (t))| ) 0 0 0 1 0 0 0 0 1 ∂a 0 a=0 ∂f = div f (h0(t))∆s(t) + f (h (t)) × a (h (t))| . 0 0 0 ∂a 0 a=0 The variation of constants formula gives ∞ Z R t ∂ s s − div f0(h0(s)) ds ∆ (∞) − ∆ (0) = e 0 f0(h0(t)) × fa(h0(t))|a=0 dt 0 ∂a Combined with an analogous formula for ∆u(0) − ∆u(∞), this proves the lemma. 84 CHAPTER 4. NONLOCAL BIFURCATIONS

4.2 Shil’nikov homoclinic loop

Consider a vector ﬁeld on R3, with a hyperbolic equilibrium so that the linearization about it has one real unstable eigenvalue λ > 0 and two complex conjugate stable eigenvalues µ ± α i. A homoclinic orbit to such an equilibrium is called a saddle-focus homoclinic orbit. A systematic study of the dynamics near saddle-focus homoclinic orbits was pioneered by Leonid Shil’nikov since the mid 1960s.

Theorem 2.1. Assume that the system u˙ = f(u, µ) on Rn has a Shil’nikov saddle-focus homoclinic orbit for µ = 0 which satisﬁes

(i) µ/λ < 1

At µ = 0, for each neighborhood U of the homoclinic solution, there are inﬁnitely many periodic solutions in U. Furthermore, for each neighborhood U of the homoclinic solution and each µ0 > 0, there are inﬁnitely many bifurcation values in (0, µ0) with homoclinic solutions contained in U.

The dynamics near saddle-focus homoclinic solutions is considerably more complex than hinted at by the existence of infinitely many periodic solutions. Dynamics and bifurcations are studied through the first-return map Π on a cross section Σ for which expressions can be derived by transforming the differential equation near the equilibrium into normal form and deriving expansions for solutions near the equilibrium by estimating integral formulas. This procedure gives:

Proposition 2.1. Assume µ/λ < 1, then there is a cross section Σ and smooth coordinates (θ, z) on Σ so that Π(·, µ) has the following asymptotic expansion:

− Re νs s − Re νs s ! νu Im ν νu Im ν a + φ1(θ)z sin − νu ln z + φ2(θ)z cos − νu ln z + R1(θ, z, µ) Π(θ, z, µ) = − Re νs s − Re νs s . νu Im ν νu Im ν b + φ3(θ)z sin − νu ln z + φ4(θ)z cos − νu ln z + R2(θ, z, µ)

The functions a, b, φi, i = 1, 2, 3, 4, are smooth functions of θ and µ (the dependence on µ is suppressed ! φ φ in the notation) and satisfy det 1 2 6= 0. Furthermore, there exist η > 0 and positive constants φ3 φ4 Ci so that

k+l+m ∂ − Re νs u +η Ri(θ, z, µ) ≤ Ck+l+mz ν . ∂θk∂zl∂µm

Next, we formulate a bifurcation result that makes the existence of attractors in parameterized families more precise.

3 Theorem 2.2. Consider a one-parameter family u˙ = f(u, 0, µ2) of diﬀerential equations in R so that each diﬀerential equation has a saddle-focus homoclinic orbit. Suppose

1 (i) 2 < µ/λ < 1.

∂ − Re νs (ii) u 6= 0. ∂µ2 ν 4.3. BLUE SKY CATASTROPHE 85

Figure 2.1: A Shil’nikov saddle-focus homoclinic orbit.

Let Σ be a cross section transverse to h and denote by Un a decreasing sequence of tubular neighborhoods of h.

Let P2,n be the set of parameter values µ2 for which f has an attracting 2-periodic orbit in Un, then,

for n large enough, P2,n is open and dense in I and limn→∞ |P2,n| = 0, where | · | denotes Lebesgue

measure. Furthermore, the set of parameter values γ ∈ I for which Xγ has inﬁnitely many 2-periodic attractors is dense in I, but has zero measure.

Let A2,n be the set of parameter values in γ ∈ I for which Xγ has a H´enon-likestrange attractor in

Un that intersects Σ in 2 connected components. For n large enough, A2,n has positive measure and is

dense in I, and limn→∞ |A2,n| = 0.

4.3 Blue sky catastrophe

The question about the possibility for a periodic orbit to remain in a bounded region of the phase space while the period and length of the orbit increase with no bound as it approaches its existence boundary was raised by Jacob Palis and Charles Pugh in 1974. The problem was code-named a ”blue sky catastrophe” as the orbit, while getting longer and longer, would be virtually vanishing in the space. Figure 3.1 gives an impression of how a blue sky catastrophe may occur. The right panel depicts a stable periodic orbit of long period. In this mechanism an attracting periodic orbit develops longer and longer period, and arc length, before it disappears at a saddle-node bifurcation of another periodic orbit.

x˙ = x(2 + µ − 10(x2 + y2)) + y2 + z2 + 2y, y˙ = −z3 − (1 + y)(y2 + z2 + 2y) − 4x + µy, (3.1) z˙ = (1 + y)z2 + x2 − ε. 86 CHAPTER 4. NONLOCAL BIFURCATIONS

Figure 3.1: A saddle-node bifurcation of a periodic solution can give rise to a blue-sky catastrophe. The right panel shows a stable periodic orbit close to a blue sky catastrophe in the system (3.1) with µ = 0.456, ε = 0.0357.

4.4 Homoclinic tangencies

A saddle periodic orbit γ of a differential equation R3 has two dimensional stable manifold W s(γ) and two dimensional unstable manifold W u(γ). An intersection of W s(γ) and W u(γ) different from γ itself, is a homoclinic orbit. Let η be a homoclinic orbit to γ, and take a local cross section Σ through the point η(0). Then W s(γ) and W u(γ) each intersect Σ in a curve through η(0) and possibly other connected components that do not contain η(0). These curves can have a transversal intersection at η(0), in which case one speaks of a transverse homoclinic orbit, or a tangency at η(0), in which case one speaks of a homoclinic tangency. To fix thoughts, consider a nonautonomous differential equationx ˙ = f(x, t), x ∈ R2, which is periodic in time with period T . The state space is R2 × T, a Poincaréreturn map on {t = 0} is the time T map. Suppose p is a hyperbolic fixed point of a diffeomorphism f on R2 with transversely intersecting stable an unstable manifolds W s(p) and W u(p). Henri Poincaréwas aware of the complex geometry of such transversely intersecting stable an unstable manifolds.

It may be futile to even try to represent the figure formed by these two curves and their infinite intersections, each of which corresponds to a doubly asymptotic solution to the equations of motion. The intersections form a kind of trellis, a tissue, an infinitely tight lattice; each of the curves must never self-intersect, but it must fold itself in a very complex way, so as to return and cut the lattice an infinite number of times. The complexity of this figure is so astonishing that I cannot even attempt to draw it. (Henri Poincaré)

The work of George Birkhoﬀ and Stephen Smale made clear that the transverse intersections imply the existence of complicated dynamics nearby. We say that a map f : B → R2 on a square B ⊂ R2 is a horseshoe map, if it the maximal invariant set of f in B is hyperbolic and f restricted to it is topologically conjugate to the Smale horseshoe map. Figure 4.1 illustrates the geometry of how an iterate of a diﬀeomorphism with a transverse homoclinic point restricted to a thin box is a horseshoe 4.4. HOMOCLINIC TANGENCIES 87 map.

Figure 4.1: The idealization of this picture is the Smale horseshoe map: a diﬀeomorphism with a transverse homoclinic orbit admits a horseshoe map for an iterate.

Theorem 4.1. Let P : R2 → R2 be a diﬀeomorphism with a hyperbolic saddle ﬁxed point p, and transversely intersecting stable and unstable manifolds W s(p) and W u(p). Then there is a box B and an integer n so that P n is a horseshoe map on B.

We continue with the bifurcation that occurs with a homoclinic tangency. We will discuss a method to detect homoclinic tangencies in forced dynamical systems in the speciﬁc setting of nonautonomous equations of the form x˙ = y, (4.1) y˙ = −f(x) + εh(x, y, t), where h(x, y, t) is T -periodic in t and ε is a small parameter. The state space of the system is R2 × T. It is assume that the unperturbed system,

x˙ = y, (4.2) y˙ = −f(x), in the plane, satisﬁes f(0) = 0, so has a saddle equilibrium at the origin, and admits a homoclinic orbit

(x0(t − t0), y0(t − t0)) to the origin. We assume that the homoclinic orbit lies in the half-plane {x ≥ 0} As ε increases from zero, an application of the implicit function theorem gives a saddle periodic orbit

(xε(t), yε(t)) continuing (0, 0). Consider the Poincar´ereturn map Pε on {t = 0}. Notice that Pε is the time T map. Melnikov’s method provides expressions for the distance between stable and unstable s u manifolds W (pε) and W (pε) measured in a line piece in {t = 0} that is transverse to the homoclinic orbit for ε = 0. Expand an orbit in the stable manifold as

s s 2 xε(t, t0) = x0(t − t0) + εx1(t, t0) + O(ε ), s s 2 yε(t, t0) = y0(t − t0) + εy1(t, t0) + O(ε ), 88 CHAPTER 4. NONLOCAL BIFURCATIONS

s s for t ≥ 0. As the orbit lies in the stable manifold of pε, xε(t, t0) − xε(t) → 0 and yε(t, t0) − yε(t) → 0 as t → ∞. Substitute the expansions into (4.1) and equate coeﬃcients of ε. Using also

s 0 s 2 f(xε(t, t0)) = f(x0(t − t0)) + εf (x0(t − t0))x1(t, t0) + O(ε ), is follows that

s s x˙ 1(t, t0) = y1(t, t0), s 0 s y˙1(t, t0) = −f (x0(t − t0))x1(t, t0) + h(x0(t − t0), y0(t − t0), t).

In a similar manner, expand an an orbit in the unstable manifold as

u u 2 xε (t, t0) = x0(t − t0) + εx1 (t, t0) + O(ε ), u u 2 yε (t, t0) = y0(t − t0) + εy1 (t, t0) + O(ε ),

u u for t ≤ 0. Here xε (t, t0) − xε(t) → 0 and yε (t, t0) − yε(t) → 0 as t → −∞. Then

u u x˙ 1 (t, t0) = y1 (t, t0), u 0 u y˙1 (t, t0) = −f (x0(t − t0))x1 (t, t0) + h(x0(t − t0), y0(t − t0), t).

Figure 4.2: The deﬁnition of the distance function D(t0).

The displacement vector at t = 0 is deﬁned as ! ! xu(0, t ) xs(0, t ) d(t ) = ε 0 − ε 0 0 u s yε (0, t0) yε(0, t0) ! !! xu(0, t ) xs(0, t ) = ε 1 0 − 1 0 + O(ε2) u s y1 (0, t0) y1(0, t0) 4.4. HOMOCLINIC TANGENCIES 89

We use a distance function D(t0) which is obtained by projecting the displacement vector d(t0) onto the unit normal vector n(0, t0) to the unperturbed homoclinic orbit at Q0 = (x0(−t0), y0(−t0)) on the homoclinic orbit. Thus 2 D(t0) = hd(0, t0), n(0, t0)i + O(ε ). Since the tangent vector to the unperturbed homoclinic orbit at t = 0 is ! ! x˙ (−t ) y (−t ) 0 0 = 0 0 , y˙0(−t0) −f(x0(−t0)) the unit normal vector is ! f(x (−t )) 0 0 p 2 2 / f(x0(−t0)) + y0(−t0) . y0(−t0) Hence u s u s (x1 (0, t0) − x1(0, t0))f(x0(−t0)) + (y1 (0, t0) − y1(0, t0))y0(−t0) 2 D(t0) = ε + O(ε ). p 2 2 f(x0(−t0)) + y0(−t0) Write

u u u ∆ (t, t0) = x1 (t, t0)f(x0(t − t0)) + y1 (t, t0)y0(t − t0), u s s ∆ (t, t0) = x1(t, t0)f(x0(t − t0)) + y1(t, t0)y0(t − t0), and note that u s ∆ (0, t0) − ∆ (0, t0) 2 D(t0) = ε + O(ε ). p 2 2 f(x0(−t0)) + y0(−t0) s To obtain simpler expressions, diﬀerentiate ∆ (t, t0) with respect to t, d∆s(t, t ) 0 =x ˙ s(t, t )f(x (t − t )) + xs(t, t )f 0(x (t − t ))x ˙ (t − t )+ dt 1 0 0 0 1 0 0 0 0 0 s s y˙1(t, t0)y0(t − t0) + y1(t, t0)y ˙0(t − t0) s s 0 = y1(t, t0)f(x0(t − t0)) + x1(t, t0)f (x0(t − t0))x ˙ 0(t − t0)+ 0 s s y0(t − t0)(−f (x0(t − t0))x1(t, t0) + h(x0(t − t0), y0(t − t0), t)) − y1(t, t0)f(x0(t − t0))

= y0(t − t0)h(x0(t − t0), y0(t − t0), t). (4.3) s Now integrate (4.3) between 0 and ∞ with respect to t noting that ∆ (0, t0) → 0 as t → ∞ because y0(t − t0) → 0 and f(x0(t − t0)) → 0 as t → ∞. Hence Z ∞ s ∆ (0, t0) = − y0(t − t0)h(x0(t − t0), y0(t − t0), t) dt. 0 Similarly, Z 0 u ∆ (0, t0) = − y0(t − t0)h(x0(t − t0), y0(t − t0), t) dt. −∞ Now let

u s M(t0) = ∆ (0, t0) − ∆ (0, t0) Z ∞ = y0(t − t0)h(x0(t − t0), y0(t − t0), t) dt. −∞ 90 CHAPTER 4. NONLOCAL BIFURCATIONS

The function M(t0) is known as the Melnikov function.

Example. Consider the forced Duﬃng equation

x¨ + εκx˙ − x + x3 = εγ cos(ωt). (4.4)

Put f(x) = −x + x3, h(x, y, t) = −κy + γ cos(ωt) and T = 2π/ω. The homoclinic orbit to the origin of the unperturbed system

x¨ − x + x3 = 0 √ in x > 0 is given explicitly as x0(t) = 2sech (t). Hence Z ∞ M(t0) = x˙ 0(t − t0)(−κx˙ 0(t − t0) + γ cos(ωt)) dt −∞ Z ∞ √ Z ∞ 2 2 = −2κ sech (t − t0) tanh (t − t0) dt − γ 2 sech (t − t0) tanh(t − t0) cos(ωt) dt −∞ −∞ Z ∞ √ Z ∞ 2 2 = −2κ sech (u) tanh (u) du + 2γ sin(ωt0) sech (u) tanh(u) sin(ωu) du. −∞ −∞ The integrals can be evaluated using residue theory to give 4 √ 1 M(t ) = − κ + 2ωπγsech ( ωπ) sin(ωt ). 0 3 2 0

For given κ, γ, ω, M(t0) will vanish if a real solution can be found for t0. Assuming that κ > 0, γ > 0, this condition will be met if | sin(ωt0)| ≤ 1, that is, if √ 2 2κ 1 cosh ( ωπ) ≤ 1. 3πγω 2 If κ and ω are ﬁxed, a homoclinic tangency will ﬁrst occur for increasing forcing amplitude γ when √ 2 2κ 1 γ = γ = cosh ( ωπ) c 3πω 2 and transverse homoclinic orbits will be present if γ exceeds this value.

2 To fix thoughts, we will treat diffeomorphisms on R . Let fµ be a one parameter family of diffeomorphisms on the plane, depending on a single real parameter µ. We take the following assumptions.

(i) fµ has a hyperbolic saddle ﬁxed point pµ.

s u (ii) For µ = 0, the stable and unstable manifold W (p0),W (p0) are tangent at a point q.

Theorem 4.2. Let fµ be a one parameter family of diﬀeomorphisms satisfying ...

2 (i) there are parameter dependent linearizing coordinates near pµ, that are C jointly in state space coordinates and parameter. 4.4. HOMOCLINIC TANGENCIES 91

Figure 4.3: Stable and unstable manifolds of the saddle ﬁxed point of the return map on the (x, x˙)-plane of the Duﬃng equations (4.4). Parameters are εκ = 0.3, ω = 1.2 and (a) εγ = 0.2, (b) εγ = 0.256, (c) εγ = 0.28.

(ii) The eigenvalues of the linearized map at p0 are λ, σ with 0 < λ < 1 < σ and λσ < 1.

Then there are a constant N and for for each positive integer n, a reparametrization

µ = Mn(¯µ) and a µ¯ dependent coordinate change

(x, y) = Ψn,µ¯(¯x, y¯) such that

(i) for each compact set K in (¯µ, x,¯ y¯) space, the image of K under the map

(¯µ, x,¯ y¯) 7→ (Mn(¯µ), Ψn,µ¯(¯x, y¯))

converges as n → ∞ to (0, q) in (µ, x, y) space,

(ii) the domains of the maps −1 n+N (¯µ, x,¯ y¯) 7→ (¯µ, Ψ ◦ f ◦ Ψn,µ¯) n,µ¯ Mn(¯µ) 92 CHAPTER 4. NONLOCAL BIFURCATIONS

converge for n → ∞ to all of R3 and the maps converge in the C2 topology to

(¯µ, x,¯ y¯) 7→ (¯µ, y¯2 +µ ¯).

Proof. We start by choosing parameter dependent linearizing coordinates near pµ. We denote them by

(x, y). For µ = 0, we have f0(x, y) = (λx, σy) with 0 < λ < 1 < σ and λσ < 1. Let q be a point in the orbit of tangency in the local stable manifold and r such point in the local unstable manifold. By rescaling we may assume q = (1, ) and r = (0, 1). Since q and r are in the orbit of tangency, there is N N > 0 with f0 (r) = q. For µ near 0 we adapt the linearizing coordinates so that

N u (i) fµ (0, 1) is a local maximum of the y coordinate restricted to W (pµ),

N (ii) the x coordinate of fµ (0, 1) is 1.

N We reparametrize µ so that the y coordinate of fµ (0, 1) is µ. After these preliminary steps we can write N fµ near (0, 1) as

(x, 1 + y) 7→ (1, 0) + (H1(µ, x, y),H2(µ, x, y)) with

H1(µ, x, y) = αy + H˜1(µ, x, y), 2 H2(µ, x, y) = βy + µ + γx + H˜2(µ, x, y), where α, β, γ are nonzero constants and where

H˜1,DyH˜1,DµH˜1 = 0 at (µ, x, y) = (0, 0, 0) and

DxH˜2(0, 0, 0) = 0, H˜2(µ, 0, 0) = 0,DyH˜2(µ, 0, 0) = 0.

Next we deﬁne an n-dependent reparametrization of µ and a µ-dependent coordinate transformation by the following formulas:

µ = σ−2nµ¯ − γλn + σ−n, x = 1 + σ−nx,¯ y = σ−n + σ−2ny.¯

The inverse transformation is given by

µ¯ = σ2nµ + γλnσ2n − σn, x¯ = σn(x − 1), y¯ = σ2ny − σn.

Note that σ and λ depend on µ and hence onµ ¯ although this is not expressed in the above formulas. 4.4. HOMOCLINIC TANGENCIES 93

n+N We start with the calculation to express fµ in terms ofµ, ¯ x,¯ y¯. Let (¯µ, x,¯ y¯) denote a point. After N applying fµ to this point we get as (x, y) coordinates (µ does not change):

x = λn(1 + σ−nx¯), y = 1 + σ−ny.¯

N Next we apply fµ and ﬁnd

−n n −n −n x = 1 + ασ y¯ + H˜1(µ, λ (1 + σ x¯), σ y¯), −2n 2 −2n n −n n −n −n −n y = βσ y¯ + (σ µ¯ − γλ + σ ) + γλ (1 + σ x¯) + H˜2(µ, λ(1 + σ x¯), σ y¯).

Transforming this back to the (¯x, y¯) coordinates, and denoting the values of these coordinates of the new point (˘x, y˘) we have

n −2n n −n n −n −n x˘ = αy¯ + σ H˜1(σ µ¯ − γλ + σ , λ (1 + σ x¯), σ y¯), 2 n n −2n n −n −n −n y˘ = βy¯ +µ ¯ + γλ σ x¯ + H˜2(σ µ¯ − γλ + σ , λ(1 + σ x¯), σ y¯).

In the expression fory ˘, the term γλnσnx¯ goes to zero as n → ∞ because λσ < 1. When (¯µ, x,¯ y¯) remains bounded, the corresponding values of (µ, x, y − 1) which are substituted in H˜1, H˜2, satisfy

µ = O(σ−n), x = O(λn), y − 1 = O(σ−n) as n → ∞. Deﬁne

n −2 n −n n −n −n H¯1(¯µ, x,¯ y¯) = σ H˜1(σ µ¯ − γλ + σ , λ (1 + σ x¯), σ y¯).

Then n n −n n n n H¯1(0, 0, 0) = σ H˜1(−γλ + σ , λ , 0) = σ O(λ )

converges to zero for n → ∞. Likewise, the ﬁrst and second order derivatives of H¯ (¯µ, x,¯ y¯) converge to zero, uniform on compact sets. The same procedure works for the expressions in the formula fory ˘. So, for n → ∞, the transformation formulas converge to

x˘ = αy,¯ y˘ = βy¯2 +µ. ¯

By a ﬁnal substitution µ¯ =µ/β, ˜ x¯ = αx/β,˜ y¯ =y/β, ˜

the limiting transformation becomes (˜x, y˜) 7→ (˜y, y˜2 +µ ˜). 94 CHAPTER 4. NONLOCAL BIFURCATIONS

4.5 Exercises

Exercise 4.5.1. Consider the system

x˙ = y, (5.1) y˙ = −2x − ay − 3x2 with a > 0.

(i) Show that (0, 0) is asymptotically stable and (−2/3, 0) is a saddle point.

(ii) Draw typical level sets of V (x, y) = y2/2 + x2 + x3. This function is conserved when a = 0. Notice especially that V (x, y) = 0 contains a homoclinic loop for a = 0.

2 (iii) For every initial condition (x0, y0) ∈ R for which the positive orbit is bounded, show that the ω-limit set is one of the equilibria.

(iv) Estimate the basin of attraction of the origin.

s (v) Show that for (x0, y0) in the stable manifold W ((−2/3, 0)), the negative orbit of (x0, y0) diverges to inﬁnity.

(vi) Observe that there are orbits that are unbounded for positive as well as negative time.

(vii) Sketch the phase portrait of the system in the plane.

Exercise 4.5.2. Consider Fisher’s equation; the partial differential equation ∂u ∂2u = + ru(1 − u) ∂t ∂x2 for functions u : R × R → R and a constant r > 0. Ronald Fisher proposed this equation to describe the spatial spread of an advantageous allele and explored its traveling wave solutions; solutions of the form u(x, t) = v(x − ct). Show that a traveling wave must satisfy the second order ordinary differential equation v00 + cv0 + rv(1 − v) = 0. √ Show that, for every c ≥ 2 r, there is a traveling wave solution satisfying v(s) → 1 as s → −∞ and v(s) → 0 as s → ∞, with v0 < 0. Hint: consider the corresponding first order system, discuss stability properties of the equilibrium at (1, 0). Then show that there is a triangular region in the (v, v0)-plane bounded by the lines v0 = 0, v = 1, v0 = −µv, µ appropriate, which is positively invariant.

Exercise 4.5.3. Consider the two dimensional system

x˙ = µ + x2 − xy, (5.2) y˙ = y2 − x2 − 1.

(i) Take µ = 0. Show that (5.2) has two saddle points P +(0) and P −(0) and these points are connected by a heteroclinic orbit. Give a sketch of the phase portrait. 4.5. EXERCISES 95

(ii) Now consider µ 6= 0 and small. Determine a Taylor expansion of the saddles P +(µ) and P −(µ) up to quadratic terms in µ. What happens to the heteroclinic orbit? Give sketches of the phase portraits for µ < 0 and µ > 0.

Exercise 4.5.4. Consider the integrable system

x¨ + x − x3 + C = 0, (5.3)

C ∈ R.

(i) Determine the set Ihom such that (5.3) has a homoclinic orbit if C ∈ I. For which values of

C = Chet does (5.3) have heteroclinic orbits? Give sketches of the phase portraits of (5.3) for C

such that Chet > C ∈ Ihom, C = Chet, Chet < C ∈ Ihom and Chet 6= C 6∈ Ihom.

Now consider a more general version of (5.3),

x¨ + x + Ax2 + Bx3 + C = 0, (5.4)

A, B, C ∈ R. For which A, B, C does (5.4) have heteroclinic orbits.

Exercise 4.5.5. Let V : R2 → R be a smooth function and consider the associated gradient ﬂow

x˙ = −∇V (x). (5.5)

Prove that (5.5) cannot have a homoclinic orbit.

Exercise 4.5.6. Consider the two dimensional system

x˙ = 1 + y − x2 − y2, (5.6) y˙ = 1 − x − x2 − y2.

(i) Determine the equilibria of (5.6) and their local character. Show that the ﬂow generated by (5.6) is symmetric with respect to the line {x + y = 0}.

(ii) Show that (5.6) is integrable by constructing an integral K(x, y). Hint: introduce new variables u = x − y and v = x + y that exploit the symmetry, write (5.6) as a system in u and v, and ˜ 2 dw determine an integral K(u, v) for this by introducing w = v and solving the equation for du .

(iii) Sketch the phase portrait associated to (5.6) and conclude that (5.6) has a homoclinic orbit.

Now consider a more general version of (5.6),

x˙ = 1 + y − x2 − y2 + h(x, y), (5.7) y˙ = 1 − x − x2 − y2 + h(x, y),

with h : R2 → R, h(0, 0) = 0, a smooth function. 96 CHAPTER 4. NONLOCAL BIFURCATIONS

(iv) Take h(x, y) = ε(x + y) with ε > 0 small. Show that the homoclinic orbit of (5.6) does not survive the perturbation of (5.7). Hint: determine K˙ or K˜˙ .

3 (v) Take h(x, y) = α(x − y) , α ∈ R. Show that (5.7) is integrable by deriving an integral Kα(x, y) (or K˜α(u, v)) such that K0(x, y) = K(x, y).

(vi) Take again h(x, y) = α(x − y)3, now with α > 0 small. Show that (5.7) has a homoclinic orbit and give a sketch of the phase portrait.

(vii) Take again h(x, y) = α(x − y)3, now with α > 0 large. Show that (5.7) does not have a homoclinic orbit and give a sketch of the phase portrait.

Exercise 4.5.7. Consider a planar diﬀerential equation x˙ = fa(x) with, for a = 0, a homoclinic orbit γ to a hyperbolic equilibrium p. Writing µ < 0 < λ for the eigenvalues at p, assume µ + λ < 0. Prove that there is a neighborhood U of γ ∪ {p} and a0 > 0, so that for |a| < a0 there is at most one periodic orbit in U. Such a periodic orbit is attracting. Hint: Suppose η is a periodic orbit of fa, of period T . The nonzero Floquet exponent ` of it satisﬁes 1 Z T ∂f ` = tr a (η(t)) dt. T 0 ∂x

∂fa Show that ` → tr ∂x (0) < 0 as a → 0. Conclude with an application of the Poincar´e-Bendixsontheorem. Exercise 4.5.8. Consider the time-periodic Hamiltonian 1 1 H(q, p, t) = (q2 + p2) − q3 − εq sin(t), 2 3 where ε is a small parameter.

∂H ∂H (i) For ε = 0, sketch the phase diagram for the corresponding Hamiltonian system q˙ = ∂p , p˙ = − ∂q . For this, determine equilibria together with their type, and possible homoclinic or heteroclinic orbits.

(ii) Determine an expression in terms of p and q for all homoclinic and heteroclinic solutions that exist for ε = 0. Hint: Use the Hamiltonian.

(iii) Determine the Melnikov function. For this you can use that Z ∞ cos(t) 4π 2 dt = . −∞ cosh (t/2) sinh (π) When do transverse homoclinic orbits exist?

(iv) Assume that the transverse intersection exists for t = t∗. Sketch the stable and unstable manifolds in (q, p)-space for t < t∗, t = t∗ and t > t∗.

Exercise 4.5.9. Consider the equation

θ˙ = v, (5.8) v˙ = − sin(θ) + ε(α + γ cos(t)), where ε is small and positive. 4.5. EXERCISES 97

(i) Determine the corresponding Hamiltonian.

(ii) Sketch the phase diagram for ε = 0. For this, determine equilibria together with their type, and possible homoclinic or heteroclinic orbits.

(iii) For ε = 0, determine an expression in terms of θ and v for all homoclinic and heteroclinic solutions 3 3 that satisfy − 2 π < θ < 2 π. Hint: Substitute θ = 4φ.

(iv) Use the Melnikov method to determine bifurcation curves near which quadratic homoclinic tangencies occur. Hint: used Z ∞ cos(t)n π dt = n −∞ cosh (t) cosh (π/2) for n = 0, 1.

Exercise 4.5.10. Consider a time-periodic equation

x¨ + f(x) = εg(x, t) where ε > 0 is small and g(0, t) = 0 for all t. Let F (x) be given by F 0(x) = f(x) with F (0) = 0. Assume that for ε = 0 there exists a homoclinic solution to the saddle ﬁxed point at (x, y) = (0, 0). Note that the saddle ﬁxed point at the origin persists for ε 6= 0.

(i) Determine the Hamiltonian H for ε = 0.

(ii) For ε 6= 0, determine the value Hu of H at the ﬁrst intersection of W u((0, 0)) with the x-axis and value Hs of H at the ﬁrst intersection of W s((0, 0)) with the x-axis.

(iii) Using Hs and Hu, determine a condition to test for a transversal intersection of W s((0, 0)) and W u((0, 0)). Relate it to the Melnikov method. 98 CHAPTER 4. NONLOCAL BIFURCATIONS Chapter 5

Structural stability and bifurcation

Structural stability is a fundamental property of a dynamical system which means that the qualitative behavior is unaffected by small perturbations. Structural stability of the system provides a justification for applying the qualitative theory of dynamical systems to analysis of concrete physical systems. At a bifurcation, the qualitative behavior is affected by small perturbations. In this chapter we provide mathematical definitions that make these ideas precise, and we discuss the scope of these definitions. Let M be a compact manifold. A vector field or differential equation on M gives rise to a flow n φt : M → M with t ∈ R. Note that on open manifolds such as R , orbits of differential equations may not be defined for all t ∈ R. To avoid having to discuss intervals of existence, we work on compact manifolds.

We define an equivalence relation, topological equivalence, for vector fields X,Y . Let φt, ψt be the flows of X and Y respectively.

Deﬁnition 0.1. Two vector ﬁelds X and Y are topologically equivalent if there exists a homeomorphism h : M → M that sends orbits of X to orbits of Y preserving the sense of orbits. That is, for all x ∈ M and t ∈ R there is t0 ∈ R with the same sign as t such that

h(φt(x)) = ψt0 (h(x)).

Here φt and ψt are the ﬂows of the vector ﬁelds X and Y respectively.

A local variant of this deﬁnition is as follows: two vector ﬁelds X at p ∈ M and Y at q ∈ M are locally topologically equivalent if there exists a homeomorphism h from a neighborhood V of p to a

neighborhood W of q that sends orbits of X|V to orbits of Y |W preserving the sense of orbits. The natural language to use here would be the language of germs.

Deﬁnition 0.2. A germ of a function of Rn → R at 0 is an equivalence class of mappings in which two functions are equivalent if they are identical on some neighborhood of 0 (the neighborhood depends on the choice of functions).

99 100 CHAPTER 5. STRUCTURAL STABILITY AND BIFURCATION

So one says that two germs of vector ﬁelds, X at p and Y at q, are topologically equivalent if there exists a local homeomorphism that sends orbits to orbits preserving the sense of orbits. Linked to topological equivalence is the notion of topological conjugacy.

Deﬁnition 0.3. Two vector ﬁelds X and Y are topologically conjugate if there exists a homeomorphism h : M → M such that

h(φt(x)) = ψt(h(x)).

Here φt and ψt are the ﬂows of the vector ﬁelds X and Y respectively.

The notion of structural stability originates from Alexander Andronov and Lev Pontryagin.

Deﬁnition 0.4. A vector ﬁeld X ∈ Cr(M) is called Cr-structurally stable if there is an open neighborhood U of X in Cr(M) so that each Y ∈ U is topologically equivalent to X.

n Let {Xµ} be a family of vector ﬁelds depending on a parameter µ ∈ R . If arbitrarily close to a parameter value µ0, there are values of µ so that Xµ is not topologically equivalent to Xµ0 , we say that {Xµ} has a bifurcation at µ = µ0. The set of parameter values µ for which Xµ is not structurally stable, is the bifurcation set. A bifurcation diagram of the dynamical system is a stratiﬁcation of its parameter space induced by the topological equivalence, together with representative phase portraits for each stratum.

0 k 0 r Definition 0.5. Two families of vector fields {Xµ} and {Xµ}, µ ∈ R , are (fiber − C ,C )-equivalent r if there exists a C diffeomorphism (homeomorphism if r = 0) k so that Xµ is topologically equivalent 0 to Xk(µ). That is, there is a homeomorphism hµ on the state space that maps orbits of Xµ to orbits of 0 0 Xk(µ), preserving the sense of orbits. If k can be chosen to be the identity, the families are (fiber−C , id)- 0 equivalent, or fiber − C equivalent over the identity. If hµ depends continuously on µ, the families are (C0,Cr)-equivalent.

k n For local families around (0, 0) ∈ R × R (say) one imposes the conditions that h0(0) = 0, k(0) = 0, k n hµ(x) needs only be defined for (µ, x) in a neighborhood V × W of (0, 0) ∈ R × R and k(µ) for µ ∈ V , with {(k(µ), hµ(x)) ; (µ, x) ∈ V × W } a neighborhood of (0, 0). For local studies, in terms of germs, one should consider germs of families and not families of germs. The following definition treats comparison of families with different numbers of parameters, and seeks a minimal number of parameters.

0 l r k Deﬁnition 0.6. A family {Xν }, ν ∈ R , is called C -induced by a family {Xµ}, µ ∈ R , if there is a Cr mapping k : Rl → Rk, so that 0 Xν = Xk(ν). k 0 r 0 An unfolding {Xµ}, µ ∈ R , is called a (C ,C )-versal unfolding if all unfoldings are C -equivalent over r the identity to an unfolding that is C induced from {Xµ}.

Example. One can show the following in R: for each smooth diﬀerential equation

x˙ = f(µ, x) 5.1. HYPERBOLICITY AND PEIXOTO’S THEOREM 101

with µ ∈ Rk, f(0, 0) = 0,Dxf(0, 0) = 0,Dxxf(0, 0) 6= 0, there exist smooth functions h(x, y), α(µ), λ(µ) with h(x, y) > 0, α(0) = 0, λ(0) = 0 such that

f(x + α(µ), µ) = h(x, µ)(x2 + λ(µ)),

for (x, µ) close to (0, 0). The familyx ˙ = x2 + λ, λ ∈ R, is a versal unfolding of the singularityx ˙ = ax2 + h.o.t. with a > 0.

The minimal required number of parameters, in a versal unfolding, is called the codimension of the bifurcation. In practice the notion of codimension is deﬁned more loosely as the number of parameters which must be varied for the bifurcation to occur in a persistent way. For instance, the cusp bifurcation 3 0 00 involves zeros of f(x) = x − λ1x − λ2; for λ1, λ2 = 0 the conditions are that f (0) = 0 and f (0) = 0: the codimension is two and two parameters are included to study perturbations.

Definition 0.7. For an integer k, consider the equivalence relation for which smooth functions f, g : Rn → Rn are equivalent at 0, if Di(f − g)(0) = 0 for i ≤ k. The equivalence classes are called k-jets. The Taylor polynomial of order k of a function is the unique polynomial representative of a k-jet. In a general bifurcation study of equilibria, when one starts with a polynomial vector field Xk with Xk(0) = 0, the first questions are:

(i) Is Xk C0-determining: is any vector ﬁeld with the same k-jet as Xk, topologically equivalent to Xk in a neighborhood of 0?

(ii) Can Xk be stabilized: is there an l-jet Xl, l ≥ k, with equal k-jet as Xk, so that Xl is C0 determining?

Theorem 0.1. In dimension two, any k-jet of a vector ﬁeld at 0 can be stabilized. In dimension three and higher there are k-jets that can not be stabilized.

5.1 Hyperbolicity and Peixoto’s theorem

Hyperbolic dynamics is characterized by the presence of expanding and contracting directions for the derivative. An equilibrium p of a vector ﬁeld X is hyperbolic if the spectrum of Df(p) is disjoint from s u the imaginary axis. Then there are a splitting TpM = E (p) ⊕ E (p) and constants C ≥ 1, λ < 0, µ > 0, with

λt s kDφt(x)vk ≤ Ce kvk, v ∈ E (p), 1 kDφ (x)vk ≥ eµtkvk, v ∈ Eu(p) t C ˜ for all t > 0. Note that there is t0 > 0 and λ < λ,µ ˜ < µ, so that

λt˜ s kDφt(x)vk ≤ e kvk, v ∈ E (p), µt˜ u kDφt(x)vk ≥ e kvk, v ∈ E (p) 102 CHAPTER 5. STRUCTURAL STABILITY AND BIFURCATION

for all t > t0. A corresponding definition can be given for fixed points of diffeomorphisms; a fixed p of a s u diffeomorphism f is hyperbolic if there is splitting TpM = E (p) ⊕ E (p) and constants C ≥ 1, 0 < λ < 1, µ > 1, with

kDf n(x)vk ≤ Cλnkvk, v ∈ Es(p), 1 kDf n(x)vk ≥ µnkvk, v ∈ Eu(p) C for all n ≥ 0. As the following result stipulates, a weaker condition then hyperbolicity suﬃces for persistence of an equilibrium under perturbations.

n n m 1 Theorem 1.1. Let fµ : R → R , µ ∈ R , be a family of C maps such that

1 (i) the map (x, µ) 7→ fµ(x) is C ,

(ii) fµ0 (x0) = 0,

(iii) 0 is not an eigenvalue of Dfµ0 (x0).

m n 1 Then there are open sets U ∈ R ,V ∈ R with µ0 ∈ U, x0 ∈ V , and a C function ξ : V → U such that for each µ ∈ U, ξ(µ) is the only equilibrium of x˙ = fµ(x) in V .

Proof. The proof is an immediate consequence of the implicit function theorem applied to the map

(x, µ) 7→ fµ(x).

A similar result holds for fixed points of diffeomorphisms, here one assumes that 1 is not an eigenvalue of the linearization. Define the stable set W s(p) and the unstable set W u(p) by

s W (p) = {x ∈ M ; φt(x) → p as t → ∞}, u W (p) = {x ∈ M ; φt(x) → p as t → −∞}.

We state the fundamental stable manifold theorem, originating with work of Jacques Hadamard and Oskar Perron. The attribute fundamental is underlined by the following quote from [2].

Every ﬁve years or so, if not more often, someone ‘discovers’ the theorem of Hadamard and Perron proving it either by Hadamard’s method or Perron’s. I myself have been guilty of this. (Dmitry Anosov)

Theorem 1.2. Supppose p is a hyperbolic equilibrium of a vector ﬁeld f. Then W s(p) is a manifold, s s s u injectively immersed in M, with TpW (p) = E (p). We call W (p) the stable manifold. Likewise, W (p) u u u is a manifold, injectively immersed in M, with TpW (p) = E (p). We call W (p) the unstable manifold.

Proof. We present a variant of Perron’s proof. Consider the time one map f = φ1 of the ﬂow. We will s construct a local stable manifold Wloc(p) near p for f and show that this is local stable manifold of p for s s φt. Then W (p) = ∪t∈Rφt(Wloc) is the orbit of the local stable manifold. 5.1. HYPERBOLICITY AND PEIXOTO’S THEOREM 103

Let us make this precise. Working in a chart near p, we may assume that p is the origin in Rn. For given small δ > 0, let s n n Wloc = {u ∈ R ; kf (x)k < δ for all n ∈ N}.

s u Take coordinates u = (x, y) ∈ E (0) × E (0). Write further πs and πu for the coordinate projections µ πs(x, y) = x and πu(x, y) = y. Let e be a bound for the eigenvalues of Df(0)|Es(0); spec Df(0)|Es(0) < µ λ e < 1, Likewise, let 1 < e < spec Df(0)Eu(0). Replacing f by an iterate of f, if necessary, we may µ s λ u assume kDf(0)vsk < e kvsk for all vs ∈ E (0) and kDf(0)vuk > e kvuk for all vu ∈ E (0). Let B(N, Rn) be the space of bounded sequences N → Rn, endowed with the supnorm

kγk0 = sup kγnk, n∈N where k(x, y)k = max {kxk, kyk} is a box norm on Rn = Es(0) × Eu(0), for given norms on Es(0) and Eu(0). Deﬁne the map Γ : Es(0) × B(N, Rn) → B(N, Rn) by ( (x , π f −1(γ(1))), if n = 0, Γ(x , γ)(n) = 0 u 0 −1 (πsf(γ(n − 1)), πuf (γ(n + 1))), if n > 0.

s s n Write Dδ = {x0 ∈ E ; kx0k ≤ δ} and Bδ = {γ ∈ B(N, R ); kγk0 ≤ δ} We claim that for δ small enough,

s s (i)Γ( Dδ × Bδ) ⊂ Dδ × Bδ,

s (ii) Γ is a contraction on {x0 ∈ E ; kx0k ≤ δ} × Bδ.

By continuity of derivatives of Df and Df −1 one has the following. For ε > 0 there exists δ > 0 so that

µ kπsf(x, y) − πsf(¯x, y¯)k < (e + ε)k(x, y) − (¯x, y¯)k, −1 −1 −λ kπuf (x, y) − πuf (¯y, y¯)k < (e + ε)k(x, y) − (¯x, y¯)k for all (x, y), (¯x, y¯) with k(x, y)k, k(¯x, y¯)k < δ. Pick ε so that both eµ + ε < 1 and e−λ + ε < 1. To prove that Γ is a contraction in the second coordinate, compute

kΓ(x0, γ1) − Γ(x0, γ2)k0

= sup kΓ(x0, γ1)(n) − Γ(x0, γ2)(n)k n∈N −1 −1 ≤ sup max{kπsf(γ1(n − 1)) − πsf(γ2(n − 1))k, kπuf (γ1(n + 1)) − πuf (γ2(n + 1))k} n∈N µ −λ ≤ sup max{e + ε, e + ε}kγ1 − γ2k0 n∈N

≤ κkγ1 − γ2k0 for κ = max{eµ + ε, e−λ + ε} < 1. A similar estimate establishes the first item. By Theorem 1.6, Γ is differentiable, so that by the uniform contraction theorem, the fixed point of Γ depends differentiable on x0. The correspondence x0 7→ ηx0 (0) defines a smooth manifold. 104 CHAPTER 5. STRUCTURAL STABILITY AND BIFURCATION

Let ηx0 be the ﬁxed point of Γ(x0, ·), kx0k < δ. We claim that ηx0 is a positive orbit of f, for ε small enough (we can take ε smaller if needed). Write ηx0 (n) = (xn, yn). Given n ∈ N, we have

−1 πsf(xn, yn) = xn+1, πuf (xn+1, yn+1) = yn.

This deﬁnes (xn+1, yn+1) as function of xn, yn: with

−1 F (xn, yn, xn+1, yn+1) = (πsf(xn, yn) − xn+1, πuf (xn+1, yn+1) − yn)

we have F (0, 0, 0, 0) = (0, 0)

and

Dxn+1,yn+1 F (0, 0, 0, 0)

has bounded inverse. Hence, applying the implicit function theorem, for (xn, yn) near (0, 0), one ﬁnds

(xn+1, yn+1) as a smooth function of (xn, yn), equal to (0, 0) if (xn, yn) = (0, 0). Now let (¯xn+1, y¯n+1) =

f(xn, yn). We must show that (xn+1, yn+1) = (¯xn+1, y¯n+1). Observe that (¯xn+1, y¯n+1) = f(xn, yn) −1 implies that f (¯xn+1, y¯n+1) = (xn, yn) and thus

−1 πsf(xn, yn) =x ¯n+1, πuf (¯xn+1, y¯n+1) = yn.

So (xn+1, yn+1) and (¯xn+1, y¯n+1) satisfy the same equation. By uniqueness of solutions, it follows that

(xn+1, yn+1) = (¯xn+1, y¯n+1).

We conclude that ηx0 is a positive orbit of f that stays in a δ-neighborhood of 0. We must show

that ηx0 (n) converges to 0 as n → ∞. For this we note invariance of the cone kyk ≤ kxk: for δ > 0

small enough and k(x, y)k ≤ δ, kyk ≤ kxk, we have kπuf((x, y))k ≤ kπsf((x, y))k. This follows since 1 kπuDf(0)(x, y)k ≤ kπsDf(0)(x, y)k and f is C . Now x0 7→ ηx0 (0) is a smooth manifold that is tangent s to E (0) at the origin. So ηx0 is contained in the cone kyk ≤ kxk for x0 small. From xn+1 = πsf(xn, yn) µ µ and kynk ≤ kxnk we get kxn+1k ≤ (e + δ)k(xn, yn)k ≤ (e + δ)kxnk so that limn→∞(xn, yn) = 0.

It remains to show that the local stable manifold of f = φ1 is also the local stable manifold of φt for kn any t > 0. For t = 1/k, the previous reasoning shows that φ1/k(x0, y0) goes to 0 as n → ∞. It follows that for any positive t = j/k ∈ Q,(x0, y0) is in the local stable manifold of φt. By continuity, (x0, y0) is in the local stable manifold for all real t > 0.

Remark 1.1. The construction in the above proof applied to the time one map f of the flow φt, finding orbits in the local stable manifold for f. A variant gives a construction directly for the flow φt, finding s 0 n 0 n orbits for φt. Write f(x) = Ax+r(x), with A = Df(0). Let Γt : E (0)×C ([0, ∞), R ) → C ([0, ∞), R ) be given by

Z t Z ∞ πsAt πsA(t−u) πuA(t−u) Γt(x0, γ)(t) = e x0 + e πsr(γ(u)) du, e πur(γ(u)) du . 0 t

The expression is derived from the variation of constants formula. The ﬁxed point ν(t) of Γ(x0, ·) is the positive orbit of f in the local stable manifold of 0 with πs(ν(0)) = x0. 5.1. HYPERBOLICITY AND PEIXOTO’S THEOREM 105

The notion of hyperbolicity extends to periodic orbits and other invariant sets. Consider a vector

field f and its flow φt. Let Λ be a compact invariant set of the flow φt; φt(Λ) = Λ for all t ∈ R. Then s u Λ is called (uniformly) hyperbolic if there is a splitting TxM = E (x) ⊕ Rf(x) ⊕ E (x), depending continuously on x, and constants C ≥ 1, λ < 0, µ > 0 that

λt s kDφt(x)vk ≤ Ce kvk, v ∈ E (x), 1 kDφ (x)vk ≥ eµtkvk, v ∈ Eu(x). t C This definition without the flow direction also applies to diffeomorphisms. Let f : M → M be a diffeomorphism and Λ a compact invariant set. Then Λ is called (uniformly) hyperbolic if there is a s u splitting TxM = E (x) ⊕ E (x), depending continuously on x, and constants C ≥ 1, 0 < λ < 1, µ > 1, that

kDf n(x)vk ≤ Cλnkvk, v ∈ Es(x), 1 kDf n(x)vk ≥ µnkvk, v ∈ Eu(x). C Theorem 1.3. Let f : M → M be a diﬀeomorphism with a maximal invariant hyperbolic set Λ: there is an open neighborhood U of Λ so that

n Λ = ∩n∈Zf (U).

Then there is a neighborhood V of f in the C1 topology, so that for any g ∈ V ,

n Λ(g) = ∩n∈Zg (U)

is a hyperbolic set of g.

Proof. The theorem follows by finding a robust way of constructing the stable an unstable subspaces Es(x),Eu(x). This is done using invariant (stable and unstable) cone fields. For simplicity we assume s C = 1 in the definition of hyperbolicity. For x ∈ Λ, write v ∈ TxM as v = vs + vu with vs ∈ E (x), vu ∈ Eu(x). For α > 0 and x ∈ Λ define cones

s Kα(x) = {v ∈ TxM ; |vu| ≤ α|vs|}, u Kα(x) = {v ∈ TxM ; |vs| ≤ α|vu|}.

We may extend these cones to cone ﬁelds in TU M, possible taking a smaller neighborhood U of Λ. Note that there exists ν > 1 so that for α small enough, x ∈ Λ,

u u Df(x)Kα(x) ⊂ Kα/ν (f(x)), (1.1) u |Df(x)v| ≥ ν|v| for v ∈ Kα. (1.2)

It follows that Eu(x) is obtained as

u n −n u −n E (x) = ∩n≥0Df (f (x))Kα(f (x)) 106 CHAPTER 5. STRUCTURAL STABILITY AND BIFURCATION

for some small α > 0. Similarly,

s −n n u n E (x) = ∩n≥0Df (f (x))Kα(f (x)).

Estimates (1.1) and (1.2) apply for any g suﬃciently close to f in the C1 topology, with x ∈ Λ(g). This proves the theorem.

For two dimensional ﬂows, Peixoto’s theorem classiﬁes the structurally stable systems.

Theorem 1.4. Let M be a compact orientable surface. The C1 structurally stable ﬂows are those which satisfy the following properties:

(i) all equilibria are hyperbolic,

(ii) all periodic solutions are hyperbolic,

(iii) there are no orbits that converge to equilibria for both positive and negative time.

The set of C1-structurally stable ﬂows is open and dense in C1(M).

An orbit that converges to equilibria for both positive and negative time is called a saddle connection. It is called a homoclinic orbit if the two equilibria are the same, and a heteroclinic orbit connects two different equilibria. Figure 1.1 illustrates the effect of breaking a heteroclinic orbit. To give the flavor of

Figure 1.1: The two panels on the left show part of a phase diagram of a vector field and a small perturbation which is qualitatively the same. In contrast, the two panels on the right show part of a phase diagram of a vector field, with a saddle connection, and a small perturbation which is qualitatively different. the argumentation in the proof of Theorem 1.4, we prove the simpler one-dimensional case.

Theorem 1.5. Let M be the circle. The C1 structurally stable ﬂows are those which satisfy the following property:

(i) all equilibria are hyperbolic. 5.1. HYPERBOLICITY AND PEIXOTO’S THEOREM 107

The set of C1-structurally stable ﬂows is open and dense in C1(M).

Proof. We first prove that a vector field f with only hyperbolic equilibria is C1 structurally stable. Identifying the circle with R/Z, we can consider f as a periodic function on R. Suppose first f has no equilibria. Without loss, f > 0. The f has a positive minimum. Any g sufficiently close to f in the C1 topology is also positive. So g(x) = t(x)f(x) for a positive function h. It follows that f and g are topologically equivalent. Next suppose that f does have hyperbolic equilibria. Since hyperbolic equilibria are isolated, there is a finite number of them by compactness of the circle. Denote the hyperbolic equilibria of f by x1, . . . , xk.

Between two equilibria, f does not change sign. By the implicit function theorem, for any xk there is 1 an interval neighborhood Uk of xk and a neighborhood V of f in C (R/Z), so that if g ∈ V , then g has 1 a unique hyperbolic equilibrium yk in Uk. Outside the intervals Uk, f is nonzero. Any g sufficiently C close to f is also nonzero outside the Uk’s. We get the existence of a neighborhood V of f in C1(R/Z) so that if g ∈ V , then g has has precisely k equilibria yk, each yk ∈ Uk. Given f with equilibria xk and nearby g ∈ V with equilibria yk, we can define a homeomorphism h : R/Z → R/Z with h(xk) = yk and arbitrary increasing (e.g. piecewise affine) in between. It is clear that h defines a topological equivalence between f and g. We establish that the set of C1 structurally stable systems is dense in C1(R/Z). This relies on Sard’s theorem, Theorem 9.2. Consider the family of vector fields x 7→ f(x) + ε depending on a real parameter ε. By Sard’s theorem, there are values of ε arbitrarily close to 0, for which f + ε has no critical value at 0. The vector field f + ε then has only hyperbolic equilibria. It remains to show that if the hypothesis on hyperbolic fixed points is not satisfied, then f is not structurally stable. Sard’s theorem shows that a system with infinitely many equilibria is not C1 structurally stable. A homeomorphism defining topological equivalence must map equilibria onto equilibria, and by Sard’s theorem there are arbitrary small perturbations with only finitely many equilibria. So assume that f has a finite number of isolated equilibria, with a nonhyperbolic equilibrium x1. Observe that x1 is a critical point of f. If f has the same sign on either side of x1, say f > 0 on U \{x1} for a small neighborhood U of x1, then for any ε > 0, f + ε > 0 on U.

If f has different signs on both sides of x1, we apply a different perturbation. Assume f(x) > 0 for x > x1, x ∈ U, and f < 0 for x < x1, x ∈ U. Here U is a small interval neighborhood of x1. Let χ : R → [0, 1] be a bump function: χ is a smooth nonnegative function with χ(x) = 1 on [−1, 1] and χ = 0 on R \ [−2, 2]. Define j(x) = −2xχ(x) and, for ε > 0,

2 jε(x) = −2ε xχ(x/ε).

1 0 2 Note that jε converges to 0 in the C topology if ε → 0. As jε(0) = −2ε , it is easily seen that f(x) + jε(x − x1) possesses three equilibria close to x1, for ε > 0 small. 108 CHAPTER 5. STRUCTURAL STABILITY AND BIFURCATION

5.2 The Hartman-Grobman theorem

Let A be a nonsingular n × n matrix. Suppose that Rn = Es ⊕ Eu is an invariant splitting for A. For n s u n any x ∈ R , we let x = xs + xu with xs ∈ E and xu ∈ E . On R we can take a product norm s u |x| = max{|xs|, |xu|}, for given norms on E , E . We suppose that the eigenvalues of As = A|Es have moduli less than one and that the eigenvalues of Au = A|Eu have moduli greater than one. By a linear coordinate change we may assume that −1 |As|, |Au | < 1.

Let Cj be the space of maps from Rn to Rn whose derivatives up to order j are bounded and uniformly j continuous. The norm | · |j on C is the sup norm of all derivatives up through order j;

j n |f|j = sup {|f(x)|, |Df(x)|,..., |D f(x)|;; x ∈ R }.

Theorem 2.1. There is µ > 0 such that, for any f ∈ C1 = {f ∈ C1 ; |f| < µ }, there is a unique 0 µ0 1 0 homeomorphism h with f 7→ h−id ∈ C0 depending continuously on f, h(0) = id, and h◦(A+f) = A◦h.

Proof. Let a = max(|A |, |A−1|) < 1 and choose µ > 0 so that a − µ > 0, and for any f ∈ C1 , s u 0 0 µ0 (A + f)−1 exists and belongs to C1. If f ∈ C1 , the equation h ◦ (A + f) = A ◦ h is equivalent to either µ0 of the equations

h = A ◦ h ◦ (A + f)−1, h = A−1 ◦ h ◦ (A + f).

We will use the first equation to define hs and the second to define hu (where h = hs + hu). For any continuous h, f ∈ C1 , define T (h, f) = T (h, f) + T (h, f) by the relations µ0 s u

−1 T (h, f)s = hs − As ◦ hs ◦ (A + f) , −1 T (h, f)u = hu − Au ◦ hu ◦ (A + f).

Denote D0 = {h : n → n ; h − id ∈ C0}. It is easy to verify that T : D0 × C1 → C0 is continuous in R R µ0 h, f, and T (id, 0) = 0. Furthermore, DhT (h, f) exists and is continuous in h, f with

−1 (DhT (h, 0)g)s = gs − As ◦ gs ◦ A , −1 (DhT (h, 0)g)u = gu − Au ◦ gu ◦ A.

0 −1 For any w ∈ C , the equation DhT (h, 0)g = w has a unique solution bounded above by (1 − a) |w| −1 (note that DhT (h, 0) is of the form I + L with |L| < a, so that DhT (h, 0) = (I − L) exists and equals 2 3 I + L + L + L + ...). Thus, DhT (h, 0) is an isomorphism. The implicit function theorem implies there is a unique function h = h(f), continuous in f in C1 (we may have to take µ smaller to get this), µ0 0 h(0) = id, and T (h(f), f) = 0. It remains to show that h is a homeomorphism. Consider the equation (A + f) ◦ g = g ◦ A for g ∈ C0, f ∈ C1. We can repeat the same type of argument as above to obtain a unique function g = g(f) ∈ C0, 5.3. THE SMALE HORSESHOE AND THE ARNOLD CAT MAP 109

f ∈ C1 , g(f) − id ∈ C0 continuous in f, g(0) = id and (A + f) ◦ g = g ◦ A. The same type of argument µ0 also gives unique solutions to equations of the form (A + f1) ◦ g = g ◦ (A + f2) for diﬀerentiable and small

f1, f2. The combination of solutions g and h of (A + f) ◦ g = g ◦ A and A ◦ h = h ◦ (A + f) provides a solution h ◦ g of h ◦ g ◦ A = Ah ◦ g and a solution g ◦ h of g ◦ h ◦ (A + f) = (A + f) ◦ g ◦ h. These solutions are unique. Since the identity map also solves these equations, we have h ◦ g = id and g ◦ h = id. Hence h is a homeomorphism.

If one only is interested in local structural stability of a hyperbolic fixed point, then the uniqueness of the local homeomorphism is lost. We now give the corresponding theorem for flows. Consider the differential equations

x˙ = Lx, (2.1) x˙ = Lx + R(x), (2.2)

1 s u n where R ∈ C . Assume there are invariant subspaces E ,E ⊂ R such that the restrictions Ls = L|Es and Lu = L|Eu have all their eigenvalues with negative and positive part respectively. Let φt(x) and

ψt(x) be the ﬂows generated by (2.1) and (2.2) respectively.

Theorem 2.2. There is a µ > 0 such that, for any r ∈ C1 , there exists a unique homeomorphism 0 µ0 h ∈ C0, depending continuously on r, such that, for all x ∈ Rn and t ∈ R,

h ◦ ψt(x) = φt(h(x)).

Proof. Consider the time one maps φ1 and ψ1. By the variation of constants formula,

L φ1(x) = e x, L ψ1(x) = e x + R(x),

with R ∈ C1 for some µ > 0. Replacing A and A + f by eL and eL + R in Theorem 2.1, we obtain a µ1 1 unique homeomorphism h such that h ◦ (eL + R) = eL ◦ h.

n We claim that h satisfies h ◦ ψt(x) = φt(x) ◦ h for all x ∈ R and t ∈ R. This is equivalent to proving φt(h(ψ−t(x)) = h(x). Let t ∈ R be fixed but arbitrary and α(x) = φt(h(ψ−t(x))). It is not difficult to 0 see that α ∈ C and α ◦ ψ1 = φ1 ◦ α. By the uniqueness of h in Theorem 2.1, α ≡ h.

5.3 The Smale horseshoe and the Arnold cat map

The Smale horseshoe is the hallmark of chaos. It is given as a geometric construction that clarifies chaotic dynamics encountered by Henri Poincaréand studied by George Birkhoff. Smale’s motivation was to gain a geometric understanding of dynamics found in driven oscillators considered by Mary Cartwright and John Littlewood. 110 CHAPTER 5. STRUCTURAL STABILITY AND BIFURCATION

Figure 3.1: The Smale horseshoe map.

Consider a square D ⊂ R2 and a diﬀeomorphism f : R2 → R2 that stretches and bends D into a horseshoe shape as shown in Figure 3.1. Assume that f stretches uniformly in the horizontal direction by a factor µ > 2 and contracts uniformly in the vertical direction by λ < 1/2. Note that f n(D) ∩ D consists of 2n horizontal strips of height λn. In the limit n → ∞ the image f n(D) ∩ D converges a set C+, which is the product of Cantor set and a horizontal line piece. In the same way, there are 2n vertical strips in f −n(D) ∩ D, that are mapped onto f n(D) ∩ D. The vertical strips have width µ−n. In the limit the vertical strips converge to the product C− of a Cantor set and a vertical line piece. The intersection C = C+ ∩ C− is a Cantor set which is the maximal invariant set

n C = ∩n∈Zf (D) in D. It is easily seen that a connected component in f −m(D) ∩ f n(D) ∩ D, a rectangle of height λn −m and width µ contains a periodic point. Periodic points are therefore dense in C. Indeed, let D1 be −m n m such a rectangle in f (D) ∩ f (D) ∩ D. Then f (D1) is a horizontal strip V1 and we get a decreasing m+n sequence of strips Vk+1 = f (Vk) ∩ Vk converging to a line as k → ∞. This line contains a periodic

point which lies inside D1. Although this is a specific model, it is robust. A diffeomorphism which is C1 close to f admits an invariant set which is a Cantor set close to C in the Hausdorff metric. We include a topological approach to Smale horseshoes. Take a box B = [0, 1]×[0, 1]n−1 ⊂ Rn. Write n−1 n−1 B0 = {0} × [0, 1] and B1 = {1} × [0, 1] . A connection is a continuous curve in B that connects

B0 to B1. A preconnection is a compact connected subset γ of B for which f(γ) is a connection. We deﬁne the crossing number m to be the largest number such that every connection contains at least m mutually disjoint preconnections. The following theorem is a special case of a result by Judy Kennedy and James Yorke [27].

Theorem 3.1. Let f : Rn → Rn be a diﬀeomorphism with crossing number 2 on the B = [0, 1]×[0, 1]n−1. Then there is an invariant set S ⊂ B that admits a surjective continuous map on the Cantor set C.

A second example of chaotic dynamics is given by hyperbolic torus automorphisms. Consider a matrix A ∈ GL(2, Z), i.e. a 2 × 2 matrix with integer coeﬃcients, whose inverse also has integer coeﬃcients. 5.3. THE SMALE HORSESHOE AND THE ARNOLD CAT MAP 111

! a b Note that det A = ±1. A matrix A = ∈ GL(2, Z) induces an automorphism on the torus c d T2 = R2/Z2. The induced map on T2, also denoted by A, is given by

A(x, y) = (ax + by, cx + dy) mod 1.

2 s u 2 Suppose that R = E ⊕ E is an invariant splitting for A. For any x ∈ R , we let x = xs + xu with s u xs ∈ E and xu ∈ E . We suppose that the eigenvalue of As = A|Es has modulus less than one and that the eigenvalue of Au = A|Eu has modulus greater than one. Hence

−1 |As|, |Au | < 1.

The induced map on T2 is called hyperbolic. Periodic points of A lie dense in T2, in fact, every point in T2 with rational coordinates is periodic. An example is ! 2 1 A = 1 1 √ 1 with eigenvalues 2 (3 + 5) and its reciprocal. This example is known as Arnold’s cat map, because its dynamics was illustrated using the image of a cat in [6], as in Figure 3.2.

Figure 3.2: The Arnold cat map.

Let Cj(T2) be the space of maps from T2 to T2 whose derivatives up to order j are bounded and j 2 uniformly continuous. The norm | · |j on C (T ) is the sup norm of all derivatives up through order j;

j n |f|j = sup {|f(x)|, |Df(x)|,..., |D f(x)|;; x ∈ R }.

Theorem 3.2. There is µ > 0 such that, for any f ∈ C1 ( 2) = {f ∈ C1( 2); |f| < µ }, there is a 0 µ0 T T 1 0 unique homeomorphism h in C0, depending continuously on f, h(0) = id, such that h ◦ (A + f) = A ◦ h.

Proof. The proof follows the argument in Section 5.2 to prove the Hartman-Grobman theorem. We lift functions on T2 to functions on its universal cover R2. 112 CHAPTER 5. STRUCTURAL STABILITY AND BIFURCATION

Following the proof of Theorem 2.1, deﬁne T : C0 × C1 as T (h, f) = T (h, f) + T (h, f) by the µ0 s u relations

−1 T (h, f)s = hs − As ◦ hs ◦ (A + f) , −1 T (h, f)u = hu − Au ◦ hu ◦ (A + f).

Write

j j 2 D = {f ∈ C ; f(x, y) + (k, l) = f(x, y) for all (k, l) ∈ Z }, Ej = {f ∈ Cj ; f − id ∈ Dj}.

One checks that T : E0 × D1 → E0 is well deﬁned. To see this, note that because A−1((x, y) + (k, l)) = µ0 A−1((x, y)) + A−1((k, l)), f is small in the C1 topology, and (A + f)−1 is Z2-periodic, we have

(A + f)−1((x, y) + (k, l)) = (A + f)−1((x, y)) + A−1((k, l)).

Likewise, for h − id ∈ Dj,

−1 −1 −1 hs((A + f) ((x, y)) + A ((k, l))) = hs((A + f) ((x, y))).

−1 −1 −1 As ◦ hs ◦ (A + f) ((x, y) + (k, l)) = As ◦ hs((A + f) ((x, y)) + A ((k, l))) −1 = As ◦ hs ◦ (A + f) ((x, y)) and, likewise,

−1 2 −1 Au ◦ hu ◦ (A + f)((x, y) + Z ) = Au ◦ hu ◦ (A + f)((x, y)).

The remainder of the proof follows the arguments for Theorem 2.1.

When Stephen Smale started to develop the theory of hyperbolic dynamical systems, he hoped that structurally stable systems would be typical. This would have been consistent with the situation in dimension one and two. However, he soon found examples of vector ﬁelds on higher-dimensional manifolds that cannot be made structurally stable by an arbitrarily small perturbation. This means that in higher dimensions, structurally stable systems are not dense. This diminished the importance of the notion of structural stability. We discuss the example by Stephen Smale on the four dimensional torus. Other examples were later found on three dimensional manifolds. In fact, Lorenz equations for parameter values with a Lorenz attractor, constitute examples.

Theorem 3.3. There is an open set of vector ﬁelds in C2(T4) that is not C1 structurally stable.

Proof. The construction involves diffeomorphisms on the three dimensional torus T3 = R3/Z3. This gives an example of a flow on the four dimensional torus. On T3 we have coordinates (x, y, z). Let f be a diffeomorphism on T3 with the following properties. 5.4. STABILITY THEOREMS 113

(i) On a small neighborhood U of the two-torus {z = 0}, we have

f(x, y, z) = (2x + y, x + y, z/2).

(ii) On a small neighborhood V of p = (0, 0, 1),

f(x, y, z) = (x/2, y/2, −1 + 2z).

On the invariant two-torus T2 = {z = 0}, there is a foliation: T2 is filled with a collection of curves ! 2 1 parallel to the stable eigendirection of . Inside U there is therefore a foliation with two 1 1 dimensional leaves W s(q), q ∈ T2. The fixed point p has a one dimensional unstable manifold W u(p). Assume that W u(p) is tangent to a stable leave W s(q) for some q ∈ T2. Now if we perturb f to a nearby diffeomorphism g, then g admits an invariant two-torus which is close to T2. Further, g has a stable foliation inside U. These statements can be proved with the method used for Theorems 2.1 and 3.2. To apply this method, note that we can consider g on T2 × R and replace g by a map that is equal to g on a small neighborhood of T2 × {0} and globally close to (x, y, z) 7→ (2x + y, x + y, z/2). In this setting the method works. Further, by the implicit function theorem, g has a saddle point near p. For f as well as for g, both stable manifolds of periodic points and stable manifolds not containing periodic points are dense in the stable foliation in U. An arbitrary small perturbation will make W u(p) tangent to either a stable manifold of a periodic point or a stable manifold not containing a periodic point. However, the property that W u(p) is tangent to a stable manifold of a periodic point is an invariant under topological conjugacy. So any diffeomorphism sufficiently close to f is not structurally stable.

5.4 Stability theorems

We state some key results on global stability of vector fields, and continue with a discussion of open problems. Recall Theorem 1.3 on robustness of maximal invariant hyperbolic sets. The next result yields structural stability of diffeomorphisms restricted to maximal invariant hyperbolic sets, and generalizes Theorem 3.2 for hyperbolic torus automorphisms. Consider a diffeomorphism f : M → M on a compact manifold M. Recall that a compact invariant s u set Λ is called (uniformly) hyperbolic if there is a splitting TxM = E (x)⊕E (x), depending continuously on x, and constants C ≥ 1, 0 < λ < 1, µ > 1, that

kDf n(x)vk ≤ Cλnkvk, v ∈ Es(x), 1 kDf n(x)vk ≥ µnkvk, v ∈ Eu(x). C

n The hyperbolic set Λ is called maximal invariant if Λ = ∩n∈Zf (U) for some open neighborhood U of Λ. 114 CHAPTER 5. STRUCTURAL STABILITY AND BIFURCATION

Theorem 4.1. Let f : M → M be a diﬀeomorphism with a maximal invariant set

n Λ = ∩n∈Zf (U)

which is hyperbolic. There exists ε > 0 so that any g which is ε close to f in the C1 topology, possesses a

maximal invariant hyperbolic set Λ(g) in U. Moreover, f|Λ and g|Λ(g) are topologically conjugate: there exists a homeomorphism h = h(g):Λ → Λ(g) with

g ◦ h = h ◦ f.

The dependence g 7→ h(g) is continuous in the C0 topology and h(f) = id.

Proof. We follow ideas that are contained in the proof of Theorem 1.2. Along Λ, there is a continuous s u splitting TΛM = E ⊕E in stable and unstable subspaces. For simplicity assume C = 1 in the deﬁnition

of hyperbolicity. For x ∈ M and δ > 0 small, the ball Bδ(x) of radius δ about x is well deﬁned. Cover

Λ with ﬁnitely many balls Uj = Bδ(yj) of small radius δ > 0 with the property that for any x ∈ Λ, the m ball Bδ/2(x) is contained in one of Uj’s. Working in a chart, we may assume Uj ⊂ R . Then we have s u u s coordinate projections πs to E (yj) with kernel E (yj) and πu to E (yj) with kernel E (yj).

Fix an orbit x(i + 1) = f(x(i)), i ∈ Z, of f in Λ. Suppose x(i) ∈ Uj(i) and Bδ/2(x(i)) ⊂ Uj(i). Write B(Z,M) for the space of sequences Z 7→ M endowed with the C0 topology and let B = {γ ∈ B(Z,M); γ(i) ∈ Bδ/2(x(i))}. Deﬁne Γ : B → B(Z,M) by

−1 Γ(γ)(n) = (πsf(γ(n − 1)), πuf (γ(n + 1))).

1 Likewise, for a diﬀeomorphism g : M → M close to f in the C topology, deﬁne Γg : B → B(Z,M) by

−1 Γg(γ)(n) = (πsg(γ(n − 1)), πug (γ(n + 1))).

We make the following claims. For suﬃciently small ε > 0, there exists δ > 0 so that for any diﬀeomor- phism g : M → M that is ε close to f in the C1 topology,

(i)Γ g(B) ⊂ B,

(ii)Γ g is a contraction on B,

(iii) the ﬁxed point of Γg is the orbit z(i + 1) = g(z(i)) of g with z(i) ∈ Bδ/2(x(i)).

s u s The claims rely on closeness of f to Df(x(i)), closeness of E (x(i)) and E (x(i)) to E (yj(i)) and u 1 m s E (yj(i)), and closeness of g to f in the C topology. Consider first projectionsπ ˆs : R → E (x(i)) and m u πû : R → E (x(i)) withπ ˆs +π û = id. For ε > 0 there is δ > 0 so that for x1, x2 ∈ Bδ/2(x(i − 1)),

|πˆsf(x1) − πˆsf(x2)| ≤

|πˆs(f(x1) − x(i) − Df(x(i − 1))(x(i − 1) − x1))| + |πˆs(f(x2) − x(i) − Df(x(i − 1))(x(i − 1) − x2))|

+|πˆsDf(x(i − 1))(x1 − x2)| ≤

2ε|x1 − x2| + λ|x1 − x2|. 5.4. STABILITY THEOREMS 115

−1 A similar estimate applies toπ ˆuf . We conclude that for ε > 0 there is δ > 0 so that for x1, x2 ∈

Bδ/2(x(i − 1)), y1, y2 ∈ Bδ/2(x(i + 1)),

|πˆsf(x1) − πˆsf(x2)| ≤ (λ + 2ε)|x1 − x2|, −1 −1 |πˆuf (y1) − πˆuf (y2)| ≤ (1/µ + 2ε)|y1 − y2|.

s u We replace the projectionsπ ˆs, πˆu by πs, πu. From the continuous dependence of the splitting E ,E

along Λ, we get the following. For ε > 0 there exists δ > 0 so that for x1, x2 ∈ Bδ/2(x(i − 1)),

|πsf(x1) − πsf(x2)| ≤

|πsf(x1) − πˆsf(x1)| + |πsf(x2) − πˆsf(x2)| + |πˆsf(x1) − πˆsf(x2)| ≤

2ε|x1 − x2| + (λ + 2ε)|x1 − x2|.

With a similar estimate for the unstable coordinate, we get that for ε > 0 there is δ > 0 so that for

x1, x2 ∈ Bδ/2(x(i − 1)), y1, y2 ∈ Bδ/2(x(i + 1)),

|πsf(x1) − πsf(x2)| ≤ (λ + 4ε)|x1 − x2|, −1 −1 |πuf (y1) − πuf (y2)| ≤ (1/µ + 4ε)|y1 − y2|.

1 Finally and in the same manner, for g within ε of f in the C topology, again with x1, x2 ∈ Bδ/2(x(i−1)),

y1, y2 ∈ Bδ/2(x(i + 1)),

|πsg(x1) − πsg(x2)| ≤ (λ + 6ε)|x1 − x2|, −1 −1 |πug (y1) − πsg (y2)| ≤ (1/µ + 6ε)|y1 − y2|.

For ε small, both λ + 6ε < 1 and 1/µ + 6ε < 1. The ﬁrst two claims follow. The observation that the ﬁxed point is an orbit, follows as in the proof of Theorem 1.2.

Deﬁne a map h :Λ → U that maps x(k) ∈ Λ to the ﬁxed point of Γg({x(i)})(k). The construction shows that h ◦ f = g ◦ h. We leave it to the reader to show that h is continuous. Vice versa, recall that the maximal invariant set of g in U is hyperbolic by Theorem 1.3. Given an orbit {y(i)} of g in h(Λ) for

g suﬃciently close to f,Γf ({y(i)}) provides a nearby orbit of f in Λ. So the map j : g(Λ) → M that

maps y(k) ∈ g(Λ) to the ﬁxed point of Γf ({y(i)})(k) is the inverse of h and is continuous. It follows that h is a homeomorphism.

Remark 4.1. The construction yields as a byproduct a shadowing property for orbits in maximal in-

variant hyperbolic sets: for any sequence of points yi with yi+1 close to f(yi) and close to Λ, there is a

nearby real orbit xi with xi+1 = f(xi). We say that yi is shadowed by xi. See e.g. [7].

Deﬁnition 4.1. A point x is called nonwandering for a ﬂow φt if for any neighborhood U of x, there is

t > 0 with φt(U) ∩ U 6= ∅. The nonwandering set is the set of nonwandering points.

Although this definition does not make a statement on the positive orbit of x, it means that close to x there are points that return close to x. The nonwandering set forms a closed invariant set. A vector field f is Kupka-Smale if it satisfies the following properties. 116 CHAPTER 5. STRUCTURAL STABILITY AND BIFURCATION

(i) The critical elements, i.e. the equilibria and periodic orbits of f, are hyperbolic.

s u (ii) If σ1, σ2 are critical elements of f, then the invariant manifolds W (σ1) and W (σ2) are transversal.

The Kupka-Smale theorem states that any smooth vector ﬁeld can be perturbed to a Kupka-Smale vector ﬁeld.

Theorem 4.2. For any compact manifold M, the set of Kupka-Smale vector ﬁelds is dense in Cr(M).

There are open sets of vector fields that are not structurally stable. Among the class of structurally stable systems are Morse-Smale systems with simple dynamics. A vector field f is Morse-Smale if it satisfies the following properties.

(i) f has a ﬁnite number of critical elements, all of which are hyperbolic.

s u (ii) If σ1, σ2 are critical elements of f, then the invariant manifolds W (σ1) and W (σ2) are transversal.

(iii) The nonwandering set of f equals the union of the critical elements.

Structural stability of Morse-Smale vector ﬁelds was proved by Stephen Smale and Jacob Palis.

Theorem 4.3. A Morse-Smale vector ﬁeld is structurally stable in Cr(M).

Among systems with simple dynamics, not necessarily structurally stable, are gradient vector fields. Given a manifold M ⊂ Rn and a function V : Rn → R, a gradient vector field is defined by

f(x) = −πx∇V (x),

n where πx is the orthogonal projection of vectors at x in R to tangent vectors in TxM. The vector field is determined by V restricted to M. The corresponding flow is called a gradient flow. Different transitions from Morse-Smale systems to systems with chaotic dynamics exist. Cascades of period-doubling bifurcations provide one route to chaos in one parameter families. Direct transitions from Morse-Smale systems to systems with strange attractors also exist. Figure 4.1 depicts a phase diagram illustrating a direct transition from a Morse-Smale systems to a system with an attractor that is topologically conjugate to a Lorenz attractor; the bifurcation is a nonlocal bifurcation involving connecting orbits between an equilibrium and a periodic orbit. A vector field f satisfies Axiom A and strong transversality, if

(i) The nonwandering set Ω(f) is hyperbolic,

(ii) For any x, y ∈ Ω(f), the stable and unstable manifolds W s(x) and W u(y) are transversal.

Combined efforts of Robbin, Robinson, Mañé,Hu, Hayashi yield the following characterization of structural stability.

Theorem 4.4. A vector ﬁeld is C1 structurally stable if and only if satisﬁes Axiom A and strong transversality. 5.5. GLOBAL ASPECTS OF BIFURCATIONS 117

Figure 4.1: Direct transitions from Morse-Smale systems to systems with chaotic dynamics exist.

This result is restricted to the C1 topology. Similar characterizations in higher degrees of regularity are unknown. This, and the existence of open sets of systems that are not structurally stable, leaves the question of typical dynamics open. Here one should think of properties of a set of systems that one considers to include most dynamical systems. Jacob Palis formulated a set of problems that, if resolved, would permit a global view on chaos. Paraphrased he conjectured that

(i) For r ≥ 1, there is a Cr dense set D of dynamical systems such that any system in D has a ﬁnite number of attractors.

(ii) For any system f in D and generic k-parameter families of small Cr perturbations, almost all systems in the family have ﬁnitely may attractors.

Note that the conjectures predict that typical systems have a finite number of attractors, where typical is expressed in families with a finite number of parameters. The Newhouse theorem makes clear that it is false that generic systems have finitely many attractors. The conjectures gives further details on the attractors. For instance, it is stated that they support a physical measure: a measure that gives the distribution of points in the orbits converging to the attractor. The conjectures also state stochastic stability of the attractors: the distribution of points in orbits when small random perturbations are added each iterate, is close to the physical measure.

5.5 Global aspects of bifurcations

In the supercritical Hopf bifurcation, the attractor changes from an equilibrium to a small limit cycle. Although this is an important change in the dynamics in the sense that it is the moment that oscillations are created, the attractor changes little in the sense that close to the bifurcation it is close to the equilibrium. In contract, in the saddle-node bifurcation, the attractor disappears altogether. Following these examples one can distinguish hard and soft bifurcations. The saddle-node bifurcation and the subcritical Hopf bifurcation are examples of hard bifurcations. The supercritical Hopf bifurcation is an example of a soft bifurcation. 118 CHAPTER 5. STRUCTURAL STABILITY AND BIFURCATION

Hard bifurcations can have a profound eﬀect on the dynamics. This section treats the phenomenon of intermittency, ﬁrst described by Yves Pomeau and Paul Mannevile, that can accompagny a hard bifurcation such as a saddle-node bifurcation. Intermittency is the alternation of phases of apparently periodic and chaotic dynamics. Phases with nearly periodic dynamics are to referred to as laminar phases. In the other phase, the relaminarization phase, the orbit makes large, seemingly chaotic excursions away from the periodic region. These excursions are called chaotic bursts.

One context in which intermittent dynamics can occur is close to a saddle-node bifurcation. Let fµ

be a one parameter family of vector ﬁelds unfolding a saddle-node bifurcation at µ = µ0. For µ < µ0,

say, the family possesses an attracting equilibrium. This has disappeared for µ > µ0. Let U be a small

neighborhood of the equilibrium at µ = µ0. For µ slightly larger than µ0, an orbit starting in U will take a long time to leave U, slowed down by the “ghost of the equilibrium”. Intermittent dynamics occurs if orbits that leave U return to U at a later time, creating other periods of almost constant dynamics. We discuss one instance of intermittent dynamics, in the context of interval maps, where quantitative characteristics have been derived [21]. Consider the logistic family f (x) = µx(1 − x) for µ close to √ µ the bifurcation value µ0 = 1 + 8 of the saddle-node bifurcation of the period-three orbit O(a) = 3 {a, f (a), f (a)}. Let E¯ be a small neighborhood O(a) not containing a critical point of f . Let χ ¯ µ0 µ0 µ0 E be deﬁned as n−1 1 X i χE¯ (x, µ) = lim 1E¯ (fµ(x)), (5.1) n→∞ n i=0 ¯ whenever the limit exists, where 1E¯ is the usual indicator function of the set E. That is, χE¯ (x, γ) is the relative frequency with which the orbit O(x) visits E¯ (for those x for which the limit exists). The

following theorem discusses χE¯ (x, γ) for γ near 0.

Figure 5.1: The left panel shows time series for the quadratic map x 7→ µx(1 − x) for µ = 3.828 near the saddle-node bifurcation of the period three orbit. Both the laminar phase (nearly periodic) and chaotic bursts are clearly seen. The right panel shows the dependence on µ of the frequency of visits of orbits to

a small neighborhood E¯ of the period three orbit at µ0.

Theorem 5.1. Let {fµ} be as above, unfolding the saddle-node bifurcation of the 3-periodic orbit at

µ = µ0. There exist sets Ω ⊂ Γ of parameter values with positive density at µ = µ0, so that when 5.6. EXERCISES 119

restricting to µ ∈ Ω, χE¯ (x, µ) is a constant, χE¯ (µ), almost everywhere on [0, 1] and χE¯ (µ) depends

continuously on µ at µ0. Moreover, there exist K1,K2 > 0 so that

1 − χE¯ (µ) K1 ≤ lim √ ≤ K2. (5.2) µ∈Ω,µ&µ0 µ

We note that the function χE¯ (µ) is not continuous in µ at µ = µ0. Restricted to µ ∈ Ω it becomes continuous.

5.6 Exercises

Exercise 5.6.1. Consider real polynomials in one real variable.

(i) How many topologically different real fifth-degree polynomials are there with four different real critical values? Two polynomials are topologically the same if one can be transformed into the other by a continuous and orientation preserving change of dependent and independent real variables.

(ii) In the space of ﬁfth-degree polynomials consider those with four diﬀerent real critical values. How many connected components does this region have?

Exercise 5.6.2. Let fλ be a one parameter family of vector ﬁelds. Assume that for each λ ∈ [a, b], fλ is structurally stable. Prove that fa and fb are topologically equivalent.

Exercise 5.6.3. Show the following invariance properties under conjugacies for ﬂows.

(i) The period of a periodic orbit is invariant under a topological conjugacy.

(ii) Eigenvalues of the linearization about an equilibrium are invariant under a C1 conjugacy.

(iii) Floquet multipliers of a periodic solution are invariant under a C1 conjugacy.

Exercise 5.6.4. Consider the constant differential equation x˙ = f(x) = 1 on R. Show that there exists ε > 0 so that each g : R → R that is ε close in the uniform C1 topology ( |f(x) − g(x)| < ε and |f 0(x) − g0(x)| < ε for all x ∈ R), then the differential equations given by f and g are differentiably conjugate. That is, there is a C1 diffeomorphism h : R → R that conjugates the flows.

Exercise 5.6.5. Consider an automorphism of T2 = R2/Z2 corresponding to a matrix in GL(2, Z) whose eigenvalues are not roots of 1.

(i) Prove that every point with rational coordinates is periodic.

(ii) Prove that every periodic point has rational coordinates.

Exercise 5.6.6. Consider the class of linear diﬀerential equations on Rn. How many parameters are needed to unfold the zero matrix? 120 CHAPTER 5. STRUCTURAL STABILITY AND BIFURCATION

Exercise 5.6.7. Prove that, for r = 4, the map f4(x) = 4x(1 − x) is topologically conjugate to the tent map ( 2x, 0 ≤ x ≤ 1/2, T (x) = 2 − 2x, 1/2 ≤ x ≤ 1 on [0, 1], with conjugation given by h(x) = sin2(xπ/2).

Use this to ﬁnd a period two point for f4.

3 2 Exercise 5.6.8. Consider the diﬀerential equation x˙ = f(x) = x − λ0x − λ1xλ2 on R, depending on real parameters λ0, λ1, λ2. Show that the quadratic term can be removed by a translation y = x + a for suitable a.

Exercise 5.6.9. Let 0 be a hyperbolic stable equilibrium for a vector ﬁeld f. Give an example where Df(0) has only one eigenvalue λ < 0, but kU −1eDf(0)tUk > eλt for t > 0. For this example, provide λ˜ < 0, C > 0, so that keDf(0)tk < Ceλt˜ .

Exercise 5.6.10. Consider a diﬀerential equation on a compact manifold. Prove that the ω-limit set of a point is closed, nonempty, and connected.

Exercise 5.6.11. Consider a diﬀerential equation on a compact manifold. Prove that the nonwandering set is closed and invariant. Provide an example to show that the nonwandering and the limit set (the union of the ω-limit sets of points) can be diﬀerent. Is one always contained in the other?

Exercise 5.6.12. Show that an orientation reversing circle homeomorphism has rotation number 0.

Exercise 5.6.13. Let V : M → R be a C2 function on a compact manifold M such that each critical point is nondegenerate, i.e. at each point where ∇V (x) = 0, the Hessian matrix of second order partial derivatives is nonsingular. Prove that all fixed points of the gradient vector field are hyperbolic. Prove that the gradient vector field has no periodic orbits. p Exercise 5.6.14. For 0 < r < R, consider the torus ( y2 + z2 − R)2 + x2 = r2 in R3. Let V : R3 → R be the height function V (x, y, z) = z. Show that the gradient vector field ∇V is not Morse-Smale.

Exercise 5.6.15. Morse-Smale diffeomorphisms are defined analogous to Morse-Smale flows, they are diffeomorphisms with a finite number of hyperbolic peridic points and transversally intersecting stable and unstable manifolds. Let φt be a Morse-Smale flow for a vector field.

(i) Suppose φt has no periodic solutions. Prove that φ1 is a Morse-Smale diﬀeomorphism.

(ii) Suppose φt has a periodic solution. Prove that φ1 is not a Morse-Smale diﬀeomorphism. Bibliography

[1] R. Abraham and J. E. Marsden. Foundations of mechanics. The Benjamin/Cummings Publishing Company, 1978.

[2] D.V. Anosov. Geodesic ﬂows on closed Riemann manifolds with negative curvature. Proc. Steklov Inst. Math., 1967.

[3] V.I. Arnold. Mathematical methods of classical mechanics. Springer-Verlag, 1978.

[4] V.I. Arnold. Geometrical methods in the theory of ordinary diﬀerntial equations. Springer-Verlag, 1988.

[5] V.I. Arnold. Catastrophe theory. Springer-Verlag, 1992.

[6] V.I. Arnold and A. Avez. Ergodic problems of classical mechanics. W.A. Benjamin, 1968.

[7] M. Brin and G. Stuck. Introduction to dynamical systems. Cambridge University Press, 2002.

[8] T. Br¨ocker. Diﬀerentiable germs and catastrophes. Cambridge University Press, 1975.

[9] B. Buﬀoni and J. Toland. Analytic theory of global bifurcation. An introduction. Princeton University Press, 2003.

[10] C. Chicone. Ordinary diﬀerential equations with applications. Springer-Verlag, 2006.

[11] P. Chossat and R. Lauterbach. Methods in equivariant bifurcations and dynamical systems. World Scientiﬁc, 2000.

[12] S.-N. Chow and J. K. Hale. Methods of bifurcation theory. Springer-Verlag, 1982.

[13] M. J. Feigenbaum. Quantitative universality for a class of nonlinear transformations. J. Stat. Phys., 19:25–52, 1978.

[14] M. J. Field. Dynamics and symmetry. World Scientiﬁc, 2007.

[15] M. Golubitsky and V. Guillemin. Stable mappings and their singularities. Springer-Verlag, 1980.

[16] M. Golubitsky and D. G. Schaeﬀer. Singularities and groups in bifurcation theory. Volume I. Springer-Verlag, 1985.

[17] M. Golubitsky, I. Stewart, and D. G. Schaeﬀer. Singularities and groups in bifurcation theory. Volume II. Springer-Verlag, 1988.

121 122 BIBLIOGRAPHY

[18] J. Graczyk and G. Swiatek.´ Generic hyperbolicity in the logistic family. Ann. Math. (2), 146:1–52, 1997.

[19] J. Guckenheimer and P. Holmes. Nonlinear oscillations, dynamical systems, and bifurcations of vector ﬁelds. Springer-Verlag, 1983.

[20] M. W. Hirsch, S. Smale, and R. L. Devaney. Diﬀerential equations, dynamical systems, and an introduction to chaos. Academic Press, 2013.

[21] A. J. Homburg and T. Young. Intermittency in families of unimodal maps. Ergodic Theory Dyn. Syst., 22:203–225, 2002.

[22] E. Hopf. Abzweigung einer periodischen Lösungvon einer stationärenLösungeines Differentialsystems. Ber. Verh. Sächs. Akad. Wiss. Leipzig, Math.-Naturw. Kl. 94, No.1, 3-22, 1943.

[23] R. B. Hoyle. Pattern formation. An introduction to methods. Cambridge University Press, 2006.

[24] Yu. Ilyashenko and W. Li. Nonlocal bifurcations. American Mathematical Society, 1999.

[25] M. V. Jakobson. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Commun. Math. Phys., 81:39–88, 1981.

[26] A. Katok and B. Hasselblatt. Introduction to the modern theory of dynamical systems. Cambridge University Press, 1997.

[27] J. Kennedy and J. A. Yorke. Topological horseshoes. Trans. Am. Math. Soc., 353:2513–2530, 2001.

[28] H. Kielh¨ofer. Bifurcation theory. An introduction with applications to partial diﬀerential equations. Springer- Verlag, 2012.

[29] Y. A. Kuznetsov. Elements of applied bifurcation theory. Springer-Verlag, 2004.

[30] S. Lang. Real and functional analysis. Springer-Verlag, 1993.

[31] E.N. Lorenz. Deterministic Nonperiodic Flow. J. Atmos. Sci., 20:130–141, 1963.

[32] M. Lyubich. Feigenbaum-Coullet-Tresser universality and Milnor’s hairiness conjecture. Ann. Math. (2), 149:319–420, 1999.

[33] B. Malgrange. Ideals of diﬀerentiable functions. Oxford University Press, 1966.

[34] J.E. Marsden and M. McCracken. The Hopf bifurcation and its applications. Springer-Verlag, 1976.

[35] J. Martinet. Singularities of smooth functions and maps. Cambridge University Press, 1982.

[36] J. Palis and W. de Melo. Geometric theory of dynamical systems. Springer-Verlag, 1982.

[37] J. Palis and F. Takens. Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations. Fractal dimensions and inﬁnitely many attractors. Cambridge University Press, 1993.

[38] C. Robinson. Dynamical systems. Stability, symbolic dynamics, and chaos. CRC Press, 1999.

[39] O. E. R¨ossler.An equation for continuous chaos. Physics Letters A, 57:397–398, 1976. BIBLIOGRAPHY 123

[40] D. Ruelle. Elements of diﬀerentiable dynamics and bifurcation theory. Academic Press, 1989.

[41] L.P. Shilnikov, A.L. Shilnikov, D.V. Turaev, and L. O. Chua. Methods of qualitative theory in nonlinear dynamics. Part I. World Scientiﬁc, 1998.

[42] L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev, and L. O. Chua. Methods of qualitative theory in nonlinear dynamics. II. World Scientiﬁc, 2001.

[43] M. Shub. Global stability of dynamical systems. Springer-Verlag, 1987.

[44] G. Teschl. Ordinary diﬀerential equations and dynamical systems. American Mathematical Society, 2012.

[45] R. Thom. Stabilitéstructurelle et morphogenese. Essai d’une théoriegénéraledes modeles. Paris: Inter Editions. XIX, 1977.

[46] C. Tresser and P. Coullet. It´erationsd’endomorphismes et groupe de renormalisation. C. R. Acad. Sci., Paris, S´er.A, 287:577–580, 1978.

[47] A. M. Turing. The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond., Ser. B, 237:37–72, 1952

[48] H. Whitney. A function not constant on a connected set of critical points. Duke Math. J., 1:514–517, 1935.

[49] H. Whitney. The general type of singularity of a set of 2n − 1 smooth functions of n variables. Duke Math. J., 10:161–172, 1943.

[50] H. Whitney. On singularities of mappings of Euclidean spaces. I. Mappings of the plane into the plane. Ann. Math. (2), 62:374–410, 1955.