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Bifurcation Theory

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Bifurcation Theory 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 Poincar´ewho 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 dynam- ical 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 Poincar´eto 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 2 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 p p 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) = p p p p p −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(jxj3 + jµxj + jµj2) 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(jxj + jµj) 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(jµj2) 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 2 R, λ = (λ1; λ2) 2 R and consider the equation 3 4 2 2 2 f(λ, x) = x − λ1x − λ2 + O(jxj + jλ1x j + jλ1xj + jλ2xj + jλ2j + jλ1λ2j) = 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(jxj ) and λ2 = 2x + O(jxj ) 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) 2 U for each t ≥ 0; x 2 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 ! 1. (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.
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