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Homoclinic and Heteroclinic Bifurcations in Vector Fields

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Homoclinic and Heteroclinic Bifurcations in Vector Fields Homoclinic and Heteroclinic Bifurcations in Vector Fields Ale Jan Homburg Bj¨ornSandstede Korteweg{de Vries Institute for Mathematics Division of Applied Mathematics University of Amsterdam Brown University 1018 TV Amsterdam, The Netherlands Providence, RI 02906, USA June 17, 2010 Abstract An overview of homoclinic and heteroclinic bifurcation theory for autonomous vector fields is given. Specifically, homoclinic and heteroclinic bifurcations of codimension one and two in generic, equivariant, reversible, and conservative systems are reviewed, and results pertaining to the existence of multi-round homoclinic and periodic orbits and of complicated dynamics such as suspended horseshoes and attractors are stated. Bifurcations of homoclinic orbits from equilibria in local bifurcations are also considered. The main analytic and geometric techniques such as Lin's method, Shil'nikov variables and homoclinic center manifolds for analyzing these bifurcations are discussed. Finally, a few related topics, such as topological moduli, numerical algorithms, variational methods, and extensions to singularly perturbed and infinite- dimensional systems, are reviewed briefly. 1 Contents 1 Introduction 4 2 Homoclinic and heteroclinic orbits, and their geometry5 2.1 Homoclinic orbits to hyperbolic equilibria.............................6 2.2 Homoclinic orbits to nonhyperbolic equilibria........................... 10 2.3 Heteroclinic cycles with hyperbolic equilibria........................... 12 3 Analytical and geometric approaches 13 3.1 Normal forms and linearizability.................................. 14 3.2 Shil'nikov variables.......................................... 15 3.3 Lin's method............................................. 17 3.4 Homoclinic center manifolds..................................... 18 3.5 Stable foliations............................................ 19 3.6 Case study: The creation of periodic orbits from a homoclinic orbit............... 20 3.6.1 Shil'nikov variables...................................... 20 3.6.2 Lin's method......................................... 22 3.6.3 Homoclinic center manifolds................................. 22 4 Phenomena 23 4.1 N-pulses and N-periodic orbits................................... 23 4.2 Robust singular dynamics...................................... 24 4.3 Singular horseshoes.......................................... 26 4.4 The boundary of Morse{Smale flows................................ 29 4.5 Homoclinic-doubling cascades.................................... 31 4.6 Intermittency............................................. 33 5 Catalogue of homoclinic and heteroclinic bifurcations 34 5.1 Homoclinic orbits in generic systems................................ 34 5.1.1 Creation of 1-periodic orbits................................. 34 5.1.2 Shil'nikov's saddle-focus homoclinic orbits......................... 36 5.1.3 Bi-focus homoclinic orbits.................................. 39 5.1.4 Belyakov transitions..................................... 40 5.1.5 Resonant homoclinic orbits................................. 41 5.1.6 Inclination flips........................................ 43 5.1.7 Orbit flips........................................... 44 5.1.8 Coexisting homoclinic orbits................................ 45 5.1.9 Degenerate homoclinic orbits................................ 48 5.1.10 Homoclinic orbits to nonhyperbolic equilibria....................... 48 5.2 Heteroclinic cycles in generic systems................................ 50 2 5.2.1 Heteroclinic cycles with saddles of identical Morse index................. 51 5.2.2 T-points: Heteroclinic cycles with saddles of different index............... 54 5.2.3 Heteroclinic cycles with nonhyperbolic equilibria..................... 56 5.2.4 Heteroclinic cycles containing periodic orbits....................... 58 5.3 Conservative and reversible systems................................ 59 5.3.1 Introduction and hypotheses................................ 59 5.3.2 Bi-foci homoclinic orbits................................... 60 5.3.3 Belyakov{Devaney transition................................ 61 5.3.4 Homoclinic orbits to equilibria with semisimple spectrum................ 62 5.3.5 Reversible systems with SO(2)-symmetry......................... 63 5.3.6 Reversible and Hamiltonian flip bifurcations........................ 64 5.3.7 Coexisting homoclinic orbits................................ 65 5.3.8 Degenerate homoclinic orbits................................ 66 5.3.9 Saddle-center homoclinic orbits............................... 67 5.3.10 Homoclinic orbits to nonhyperbolic equilibria....................... 69 5.3.11 Heteroclinic cycles and snaking............................... 70 5.4 Homoclinic orbits arising through local bifurcations........................ 71 5.4.1 Nilpotent singularities.................................... 71 5.4.2 Hopf/saddle-node bifurcations in generic and reversible systems............. 73 5.4.3 1:1 resonances in reversible systems............................ 76 5.4.4 02+i! resonances in reversible systems........................... 77 5.5 Equivariant systems......................................... 79 5.5.1 Robust heteroclinic cycles.................................. 80 5.5.2 Asymptotic stability of heteroclinic networks....................... 86 5.5.3 Bifurcations from heteroclinic cycles............................ 89 5.5.4 Forced symmetry breaking................................. 92 5.5.5 Homoclinic orbits in systems with Z2-symmetry..................... 93 6 Related topics 95 6.1 Topological indices.......................................... 95 6.2 Moduli................................................. 98 6.3 Existence results........................................... 100 6.4 Numerical techniques......................................... 101 6.5 Variational methods......................................... 102 6.6 Singularly perturbed systems.................................... 102 6.7 Infinite-dimensional systems..................................... 103 3 1 Introduction Our goal in this paper is to review the existing literature on homoclinic and heteroclinic bifurcation theory for flows. More specifically, we shall focus on bifurcations from homoclinic and heteroclinic orbits between equilibria in autonomous ordinary differential equations (ODEs) du = f(u; µ); (u; µ) 2 n × m; t 2 : (1.1) dt R R R Throughout the entire survey, we shall assume that the nonlinearity f is sufficiently smooth for the results to hold (the precise requirements can be found in the cited references). We write X (Rn) for the space of ODEs on Rn endowed with the C1 Whitney topology. Equilibria p of (1.1) are time-independent solutions that therefore satisfy f(p; µ) = 0. We say that a solution n h(t) of (1.1) is a heteroclinic orbit if h(t) ! p± as t ! ±∞ for equilibria p± 2 R . If p− = p+, we say that h(t) is a homoclinic orbit (assuming tacitly that h(t) is not the equilibrium solution itself). We will also consider heteroclinic cycles which, by definition, consist of several heteroclinic orbits hj(t) labelled by the index j = 1; : : : ; ` so that lim hj(t) = pj+1 = lim hj+1(t); j = 1 : : : ; ` t!1 t→−∞ with the understanding that h`+1 := h1 and p`+1 = p1. Illustrations of homoclinic and heteroclinic orbits can be found in Figure 1.1. Homoclinic and heteroclinic orbits play an important role in applications. For instance, we may be interested in modelling action potentials in nerve axons by an ODE of the form (1.1): in this case, we can think of u as representing the electric potential and certain ion concentrations in the nerve axon, with t being physical time. The rest state of the axon determines a natural equilibrium of (1.1), and action potentials in the axon correspond then to homoclinic orbits to this equilibrium. Heteroclinic orbits typically arise when a system can cycle through several different states. For instance, in certain Rayleigh{B´enardconvection experiments, roll patterns may arise that can orient themselves at angles of 0, 120 or 240 degrees: as time progresses, the roll pattern cycles through this set of angles, but stays at each angle for a long time, followed by a fast transition to the next angle. Thus, in an appropriate ODE model with three equilibria corresponding to the rolls oriented at angles of 0, 120 and 240 degrees, heteroclinic orbits correspond to transitions from one equilibrium to another. In the above examples, the independent variable t corresponds to physical time, whence we refer to such applications as describing temporal dynamics. Another class of applications, referred to as spatial dynamics, are problems where t represents a spatial direction. An important example in this respect are travelling- wave solutions of partial differential equations (PDEs) on unbounded domains. Consider, for instance, the reaction-diffusion system N Ut = DUxx + F (u);U 2 R ; x 2 R; (1.2) where t and x represent physical time and space, respectively. A travelling wave of (1.2) is a solution of the form U(x; t) = U∗(x − ct) corresponding to a fixed profile U∗ that travels to the right (for c > 0) or the left (for c < 0) as a function of time t. Upon substituting the ansatz U(x; t) = U∗(x − ct) into (1.1) and using ξ = x − ct as the new independent variable, we find that (U; V ) = (U∗;@ξU∗) must satisfy the ODE d U V = (1.3) dξ V −D−1(cV + F (U)) Figure 1.1: The panels contain a homoclinic orbit, which connects an equilibrium to itself [left], a heteroclinic orbit that connects two different equilibria [center], and a heteroclinic cycle
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