Elements of Applied Bifurcation Theory

Transcription

1 Yuri A. Kuznetsov Elements of Applied Bifurcation Theory Second Edition With 251 Illustrations Springer

2 Preface to the Second Edition Preface to the First Edition vii ix 1 Introduction to Dynamical Systems Definition of a dynamical system State space Time Evolution operator Definition of a dynamical system Orbits and phase portraits Invariant sets Definition and types Example 1.9 (Smale horseshoe) Stability of invariant sets Differential equations and dynamical systems Poincare maps Time-shift maps Poincare map and stability of cycles Poincare map for periodically forced systems Exercises Appendix 1: Infinite-dimensional dynamical systems defined by reaction-diffusion equations Appendix 2: Bibliographical notes 37 2 Topological Equivalence, Bifurcations, and Structural Stability of Dynamical Systems Equivalence of dynamical systems 39

3 xvi 2.2 Topological classification of generic equilibria and fixed points Hyperbolic equilibria in continuous-time systems Hyperbolic fixed points in discrete-time systems Hyperbolic limit cycles Bifurcations and bifurcation diagrams Topological normal forms for bifurcations Structural stability Exercises Appendix: Bibliographical notes 76 3 One-Parameter Bifurcations of Equilibria in Continuous-Time Dynamical Systems Simplest bifurcation conditions The normal form of the fold bifurcation Generic fold bifurcation The normal form of the Hopf bifurcation Generic Hopf bifurcation Exercises Appendix 1: Proof of Lemma Appendix 2: Bibliographical notes Ill 4 One-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems Simplest bifurcation conditions The normal form of the fold bifurcation Generic fold bifurcation The normal form of the flip bifurcation Generic flip bifurcation The "normal form" of the Neimark-Sacker bifurcation Generic Neimark-Sacker bifurcation Exercises Appendix 1: Feigenbaum's universality Appendix 2: Proof of Lemma Appendix 3: Bibliographical notes Bifurcations of Equilibria and Periodic Orbits in n-dimensional Dynamical Systems Center manifold theorems Center manifolds in continuous-time systems Center manifolds in discrete-time systems Center manifolds in parameter-dependent systems Bifurcations of limit cycles Computation of center manifolds Quadratic approximation to center manifolds in eigenbasis Projection method for center manifold computation 171

4 xvii 5.5 Exercises Appendix 1: Hopf bifurcation in reaction-diffusion systems on the interval with Dirichlet boundary conditions Appendix 2: Bibliographical notes Bifurcations of Orbits Homoclinic and Heteroclinic to Hyperbolic Equilibria Homoclinic and heteroclinic orbits Andronov-Leontovich theorem Homoclinic bifurcations in three-dimensional systems: Shil'nikov theorems Homoclinic bifurcations in n-dimensional systems Regular homoclinic orbits: Melnikov integral Homoclinic center manifolds Generic homoclinic bifurcations in R Exercises Appendix 1: Focus-focus homoclinic bifurcation in four-dimensional systems Appendix 2: Bibliographical notes Other One-Parameter Bifurcations in Continuous-Time Dynamical Systems Codim 1 bifurcations of homoclinic orbits to nonhyperbolic equilibria Saddle-node homoclinic bifurcation on the plane Saddle-node and saddle-saddle homoclinic bifurcations in R "Exotic" bifurcations Nontransversal homoclinic orbit to a hyperbolic cycle Homoclinic orbits to a nonhyperbolic limit cycle Bifurcations on invariant tori Reduction to a Poincare map Rotation number and orbit structure Structural stability and bifurcations Phase locking near a Neimark-Sacker bifurcation: Arnold tongues Bifurcations in symmetric systems General properties of symmetric systems Z 2 -equivariant systems Codim 1 bifurcations of equilibria in Z 2 -equivariant systems Codim 1 bifurcations of cycles in Z 2 -equivariant systems Exercises Appendix 1: Bibliographical notes 290

5 xviii 8 Two-Parameter Bifurcations of Equilibria in Continuous-Time Dynamical Systems List of codim 2 bifurcations of equilibria Bifurcation curves Codimension two bifurcation points Cusp bifurcation Normal form derivation Bifurcation diagram of the normal form Effect of higher-order terms Bautin (generalized Hopf) bifurcation Normal form derivation Bifurcation diagram* of the normal form Effect of higher-order terms Bogdanov-Takens (double-zero) bifurcation Normal form derivation Bifurcation diagram of the normal form Effect of higher-order terms Fold-Hopf (zero-pair) bifurcation Derivation of the normal form Bifurcation diagram of the truncated normal form Effect of higher-order terms Hopf-Hopf bifurcation Derivation of the normal form Bifurcation diagram of the truncated normal form Effect of higher-order terms Exercises Appendix 1: Limit cycles and homoclinic orbits of Bogdanov normal form Appendix 2: Bibliographical notes Two-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems List of codim 2 bifurcations of fixed points Cusp bifurcation Generalized flip bifurcation Chenciner (generalized Neimark-Sacker) bifurcation Strong resonances Approximation by a flow :1 resonance :2 resonance :3 resonance :4 resonance Codim 2 bifurcations of limit cycles Exercises Appendix 1: Bibliographical notes 460

6 xix 10 Numerical Analysis of Bifurcations Numerical analysis at fixed parameter values Equilibrium location Modified Newton's methods Equilibrium analysis Location of limit cycles One-parameter bifurcation analysis Continuation of equilibria and cycles Detection and location of codim 1 bifurcations Analysis of codim 1 bifurcations Branching points Two-parameter bifurcation analysis Continuation of codim 1 bifurcations of equilibria and fixed points Continuation of codim 1 limit cycle bifurcations Continuation of codim 1 homoclinic orbits Detection and location of codim 2 bifurcations Continuation strategy Exercises Appendix 1: Convergence theorems for Newton methods Appendix 2: Detection of codim 2 homoclinic bifurcations Singularities detectable via eigenvalues Orbit and inclination flips Singularities along saddle-node homoclinic curves Appendix 3: Bibliographical notes 535 A Basic Notions from Algebra, Analysis, and Geometry 541 A.I Algebra 541 A.I.I Matrices 541 A.1.2 Vector spaces and linear transformations 543 A.1.3 Eigenvectors and eigenvalues 544 A. 1.4 Invariant subspaces, generalized eivenvectors, and Jordan normal form 545 A.1.5 Fredholm Alternative Theorem 546 A.I.6 Groups 546 A.2 Analysis 547 A.2.1 Implicit and Inverse Function Theorems 547 A.2.2 Taylor expansion 548 A.2.3 Metric, normed, and other spaces 549 A.3 Geometry 550 A.3.1 Sets 550 A.3.2 Maps 551 A.3.3 Manifolds 551 References 553 Index 577