Elements of Applied Bifurcation Theory

Transcription

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

2 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 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 A: Infinite-dimensional dynamical systems defined by reaction-diffusion equations Appendix B: Bibliographical notes 36 Topological Equivalence, Bifurcations, and Structural Stability of Dynamical Systems Equivalence of dynamical systems 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 67

3 XVIII 2.6 Exercises Appendix: Bibliographical notes 75 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 A: Proof of Lemma Appendix B: Poincare normal forms Appendix C: Bibliographical notes 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 A: Feigenbaum's universality Appendix B: Proof of Lemma Appendix C: 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 Restricted normalized equations for ODEs Restricted normalized equations for maps Exercises Appendix A: Hopf bifurcation in reaction-diffusion systems Appendix B: Bibliographical notes 194

4 XIX 6 Bifurcations of Orbits Homoclinic and Heteroclinic to Hyperbolic Equilibria 195, 6.1 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 inr Exercises Appendix A: Focus-focus homoclinic bifurcation in four-dimensional systems Appendix B: 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 E Bifurcations of orbits homoclinic to limit cycles 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 Z2-equivariant systems Codim 1 bifurcations of equilibria in Z2-equivariant systems Codim 1 bifurcations of cycles in Z2-equivariant systems Exercises Appendix: Bibliographical notes Two-Parameter Bifurcations of Equilibria in Continuous- Time Dynamical Systems List of codim 2 bifurcations of equilibria Codim 1 bifurcation curves Codim 2 bifurcation points 299

5 XX 8.2 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 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 Critical normal forms for n-dimensional systems The method Cusp bifurcation Bautin bifurcation Bogdanov-Takens bifurcation Fold-Hopf bifurcation Hopf-Hopf bifurcation Exercises Appendix A: Limit cycles and homoclinic orbits of Bogdanov normal form Appendix B: 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 Fold-flip bifurcation 466

6 XXI 9.7 Critical normal forms for n-dimensional maps Cusp Generalized flip Chenciner bifurcation Resonance 1: Resonance 1: Resonance 1: Resonance 1: Fold-flip Codim 2 bifurcations of limit cycles Exercises Appendix: Bibliographical notes 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, location, and analysis of codim 2 bifurcations Continuation strategy Exercises Appendix A: Convergence theorems for Newton methods Appendix B: Bialternate matrix product Appendix C: Detection of codim 2 homoclinic bifurcations Singularities detectable via eigenvalues Orbit and inclination flips Singularities along saddle-node homoclinic curves Appendix D: Bibliographical notes 581 A Basic Notions from Algebra, Analysis, and Geometry 587 A.I Algebra >. 587 A.1.1 Matrices 587 A. 1.2 Vector spaces and linear transformations 589 A.1.3 Eigenvectors and eigenvalues 590

7 XXII A.1.4 Invariant subspaces, generalized eigenvectors, and Jordan normal form 591 A.I.5 Fredholm Alternative Theorem 592 A.1.6 Groups 593 A.2 Analysis 593 A.2.1 Implicit and Inverse Function Theorems 593 A.2.2 Taylor expansion 594 A.2.3 Metric, normed, and other spaces 595 A.3 Geometry 596 A.3.1 Sets 596 A.3.2 Maps 597 A.3.3 Manifolds 597 References 599 Index 619