Elements of Applied Bifurcation Theory

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Preface to the Fourth Edition Preface to the Third Edition Preface to the Second Edition Preface to the First Edition Contents 1 Introduction to Dynamical Systems 1.1 Definition of a Dynamical System 1.1.1 State Space 1.1.2 Time 1.1.3 Evolution Operator 1.1.4 Definition of a Dynamical System 1.2 Orbits and Phase Portraits 1.3 Invariant Sets 1.3.1 Definition and Types 1.3.2 Smale Horseshoe 1.3.3 Stability of Invariant Sets 1.4 Differential Equations and Dynamical Systems 1.5 Poincaré Maps 1.5.1 Time-Shift Maps 1.5.2 Poincaré Map and Stability of Cycles 1.5.3 Poincaré Map for Periodically Forced Systems 1.6 Exercises 1.7 Appendix A: Planar ODE Systems 1.8 Appendix B: Reaction-Diffusion Systems 1.9 Appendix C: Differential Equations with Delays 1.10 Appendix D: Bibliographical Notes 2 Topological Equivalence, Bifurcations, and Structural Stability of Dynamical Systems 2.1 Equivalence of Dynamical Systems 2.2 Topological Classification of Generic Equilibria and Fixed Points 2.2.1 Hyperbolic Equilibria in Continuous-Time Systems 2.2.2 Hyperbolic Fixed Points in Discrete-Time Systems 2.2.3 Hyperbolic Limit Cycles 2.3 Bifurcations and Bifurcation Diagrams 2.4 Topological Normal Forms for Bifurcations 2.5 Structural Stability 2.6 Exercises 2.7 Appendix: Bibliographical Notes 3 One-Parameter Bifurcations of Equilibria in Continuous-Time Dynamical Systems 3.1 Simplest Bifurcation Conditions 3.2 The Normal Form of the Fold Bifurcation 3.3 Generic Fold Bifurcation 3.4 The Normal Form of the Hopf Bifurcation 3.5 Generic Hopf Bifurcation 3.6 Exercises 3.7 Appendix A: Proof of Lemma 3.2 3.8 Appendix B: Poincaré Normal Forms 3.9 Appendix C: Bibliographical Notes 4 One-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems 4.1 Simplest Bifurcation Conditions 4.2 The Normal Form of the Fold Bifurcation 4.3 Generic Fold Bifurcation 4.4 The Normal Form of the Flip Bifurcation 4.5 Generic Flip Bifurcation 4.6 The ``Normal Form'' of the Neimark-Sacker Bifurcation 4.7 Generic Neimark-Sacker Bifurcation 4.8 Exercises 4.9 Appendix A: Proof of Lemma 4.2 4.10 Appendix B: Proof of Lemma 4.3 4.11 Appendix C: Feigenbaum's Universality 4.12 Appendix D: Bibliographical Notes 5 Bifurcations of Equilibria and Periodic Orbits in n-Dimensional Dynamical Systems 5.1 Center Manifold Theorems 5.1.1 Center Manifolds in Continuous-Time Systems 5.1.2 Center Manifolds in Discrete-Time Systems 5.2 Center Manifolds in Parameter-Dependent Systems 5.3 Bifurcations of Limit Cycles 5.4 Computation of Center Manifolds 5.4.1 Restricted Normalized Equations for ODEs 5.4.2 Restricted Normalized Equations for Maps 5.5 Exercises 5.6 Appendix A: Periodic Normal Forms for Bifurcations of Limit Cycles 5.6.1 Periodic Center Manifolds 5.6.2 Periodic Normal Forms for Codim 1 Bifurcations of Limit Cycles 5.6.3 Critical Normal Form Coefficients 5.7 Appendix B: Hopf Bifurcation in Reaction-Diffusion Systems 5.8 Appendix C: Hopf Bifurcation in Delay Differential Equations 5.9 Appendix D: Bibliographical Notes 6 Bifurcations of Orbits Homoclinic and Heteroclinic to Hyperbolic Equilibria 6.1 Homoclinic and Heteroclinic Orbits 6.2 Andronov-Leontovich Theorem 6.3 Homoclinic Bifurcations in Three-Dimensional Systems: Shil'nikov Theorems 6.4 Homoclinic Bifurcations in n-Dimensional Systems 6.4.1 Regular Homoclinic Orbits: Melnikov Integral 6.4.2 Homoclinic Center Manifolds 6.4.3 Generic Homoclinic Bifurcations in mathbbRn 6.5 Exercises 6.6 Appendix A: Focus-Focus Homoclinic Bifurcation in Four-dimensional Systems 6.7 Appendix B: Bibliographical Notes 7 Other One-Parameter Bifurcations in Continuous-Time Dynamical Systems 7.1 Codim 1 Bifurcations of Homoclinic Orbits to Nonhyperbolic Equilibria 7.1.1 Saddle-Node Homoclinic Bifurcation on the Plane 7.1.2 Saddle-Node and Saddle-Saddle Homoclinic Bifurcations in mathbbR3 7.2 Bifurcations of Orbits Homoclinic to Limit Cycles 7.2.1 Nontransversal Homoclinic Orbit to a Hyperbolic Cycle 7.2.2 Homoclinic Orbits to a Nonhyperbolic Limit Cycle 7.3 Bifurcations on Invariant Tori 7.3.1 Reduction to a Poincaré Map 7.3.2 Rotation Number and Orbit Structure 7.3.3 Structural Stability and Bifurcations 7.3.4 Phase Locking Near a Neimark-Sacker Bifurcation: Arnold Tongues 7.4 Bifurcations in Symmetric Systems 7.4.1 General Properties of Symmetric Systems 7.4.2 mathbbZ2-Equivariant Systems 7.4.3 Codim 1 Bifurcations of Equilibria in mathbbZ2-Equivariant Systems 7.4.4 Codim 1 Bifurcations of Cycles in mathbbZ2-Equivariant Systems 7.5 Exercises 7.6 Appendix: Bibliographical Notes 8 Two-Parameter Bifurcations of Equilibria in Continuous-Time Dynamical Systems 8.1 List of Codim 2 Bifurcations of Equilibria 8.1.1 Bifurcation Curves 8.1.2 Codimension Two Bifurcation Points 8.2 Cusp Bifurcation 8.2.1 Normal Form Derivation 8.2.2 Bifurcation Diagram of the Truncated Normal Form 8.2.3 Effect of Higher-Order Terms 8.3 Bautin (Generalized Hopf) Bifurcation 8.3.1 Normal Form Derivation 8.3.2 Bifurcation Diagram of the Truncated Normal Form 8.3.3 Effect of Higher-Order Terms 8.4 Bogdanov-Takens (Double-Zero) Bifurcation 8.4.1 Normal Form Derivation 8.4.2 Bifurcation Diagram of the Truncated Normal Form 8.4.3 Effect of Higher-Order Terms 8.5 Fold-Hopf Bifurcation 8.5.1 Derivation of the Normal Form 8.5.2 Bifurcation Diagram of the Truncated Normal Form 8.5.3 Effect of Higher-Order Terms 8.6 Hopf-Hopf Bifurcation 8.6.1 Derivation of the Normal Form 8.6.2 Bifurcation Diagram of the Truncated Normal Form 8.6.3 Effect of Higher-Order Terms 8.7 Critical Normal Forms for n-Dimensional Systems 8.7.1 The Method 8.7.2 Cusp Bifurcation 8.7.3 Bautin Bifurcation 8.7.4 Bogdanov-Takens Bifurcation 8.7.5 Fold-Hopf Bifurcation 8.7.6 Hopf-Hopf Bifurcation 8.8 Exercises 8.9 Appendix A: Limit Cycles and Homoclinic Orbits of Bogdanov Normal Form 8.10 Appendix B: Bibliographical Notes 9 Two-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems 9.1 List of Codim 2 Bifurcations of Fixed Points 9.2 Cusp Bifurcation 9.3 Generalized Flip Bifurcation 9.4 Chenciner (Generalized Neimark-Sacker) Bifurcation 9.5 Strong Resonances 9.5.1 Approximation by a Flow 9.5.2 1:1 Resonance 9.5.3 1:2 Resonance 9.5.4 1:3 Resonance 9.5.5 1:4 Resonance 9.6 Fold-Flip Bifurcation 9.7 Fold-Neimark-Sacker Bifurcation 9.8 Flip-Neimark-Sacker and Double Neimark-Sacker Bifurcations 9.8.1 Poincaré Normal Forms 9.8.2 Reduction to an Amplitude Map 9.8.3 Bifurcations of the Truncated Amplitude Map 9.8.4 Bifurcations of Truncated Normal Forms 9.8.5 Effects of Higher-Order Terms 9.9 Critical Normal Forms for n-Dimensional Maps 9.9.1 Cusp 9.9.2 Generalized Flip 9.9.3 Chenciner Bifurcation 9.9.4 Resonance 1:1 9.9.5 Resonance 1:2 9.9.6 Resonance 1:3 9.9.7 Resonance 1:4 9.9.8 Fold-Flip 9.9.9 Fold-Neimark-Sacker 9.9.10 Flip-Neimark-Sacker 9.9.11 Double Neimark-Sacker 9.10 Codim 2 Bifurcations of Limit Cycles 9.11 Exercises 9.12 Appendix: Bibliographical Notes 10 Numerical Analysis of Bifurcations 10.1 Numerical Analysis at Fixed Parameter Values 10.1.1 Equilibrium Location 10.1.2 Modified Newton's Methods 10.1.3 Equilibrium Analysis 10.1.4 Location of Limit Cycles 10.2 One-Parameter Bifurcation Analysis 10.2.1 Continuation of Equilibria and Cycles 10.2.2 Detection and Location of Codim 1 Bifurcations 10.2.3 Analysis of Codim 1 Bifurcations 10.2.4 Branching Points 10.3 Two-Parameter Bifurcation Analysis 10.3.1 Continuation of Codim 1 Bifurcations of Equilibria and Fixed Points 10.3.2 Continuation of Codim 1 Limit Cycle Bifurcations 10.3.3 Continuation of Codim 1 Homoclinic Orbits 10.3.4 Detection, Location, and Analysis of Codim 2 Bifurcations 10.4 Continuation Strategy 10.5 Exercises 10.6 Appendix A: Convergence of Newton's Methods 10.7 Appendix B: Bialternate Matrix Product 10.8 Appendix C: Detection of Codim 2 Homoclinic Bifurcations 10.8.1 Singularities Detectable via Eigenvalues 10.8.2 Orbit and Inclination Flips 10.8.3 Singularities Along Saddle-Node Homoclinic Curves 10.9 Appendix D: Bibliographical Notes A Basic Notions from Algebra, Analysis, and Geometry A.1 Algebra A.1.1 Matrices A.1.2 Vector Spaces and Linear Transformations A.1.3 Eigenvectors and Eigenvalues A.1.4 Invariant Subspaces, Generalized Eigenvectors, and Jordan Normal Form A.1.5 Fredholm Alternative Theorem A.1.6 Groups A.2 Analysis A.2.1 Implicit and Inverse Function Theorems A.2.2 Taylor Expansion A.2.3 Metric, Normed, and Other Spaces A.3 Geometry A.3.1 Sets A.3.2 Maps A.3.3 Manifolds Appendix References Index