Yuri A. Kuznetsov Elements of Applied Bifurcation Theory Second Edition With 251 Illustrations Springer
Preface to the Second Edition Preface to the First Edition vii ix 1 Introduction to Dynamical Systems 1 1.1 Definition of a dynamical system 1 1.1.1 State space 2 1.1.2 Time 5 1.1.3 Evolution operator 5 1.1.4 Definition of a dynamical system 7 1.2 Orbits and phase portraits 8 1.3 Invariant sets 11 1.3.1 Definition and types 11 1.3.2 Example 1.9 (Smale horseshoe) 12 1.3.3 Stability of invariant sets 16 1.4 Differential equations and dynamical systems 18 1.5 Poincare maps 23 1.5.1 Time-shift maps 24 1.5.2 Poincare map and stability of cycles 25 1.5.3 Poincare map for periodically forced systems 30 1.6 Exercises 31 1.7 Appendix 1: Infinite-dimensional dynamical systems defined by reaction-diffusion equations 33 1.8 Appendix 2: Bibliographical notes 37 2 Topological Equivalence, Bifurcations, and Structural Stability of Dynamical Systems 39 2.1 Equivalence of dynamical systems 39
xvi 2.2 Topological classification of generic equilibria and fixed points 46 2.2.1 Hyperbolic equilibria in continuous-time systems 46 2.2.2 Hyperbolic fixed points in discrete-time systems 49 2.2.3 Hyperbolic limit cycles 54 2.3 Bifurcations and bifurcation diagrams 57 2.4 Topological normal forms for bifurcations 63 2.5 Structural stability 68 2.6 Exercises 73 2.7 Appendix: Bibliographical notes 76 3 One-Parameter Bifurcations of Equilibria in Continuous-Time Dynamical Systems 79 3.1 Simplest bifurcation conditions 79 3.2 The normal form of the fold bifurcation 80 3.3 Generic fold bifurcation 83 3.4 The normal form of the Hopf bifurcation 86 3.5 Generic Hopf bifurcation 91 3.6 Exercises 104 3.7 Appendix 1: Proof of Lemma 3.2 108 3.8 Appendix 2: Bibliographical notes Ill 4 One-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems 113 4.1 Simplest bifurcation conditions 113 4.2 The normal form of the fold bifurcation 114 4.3 Generic fold bifurcation 116 4.4 The normal form of the flip bifurcation 119 4.5 Generic flip bifurcation 121 4.6 The "normal form" of the Neimark-Sacker bifurcation 125 4.7 Generic Neimark-Sacker bifurcation 129 4.8 Exercises 138 4.9 Appendix 1: Feigenbaum's universality 139 4.10 Appendix 2: Proof of Lemma 4.3 143 4.11 Appendix 3: Bibliographical notes 149 5 Bifurcations of Equilibria and Periodic Orbits in n-dimensional Dynamical Systems 151 5.1 Center manifold theorems 151 5.1.1 Center manifolds in continuous-time systems 152 5.1.2 Center manifolds in discrete-time systems 156 5.2 Center manifolds in parameter-dependent systems 157 5.3 Bifurcations of limit cycles 162 5.4 Computation of center manifolds 165 5.4.1 Quadratic approximation to center manifolds in eigenbasis 165 5.4.2 Projection method for center manifold computation 171
xvii 5.5 Exercises 186 5.6 Appendix 1: Hopf bifurcation in reaction-diffusion systems on the interval with Dirichlet boundary conditions 189 5.7 Appendix 2: Bibliographical notes 193 6 Bifurcations of Orbits Homoclinic and Heteroclinic to Hyperbolic Equilibria 195 6.1 Homoclinic and heteroclinic orbits 195 6.2 Andronov-Leontovich theorem 200 6.3 Homoclinic bifurcations in three-dimensional systems: Shil'nikov theorems 213 6.4 Homoclinic bifurcations in n-dimensional systems 228 6.4.1 Regular homoclinic orbits: Melnikov integral 229 6.4.2 Homoclinic center manifolds 232 6.4.3 Generic homoclinic bifurcations in R 236 6.5 Exercises 238 6.6 Appendix 1: Focus-focus homoclinic bifurcation in four-dimensional systems 241 6.7 Appendix 2: Bibliographical notes 247 7 Other One-Parameter Bifurcations in Continuous-Time Dynamical Systems 249 7.1 Codim 1 bifurcations of homoclinic orbits to nonhyperbolic equilibria 250 7.1.1 Saddle-node homoclinic bifurcation on the plane 250 7.1.2 Saddle-node and saddle-saddle homoclinic bifurcations in R 3 253 7.2 "Exotic" bifurcations 262 7.2.1 Nontransversal homoclinic orbit to a hyperbolic cycle 263 7.2.2 Homoclinic orbits to a nonhyperbolic limit cycle 263 7.3 Bifurcations on invariant tori 267 7.3.1 Reduction to a Poincare map 267 7.3.2 Rotation number and orbit structure 269 7.3.3 Structural stability and bifurcations 270 7.3.4 Phase locking near a Neimark-Sacker bifurcation: Arnold tongues 272 7.4 Bifurcations in symmetric systems 276 7.4.1 General properties of symmetric systems 276 7.4.2 Z 2 -equivariant systems 278 7.4.3 Codim 1 bifurcations of equilibria in Z 2 -equivariant systems 280 7.4.4 Codim 1 bifurcations of cycles in Z 2 -equivariant systems 283 7.5 Exercises 288 7.6 Appendix 1: Bibliographical notes 290
xviii 8 Two-Parameter Bifurcations of Equilibria in Continuous-Time Dynamical Systems 293 8.1 List of codim 2 bifurcations of equilibria 294 8.1.1 Bifurcation curves 294 8.1.2 Codimension two bifurcation points 297 8.2 Cusp bifurcation 301 8.2.1 Normal form derivation 301 8.2.2 Bifurcation diagram of the normal form 303 8.2.3 Effect of higher-order terms 305 8.3 Bautin (generalized Hopf) bifurcation 307 8.3.1 Normal form derivation 307 8.3.2 Bifurcation diagram* of the normal form 312 8.3.3 Effect of higher-order terms 313 8.4 Bogdanov-Takens (double-zero) bifurcation 314 8.4.1 Normal form derivation 314 8.4.2 Bifurcation diagram of the normal form 321 8.4.3 Effect of higher-order terms 324 8.5 Fold-Hopf (zero-pair) bifurcation 330 8.5.1 Derivation of the normal form 330 8.5.2 Bifurcation diagram of the truncated normal form 337 8.5.3 Effect of higher-order terms 342 8.6 Hopf-Hopf bifurcation 349 8.6.1 Derivation of the normal form 349 8.6.2 Bifurcation diagram of the truncated normal form 356 8.6.3 Effect of higher-order terms 366 8.7 Exercises 369 8.8 Appendix 1: Limit cycles and homoclinic orbits of Bogdanov normal form 382 8.9 Appendix 2: Bibliographical notes 390 9 Two-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems 393 9.1 List of codim 2 bifurcations of fixed points 393 9.2 Cusp bifurcation 397 9.3 Generalized flip bifurcation 400 9.4 Chenciner (generalized Neimark-Sacker) bifurcation 404 9.5 Strong resonances 408 9.5.1 Approximation by a flow 408 9.5.2 1:1 resonance 410 9.5.3 1:2 resonance 415 9.5.4 1:3 resonance 428 9.5.5 1:4 resonance 435 9.6 Codim 2 bifurcations of limit cycles 446 9.7 Exercises 457 9.8 Appendix 1: Bibliographical notes 460
xix 10 Numerical Analysis of Bifurcations 463 10.1 Numerical analysis at fixed parameter values 464 10.1.1 Equilibrium location 464 10.1.2 Modified Newton's methods 466 10.1.3 Equilibrium analysis 469 10.1.4 Location of limit cycles 472 10.2 One-parameter bifurcation analysis 478 10.2.1 Continuation of equilibria and cycles 479 10.2.2 Detection and location of codim 1 bifurcations 484 10.2.3 Analysis of codim 1 bifurcations 488 10.2.4 Branching points 495 10.3 Two-parameter bifurcation analysis 501 10.3.1 Continuation of codim 1 bifurcations of equilibria and fixed points 501 10.3.2 Continuation of codim 1 limit cycle bifurcations 507 10.3.3 Continuation of codim 1 homoclinic orbits 510 10.3.4 Detection and location of codim 2 bifurcations 514 10.4 Continuation strategy 515 10.5 Exercises 517 10.6 Appendix 1: Convergence theorems for Newton methods 525 10.7 Appendix 2: Detection of codim 2 homoclinic bifurcations 526 10.7.1 Singularities detectable via eigenvalues 527 10.7.2 Orbit and inclination flips 529 10.7.3 Singularities along saddle-node homoclinic curves 534 10.8 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