Multiple time scale dynamics

목차 (Table of Contents)

• CONTENTS
• Preface = v
• 1 Introduction = 1
• 1.1 User's Guide = 1
• 1.2 Fast-Slow Systems = 8
• 1.3 The van der Pol Equation = 9
• 1.4 The FitzHugh-Nagumo Equation = 10
• 1.5 Terminology : Fast-Slow Systems = 11
• 1.6 Terminology : Asymptotic Analysis = 14
• 1.7 References = 16
• 2 General Fenichel Theory = 19
• 2.1 Invariant Manifolds = 20
• 2.2 Perturbations of Invariant Manifolds = 28
• 2.3 Normal Hyperbolicity = 40
• 2.4 Specialization to Fast-Slow Systems = 47
• 2.5 References = 50
• 3 Geometric Singular Perturbation Theory = 53
• 3.1 Fenichel's Theorem = 53
• 3.2 he Slow Flow = 58
• 3.3 Singularities = 61
• 3.4 Examples = 65
• 3.5 References = 70
• 4 Normal Forms = 71
• 4.1 The Normally Hyperbolic Case = 71
• 4.2 Fold Points = 77
• 4.3 Fold Curves = 80
• 4.4 Systems of First Approximation = 82
• 4.5 A Note on Linear Systems = 85
• 4.6 References = 88
• 5 Direct Asymptotic Methods = 91
• 5.1 Elementary Results = 91
• 5.2 Basics of Relaxation Oscillations = 95
• 5.3 Normally Hyperbolic Planar Systems = 98
• 5.4 Generic Folds in Planar Systems = 103
• 5.5 The Period of Relaxation Oscillations = 108
• 5.6 References = 111
• 6 Tracking Invariant Manifolds = 113
• 6.1 Simple Jumps and Transversality = 114
• 6.2 The FitzHugh-Nagumo Equation = 120
• 6.3 The C⁰Exchange Lemma = 124
• 6.4 The C¹Exchange Lemma = 129
• 6.5 Fast Waves in the FitzHugh-Nagumo Equation = 141
• 6.6 Strong Exchange Lemmas = 145
• 6.7 BVPs and Periodic Orbits = 149
• 6.8 References = 155
• 7 The Blowup Method = 159
• 7.1 Basics = 160
• 7.2 Local Computation = 167
• 7.3 The Quasihomogeneous Case = 171
• 7.4 Blowup Analysis of the Generic Fold = 178
• 7.5 Relaxation Oscillations inℝ² = 187
• 7.6 Relaxation Oscillations inℝ³ = 190
• 7.7 Remarks on Rescaling = 194
• 7.8 References = 196
• 8 Singularities and Canards = 197
• 8.1 Folded Singularities in Planar Systems = 198
• 8.2 Singular Hopf Bifurcation inℝ² = 205
• 8.3 The First Lyapunov Coefficient = 208
• 8.4 Canard Explosion = 210
• 8.5 Canards inℝ³ = 213
• 8.6 Secondary Canards at Folded Nodes = 223
• 8.7 Singularities beyond Folds = 228
• 8.8 Curvature = 233
• 8.9 References = 235
• 9 Advanced Asymptotic Methods = 239
• 9.1 Matched Asymptotic Expansions = 240
• 9.2 further Concepts and Terminology = 251
• 9.3 The Boundary Function Method = 255
• 9.4 WKB Theory = 259
• 9.5 Asymptotics and Blowup = 262
• 9.6 Averaging = 265
• 9.7 Gevrey Asymptotics = 268
• 9.8 Poincaré-Lindstedt and Two-Timing = 274
• 9.9 The Renormalization GroupⅠ = 279
• 9.10 The Renormalization GroupⅡ = 285
• 9.11 References = 290
• 10 Numerical Methods = 295
• 10.1 Stiff Equations = 295
• 10.2 A-Stability = 299
• 10.3 Boundary Value Problems(BVPs) = 303
• 10.4 Two Standard BVP Methods = 306
• 10.5 Conditioning = 308
• 10.6 Continuation = 312
• 10.7 Heterogeneous Multiscale Methods = 315
• 10.8 Error Analysis for HMM = 319
• 10.9 References = 322
• 11 Computing Manifolds = 327
• 11.1 Basic Techniques = 327
• 11.2 The CSP Method = 331
• 11.3 The ZDP Method = 342
• 11.4 Other Reduction Methods = 344
• 11.5 Saddle-Type Slow Manifolds = 348
• 11.6 Manifolds and Singularities = 353
• 11.7 References = 356
• 12 Scaling and Delay = 359
• 12.1 The Fold Revisited = 359
• 12.2 Delayed Hopf Bifurcation = 361
• 12.3 Delayed Bifurcation and WKB = 370
• 12.4 The Newton Polygon = 373
• 12.5 Power Transformations = 376
• 12.6 Parameterizing Critical Manifolds = 379
• 12.7 Slowly Time-Dependent Systems = 384
• 12.8 Other Newton Polygon Applications = 392
• 12.9 References = 395
• 13 Oscillations = 397
• 13.1 Overview = 398
• 13.2 Folded Nodes = 400
• 13.3 Singular Hopf and Hyperbolic Equilibria = 403
• 13.4 Tourbillon = 407
• 13.5 The Koper Model = 408
• 13.6 Square-Wave／Fold-Homoclinic Bursting = 413
• 13.7 Elliptic／subHopf-Foldcycle Bursting = 416
• 13.8 Three Time-Scale Systems = 417
• 13.9 References = 425
• 14 Chaos in Fast-Slow Systems = 431
• 14.1 A Basic Idea = 432
• 14.2 Chaotic Dynamics = 434
• 14.3 Smale Horseshoes = 436
• 14.4 Chaos in van der Pol's EquationⅠ = 442
• 14.5 Chaos in van der Pol's EquationⅡ = 448
• 14.6 Hénon-Type Maps = 455
• 14.7 Chaotic Forced Oscillations = 460
• 14.8 Constructing Chaotic Attractors = 468
• 14.9 Stochastic Replacement = 471
• 14.10 References = 473
• 15 Stochastic Systems = 477
• 15.1 Motivation and Viewpoints = 478
• 15.2 Fenichel Theory for SDEs = 482
• 15.3 Noisy Fold with Slow Drift = 488
• 15.4 Direct Asymptotics = 493
• 15.5 Stochastic Reduction = 497
• 15.6 Random Dynamical Systems = 505
• 15.7 Markov Chains = 510
• 15.8 Large Deviations = 512
• 15.9 Dynkin's Equation and WKB = 517
• 15.10 References = 520
• 16 Topological Methods = 525
• 16.1 Decomposition of Invariant Sets = 525
• 16.2 Conley Index = 530
• 16.3 Connecting Orbits = 535
• 16.4 Singular Index Pairs = 540
• 16.5 Further Techniques = 546
• 16.6 References = 550
• 17 Spatial Dynamics = 553
• 17.1 Stability of Traveling WavesⅠ = 553
• 17.2 Stability of Traveling WavesⅡ = 558
• 17.3 Conservation Laws = 562
• 17.4 Lattice Dynamical Systems = 573
• 17.5 Adaptive Neural Fields = 578
• 17.6 References = 580
• 18 Infinite Dimensions = 583
• 18.1 Semigroups = 584
• 18.2 Invariant Manifolds = 587
• 18.3 Homogenization = 595
• 18.4 Delay Equations with Small Delay = 600
• 18.5 Differential Inclusions = 605
• 18.6 Amplitude Equations = 611
• 18.7 References = 615
• 19 Other Topics = 619
• 19.1 Differential-Algebraic EquationsⅠ = 620
• 19.2 Differential-Algebraic EquationsⅡ = 624
• 19.3 Nonsmooth Dynamical Systems = 627
• 19.4 Hysteresis = 633
• 19.5 Nonstandard Analysis : Introduction = 637
• 19.6 Nonstandard Analysis : An Application = 642
• 19.7 Averaging and Adiabatic Invariants = 646
• 19.8 Singular Bifurcation Diagrams = 649
• 19.9 Critical Transitions = 654
• 19.10 References = 661
• 20 Applications = 665
• 20.1 Engineering = 665
• 20.2 Neuroscience = 668
• 20.3 Chemical Oscillations = 673
• 20.4 Lasers = 676
• 20.5 Ecology = 679
• 20.6 Pattern Formation = 681
• 20.7 Celestial Mechanics = 685
• 20.8 Systems Biology = 688
• 20.9 Fluid Dynamics = 691
• 20.10 Quantum Mechanics = 394
• 20.11 Networks = 396
• 20.12 References = 700
• Bibliography = 705
• Index = 799