ISBN: 3-540-65379-1
TITLE: Bifurcation Theory and Catastrophe Theory
AUTHOR: Arnold, V.I.; Afrajmovich, V.S.; Il'yashenko, Yu.S.; Shil'nikov, L.P.
TOC:

I. Bifurcation Theory
V.I. Arnol'd, V.S. Afrajmovich, Yu.S. Il'yashenko, L.P. Shil'nikov
Translated from the Russian by N.D. Kazarinoff
Preface 7
Chapter 1. Bifurcations of Equilibria 10
 1. Families and Deformations 11
1.1. Families of Vector Fields 11
1.2. The Space of Jets 11
1.3. Sard's Lemma and Transversality Theorems 12
1.4. Simplest Applications: Singular Points of Generic Vector Fields 13
1.5. Topologically Versal Deformations 14
1.6. The Reduction Theorem 15
1.7. Generic and Principal Families 16
 2. Bifurcations of Singular Points in Generic One-Parameter Families 17
2.1. Typical Germs and Principal Families 17
2.2. Soft and Hard Loss of Stability 19
 3. Bifurcations of Singular Points in Generic Multi-Parameter Families with Simply Degenerate Linear Parts 20
3.1. Principal Families 20
3.2. Bifurcation Diagrams of the Principal Families 3^ in Table 1 21
3.3. Bifurcation Diagrams with Respect to Weak Equivalence and Phase Portraits of the Principal Families 4^ in Table 1 21
 4. Bifurcations of Singular Points of Vector Fields with a Doubly-Degenerate Linear Part 23
4.1. A List of Degeneracies 23
4.2. Two Zero Eigenvalues 24
4.3. Reductions to Two-Dimensional Systems 24
4.4. One Zero and a Pair of Purely Imaginary Eigenvalues 25
4.5. Two Purely Imaginary Pairs 29
4.6. Principal Deformations of Equations of Difficult Type in Problems with Two Pairs of Purely Imaginary Eigenvalues (Following Zoladek) 33
 5. The Exponents of Soft and Hard Loss of Stability 35
5.1. Definitions 35
5.2. Table of Exponents 37
Chapter 2. Bifurcations of Limit Cycles 38
 1. Bifurcations of Limit Cycles in Generic One-Parameter Families 39
1.1. Multiplier 1 39
1.2. Multiplier -1 and Period-Doubling Bifurcations 41
1.3. A Pair of Complex Conjugate Multipliers 42
1.4. Nonlocal Bifurcations in One-Parameter Families of Diffeomorphisms 43
1.5. Nonlocal Bifurcations of Periodic Solutions 45
1.6. Bifurcations Resulting in Destructions of Invariant Tori 45
 2. Bifurcations of Cycles in Generic Two-Parameter Families with an Additional Simple Degeneracy 48
2.1. A List of Degeneracies 48
2.2. A Multiplier +1 or -1 with Additional Degeneracy in the Nonlinear Terms 49
2.3. A Pair of Multipliers on the Unit Circle with Additional Degeneracy in the Nonlinear Terms 49
 3. Bifurcations of Cycles in Generic Two-Parameter Families with Strong Resonances of Orders q not equal 4 51
3.1. The Normal Form in the Case of Unipotent Jordan Blocks 51
3.2. Averaging in the Seifert and the Mbius Foliations 52
3.3. Principal Vector Fields and their Deformations 53
3.4. Versality of Principal Deformations 53
3.5. Bifurcations of Stationary Solutions of Periodic Differential Equations with Strong Resonances of Orders q not equal 4 54
 4. Bifurcations of Limit Cycles for a Pair of Multipliers Crossing the Unit Circle at i 57
4.1. Degenerate Families 57
4.2. Degenerate Families Found Analytically 59
4.3. Degenerate Families Found Numerically 59
4.4. Bifurcations in Nondegenerate Families 60
4.5. Limit Cycles of Systems with a Fourth Order Symmetry 60
 5. Finitely-Smooth Normal Forms of Local Families 60
5.1. A Synopsis of Results 60
5.2. Definitions and Examples 62
5.3. General Theorems and Deformations of Nonresonant Germs 63
5.4. Reduction to Linear Normal Form 65
5.5. Deformations of Germs of Diffeomorphisms of Poincar Type 66
5.6. Deformations of Simply Resonant Hyperbolic Germs 66
5.7. Deformations of Germs of Vector Fields with One Zero Eigenvalue at a Singular Point 68
5.8. Functional Invariants of Diffeomorphisms of the Line 69
5.9. Functional Invariants of Local Families of Diffeornorphisms 70
5.10.Functional Invariants of Families of Vector Fields 71
5.11.Functional Invariants of Topological Classifications of Local Families of Diffeomorphisms of the Line 71
 6. Feigenbaum Universality for Diffeomorphisms and Flows 73
6.1. Period-Doubling Cascades 73
6.2. Perestroikas of Fixed Points 75
6.3. Cascades of n-fold Increases of Period 75
6.4. Doubling in Hamiltonian Systems 75
6.5. The Period-Doubling Operator for One-Dimensional Mappings 75
6.6. The Universal Period-Doubling Mechanism for Diffeomorphisms 77
Chapter 3. Nonlocal Bifurcations 79
 1. Degeneracies of Codimension 1. Summary of Results 80
1.1. Local and Nonlocal Bifurcations 80
1.2. Nonhyperbolic Singular Points 82
1.3. Nonhyperbolic Cycles 83
1.4. Nontransversal Intersections of Manifolds 84
1.5. Contours 85
1.6. Bifurcation Surfaces 87
1.7. Characteristics of Bifurcations 88
1.8. Summary of Results 88
 2. Nonlocal Bifurcations of Flows on Two-Dimensional Surfaces 90
2.1. Semilocal Bifurcations of Flows on Surfaces 90
2.2. Nonlocal Bifurcations on a Sphere: The One-Parameter Case 91
2.3. Generic Families of Vector Fields 92
2.4. Conditions for Genericity 94
2.5. One-Parameter Families on Surfaces different from the Sphere 95
2.6. Global Bifurcations of Systems with a Global Transversal Section on a Torus 96
2.7. Some Global Bifurcations on a Klein bottle 97
2.8. Bifurcations on a Two-Dimensional Sphere: The Multi-Parameter Case 98
2.9. Some Open Questions 101
 3. Bifurcations of Trajectories Homoclinic to a Nonhyperbolic Singular Point 102
3.1. A Node in its Hyperbolic Variables 103
3.2. A Saddle in its Hyperbolic Variables: One Homoclinic Trajectory 103
3.3. The Topological Bernoulli Automorphism 104
3.4. A Saddle in its Hyperbolic Variables: Several Homoclinic Trajectories 105
3.5. Principal Families 106
 4. Bifurcations of Trajectories Homoclinic to a Nonhyperbolic Cycle 106
4.1. The Structure of a Family of Homoclinic Trajectories 107
4.2. Critical and Noncritical Cycles 107
4.3. Creation of a Smooth Two-Dimensional Attractor 108
4.4. Creation of Complex Invariant Sets (The Noncritical Case) 109
4.5. The Critical Case 109
4.6. A Two-Step Transition from Stability to Turbulence 111
4.7. A Noncompact Set of Homoclinic Trajectories 112
4.8. Intermittency 113
4.9. Accessibility and Nonaccessibility 113
4.10.Stability of Families of Diffeomorphisms 114
4.11.Some Open Questions 116
 5. Hyperbolic Singular Points with Homoclinic Trajectories 116
5.1. Preliminary Notions: Leading Directions and Saddle Numbers 117
5.2. Bifurcations of Homoclinic Trajectories of a Saddle that Take Place on the Boundary of the Set of Morse-Smale Systems 117
5.3. Requirements for Genericity 118
5.4. Principal Families in R^3 and their Properties 119
5.5. Versality of the Principal Families 122
5.6. A Saddle with Complex Leading Direction in R^3 122
5.7. An Addition: Bifurcations of Homoclinic Loops Outside the Boundary of a Set of Morse-Smale Systems 126
5.8. An Addition: Creation of a Strange Attractor upon Bifurcation of a Trajectory Homoclinic to a Saddle 127
 6. Bifurcations Related to Nontransversal Intersections 129
6.1. Vector Fields with No Contours and No Homoclinic Trajectories 129
6.2. A Theorem on Inaccessibility 130
6.3. Moduli 131
6.4. Systems with Contours 132
6.5. Diffeomorphisms with Nontrivial Basic Sets 133
6.6. Vector Fields in R^3 with Trajectories Homoclinic to a Cycle 133
6.7. Symbolic Dynamics 134
6.8. Bifurcations of Smale Horseshoes 136
6.9. Vector Fields on a Bifurcation Surface 138
6.10.Diffeomorphisms with an Infinite Set of Stable Periodic Trajectories 138
 7. Infinite Nonwandering Sets 139
7.1. Vector Fields on the Two-Dimensional Torus 139
7.2. Bifurcations of Systems with Two Homoclinic Curves of a Saddle 140
7.3. Systems with Feigenbaum Attractors 142
7.4. Birth of Nonwandering Sets 142
7.5. Persistence and Smoothness of Invariant Manifolds 143
7.6. The Degenerate Family and Its Neighborhood in Function Space 144
7.7. Birth of Tori in a Three-Dimensional Phase Space 145
 8. Attractors and their Bifurcations 145
8.1. The Likely Limit Set According to Milnor (1985) 147
8.2. Statistical Limit Sets 147
8.3. Internal Bifurcations and Crises of Attractors 149
8.4. Internal Bifurcations and Crises of Equilibria and Cycles 149
8.5. Bifurcations of the Two-Dimensional Torus 150
Chapter 4. Relaxation Oscillations 154
 1. Fundamental Concepts 155
1.1. An Example: van der Pol's Equation 155
1.2. Fast and Slow Motions 156
1.3. The Slow Surface and Slow Equations 157
1.4. The Slow Motion as an Approximation to the Perturbed Motion 158
1.5. The Phenomenon of Jumping 159
 2. Singularities of the Fast and Slow Motions 160
2.1. Singularities of Fast Motions at Jump Points of Systems with One Fast Variable 160
2.2. Singularities of Projections of the Slow Surface 161
2.3. The Slow Motion for Systems with One Slow Variable 162
2.4. The Slow Motion for Systems with Two Slow Variables 163
2.5. Normal Forms of Phase Curves of the Slow Motion 164
2.6. Connection with the Theory of Implicit Differential Equations 167
2.7. Degeneration of the Contact Structure 168
 3. The Asymptotics of Relaxation Oscillations 170
3.1. Degenerate Systems 170
3.2. Systems of First Approximation 171
3.3. Normalizations of Fast-Slow Systems with Two Slow Variables for epsilon > 0 173
3.4. Derivation of the Systems of First Approximation 175
3.5. Investigation of the Systems of First Approximation 175
3.6. Funnels 177
3.7. Periodic Relaxation Oscillations in the Plane 177
 4. Delayed Loss of Stability as a Pair of Eigenvalues Cross the Imaginary Axis 179
4.1. Generic Systems 179
4.2. Delayed Loss of Stability 180
4.3. Hard Loss of Stability in Analytic Systems of Type 2 181
4.4. Hysteresis 181
4.5. The Mechanism of Delay 182
4.6. Computation of the Moment of Jumping in Analytic Systems 182
4.7. Delay Upon Loss of Stability by a Cycle 185
4.8. Delayed Loss of Stability and "Ducks" 185
 5. Duck Solutions 185
5.1. An Example: A Singular Point on the Fold of the Slow Surface 186
5.2. Existence of Duck Solutions 188
5.3. The Evolution of Simple Degenerate Ducks 189
5.4. A Semi-local Phenomenon: Ducks with Relaxation 190
5.5. Ducks in R^3 and R^n 191
Recommended Literature 193
References 195
Additional References 205
II. Catastrophe Theory
V.I. Arnol'd
Translated from the Russian by N.D. Kazarinoff
 1. Basic Concepts 209
1.1. Catastrophes and Bifurcations 209
1.2. Catastrophes and Singularities 210
1.3. Zeeman's Machine 210
1.4. Models of Catastrophes 212
1.5. The Verification of Models 213 
1.6. An Inadequate Model 214
1.7. Adequate Models 215
 2. The Theory of Catastrophes Before Poincar 215
2.1. Evolvents and Caustics, Involutes and Fronts 215
2.2. Families of Functions in the Work of Hamilton and His Successors 216
2.3. Points of Inflection and Swallowtails 216
2.4. The Umbrella and Umbilic Singularities of Caustics 217
2.5. Transversality 219
 3. The Theory of Bifurcations in the Work of Poincar 220
3.1. Classification of Singularities and Normal Forms 220
3.2. The Preparation Theorem, Finite Determinacy and Versal Deformations 220
3.3. Poincar and Contemporary Mathematics 221
3.4. Naive and Abstract Definitions 221
3.5. Catastrophe Theory in the Work of Poincar 222
3.6. Analyticity and Smoothness 223
 4. The Theory of Bifurcations in the Work of A.A. Andronov 224
4.1. The Point of View of Function Space 224
4.2. Structural Stability 224
4.3. Bifurcation Sets 225
4.4. Degrees of Nonroughness 227
4.5. Structural Stability and Deformational Stability 228
4.6. The Bifurcation Which Gives Birth to a Cycle 229
4.7. Delayed Loss of Stability 230
4.8. The Pleat in the Work of A.A. Andronov 231
 5. Physicists' Treatment of Catastrophes Before Catastrophe Theory 232
5.1. Thermodynamics 232
5.2. Thermal Explosions 235
5.3. Short-Wave Asymptotics 236
5.4. The Theory of Elasticity 237
5.5. The Work of L.D. Landau 238
 6. Thom's Conjecture 239
6.1. Gradient Dynamics 239
6.2. The Classification of Critical Points of Functions 240
6.3. The Classification of Gradient Systems 240
6.4. Bifurcations of Gradient Systems 242
6.5. Stating Thom's Conjecture More Precisely 242
6.6. Bifurcations of Gradient Systems of Type D_4 243
 7. Classifications of Singularities and Catastrophes 244
7.1. Codimension and Modality 244
7.2. Simple Objects 245
7.3. Functional Moduli 246
7.4. The Selection of the Classifying Group 248
7.5. Principles for Choice of Classifications 250
7.6. Recurrence of Singularities 252
7.7. The Problem of Going Around an Obstacle 255
Recommended Literature 259
References 260
Author Index 265
Subject Index 269
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