Dynamical Systems V

Bifurcation Theory and Catastrophe Theory

By: V.I. Arnold, N. Kazarinoff (Translator)

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Paperback | 20 May 1999

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Bifurcation theory and catastrophe theory are two well-known areas within the field of dynamical systems. Both are studies of smooth systems, focusing on properties that seem to be manifestly non-smooth. Bifurcation theory is concerned with the sudden changes that occur in a system when one or more parameters are varied. Examples of such are familiar to students of differential equations, from phase portraits. Understanding the bifurcations of the differential equations that describe real physical systems provides important information about the behavior of the systems. Catastrophe theory became quite famous during the 1970's, mostly because of the sensation caused by the usually less than rigorous applications of its principal ideas to "hot topics", such as the characterization of personalities and the difference between a "genius" and a "maniac". Catastrophe theory is accurately described as singularity theory and its (genuine) applications. The authors of this book, previously published as Volume 5 of the Encyclopaedia, have given a masterly exposition of these two theories, with penetrating insight.

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Non-Fiction Mathematics Geometry Differential & Riemannian Geometry Calculus & Mathematical Analysis Differential Calculus & Equations Applied Mathematics Science Physics Mathematical Physics

Preface	p. 7
Bifurcations of Equilibria	p. 10
Families and Deformations	p. 11
Families of Vector Fields	p. 11
The Space of Jets	p. 11
Sard's Lemma and Transversality Theorems	p. 12
Simplest Applications: Singular Points of Generic Vector Fields	p. 13
Topologically Versal Deformations	p. 14
The Reduction Theorem	p. 15
Generic and Principal Families	p. 16
Bifurcations of Singular Points in Generic One-Parameter Families	p. 17
Typical Germs and Principal Families	p. 17
Soft and Hard Loss of Stability	p. 19
Bifurcations of Singular Points in Generic Multi-Parameter Families with Simply Degenerate Linear Parts	p. 20
Principal Families	p. 20
Bifurcation Diagrams of the Principal Families (3^±) in Table 1	p. 21
Bifurcation Diagrams with Respect to Weak Equivalence and Phase Portraits of the Principal Families (4^±) in Table 1	p. 21
Bifurcations of Singular Points of Vector Fields with a Doubly-Degenerate Linear Part	p. 23
A List of Degeneracies	p. 23
Two Zero Eigenvalues	p. 24
Reductions to Two-Dimensional Systems	p. 24
One Zero and a Pair ofPurely Imaginary Eigenvalues	p. 25
Two Purely Imaginary Pairs	p. 29
Principal Deformations of Equations of Difficult Type in Problems with Two Pairs of Purely Imaginary Eigenvalues (Following Żol&acedil;dek)	p. 33
The Exponents of Soft and Hard Loss of Stability	p. 35
Definitions	p. 35
Table of Exponents	p. 37
Bifurcations of Limit Cycles	p. 38
Bifurcations of Limit Cycles in Generic One-Parameter Families	p. 39
Multiplier 1	p. 39
Multiplier - 1 and Period-Doubling Bifurcations	p. 41
A Pair of Complex Conjugate Multipliers	p. 42
Nonlocal Bifurcations in One-Parameter Families of Diffeomorphisms	p. 43
Nonlocal Bifurcations of Periodic Solutions	p. 45
Bifurcations Resulting in Destructions of Invariant Tori	p. 45
Bifurcations of Cycles in Generic Two-Parameter Families with an Additional Simple Degeneracy	p. 48
A List of Degeneracies	p. 48
A Multiplier +1 or -1 with Additional Degeneracy in the Nonlinear Terms	p. 49
A Pair of Multipliers on the Unit Circle with Additional Degeneracy in the Nonlinear Terms	p. 49
Bifurcations of Cycles in Generic Two-Parameter Families with Strong Resonances of Orders q &neq; 4	p. 51
The Normal Form in the Case of Unipotent Jordan Blocks	p. 51
Averaging in the Seifert and the Möbius Foliations	p. 52
Principal Vector Fields and their Deformations	p. 53
Versality of Principal Deformations	p. 53
Bifurcations of Stationary Solutions of Periodic Differential Equations with Strong Resonances of Orders q &neq; 4	p. 54
Bifurcations of Limit Cycles for a Pair of Multipliers Crossing the Unit Circle at ±i	p. 57
Degenerate Families	p. 57
Degenerate Families Found Analytically	p. 59
Degenerate Families Found Numerically	p. 59
Bifurcations in Nondegenerate Families	p. 60
Limit Cycles of Systems with a Fourth Order Symmetry	p. 60
Finitely-Smooth Normal Forms of Local Families	p. 60
A Synopsis of Results	p. 60
Definitions and Examples	p. 62
General Theorems and Deformations of Nonresonant Germs	p. 63
Reduction to Linear Normal Form	p. 65
Deformations of Germs of Diffeomorphisms of Poincaré Type	p. 66
Deformations of Simply Resonant Hyperbolic Germs	p. 66
Deformations of Germs of Vector Fields with One Zero Eigenvalue at a Singular Point	p. 68
Functional Invariants of Diffeomorphisms of the Line	p. 69
Functional Invariants of Local Families of Diffeornorphisms	p. 70
Functional Invariants of Families of Vector Fields	p. 71
Functional Invariants of Topological Classifications of Local Families of Diffeomorphisms of the Line	p. 71
Feigenbaum Universality for Diffeomorphisms and Flows	p. 73
Period-Doubling Cascades	p. 73
Perestroikas of Fixed Points	p. 75
Cascades of n-fold Increases of Period	p. 75
Doubling in Hamiltonian Systems	p. 75
The Period-Doubling Operator for One-Dimensional Mappings	p. 75
The Universal Period-Doubling Mechanism for Diffeomorphisms	p. 77
Nonlocal Bifurcations	p. 79
Degeneracies of Codimension 1. Summary of Results	p. 80
Local and Nonlocal Bifurcations	p. 80
Nonhyperbolic Singular Points	p. 82
Nonhyperbolic Cycles	p. 83
Nontransversal Intersections of Manifolds	p. 84
Contours	p. 85
Bifurcation Surfaces	p. 87
Characteristics of Bifurcations	p. 88
Summary of Results	p. 88
Nonlocal Bifurcations of Flows on Two-Dimensional Surfaces	p. 90
Semilocal Bifurcations of Flows on Surfaces	p. 90
Nonlocal Bifurcations on a Sphere: The One-Parameter Case	p. 91
Generic Families of Vector Fields	p. 92
Conditions for Genericity	p. 94
One-Parameter Families on Surfaces different from the Sphere	p. 95
Global Bifurcations of Systems with a Global Transversal Section on a Torus	p. 96
Some Global Bifurcations on a Klein bottle	p. 97
Bifurcations on a Two-Dimensional Sphere: The Multi-Parameter Case	p. 98
Some Open Questions	p. 101
Bifurcations of Trajectories Homoclinic to a Nonhyperbolic Singular Point	p. 102
A Node in its Hyperbolic Variables	p. 103
A Saddle in its Hyperbolic Variables: One Homoclinic Trajectory	p. 103
The Topological Bernoulli Automorphism	p. 104
A Saddle in its Hyperbolic Variables: Several Homoclinic Trajectories	p. 105
Principal Families	p. 106
Bifurcations of Trajectories Homoclinic to a Nonhyperbolic Cycle	p. 106
The Structure of a Family of Homoclinic Trajectories	p. 107
Critical and Noncritical Cycles	p. 107
Creation of a Smooth Two-Dimensional Attractor	p. 108
Creation of Complex Invariant Sets (The Noncritical Case)	p. 109
The Critical Case	p. 109
A Two-Step Transition from Stability to Turbulence	p. 111
A Noncompact Set of Homoclinic Trajectories	p. 112
Intermittency	p. 113
Accessibility and Nonaccessibility	p. 113
Stability of Families of Diffeomorphisms	p. 114
Some Open Questions	p. 116
Hyperbolic Singular Points with Homoclinic Trajectories	p. 116
Preliminary Notions: Leading Directions and Saddle Numbers	p. 117
Bifurcations of Homoclinic Trajectories of a Saddle that Take Place on the Boundary of the Set of Morse-Smale Systems	p. 117
Requirements for Genericity	p. 118
Principal Families in <$>{\op R}^3<$> and their Properties	p. 119
Versality of the Principal Families	p. 122
A Saddle with Complex Leading Direction in <$>{\op R}^3<$>	p. 122
An Addition: Bifurcations of Homoclinic Loops Outside the Boundary of a Set of Morse-Smale Systems	p. 126
An Addition: Creation of a Strange Attractor upon Bifurcation of a Trajectory Homoclinic to a Saddle	p. 127
Bifurcations Related to Nontransversal Intersections	p. 129
Vector Fields with No Contours and No Homoclinic Trajectories	p. 129
A Theorem on Inaccessibility	p. 130
Moduli	p. 131
Systems with Contours	p. 132
Diffeomorphisms with Nontrivial Basic Sets	p. 133
Vector Fields in <$>{\op R}^3<$> with Trajectories Homoclinic to a Cycle	p. 133
Symbolic Dynamics	p. 134
Bifurcations of Smale Horseshoes	p. 136
Vector Fields on a Bifurcation Surface	p. 138
Diffeomorphisms with an Infinite Set of Stable Periodic Trajectories	p. 138
Infinite Nonwandering Sets	p. 139
Vector Fields on the Two-Dimensional Torus	p. 139
Bifurcations of Systems with Two Homoclinic Curves of a Saddle	p. 140
Systems with Feigenbaum Attractors	p. 142
Birth of Nonwandering Sets	p. 142
Persistence and Smoothness of Invariant Manifolds	p. 143
The Degenerate Family and Its Neighborhood in Function Space	p. 144
Birth of Tori in a Three-Dimensional Phase Space	p. 145
Attractors and their Bifurcations	p. 145
The Likely Limit Set According to Milnor (1985)	p. 147
Statistical Limit Sets	p. 147
Internal Bifurcations and Crises of Attractors	p. 149
Internal Bifurcations and Crises of Equilibria and Cycles	p. 149
Bifurcations of the Two-Dimensional Torus	p. 150
Relaxation Oscillations	p. 154
Fundamental Concepts	p. 155
An Example: van der Pol's Equation	p. 155
Fast and Slow Motions	p. 156
The Slow Surface and Slow Equations	p. 157
The Slow Motion as an Approximation to the Perturbed Motion	p. 158
The Phenomenon of Jumping	p. 159
Singularities of the Fast and Slow Motions	p. 160
Singularities of Fast Motions at Jump Points of Systems with One Fast Variable	p. 160
Singularities of Projections of the Slow Surface	p. 161
The Slow Motion for Systems with One Slow Variable	p. 162
The Slow Motion for Systems with Two Slow Variables	p. 163
Normal Forms of Phase Curves of the Slow Motion	p. 164
Connection with the Theory of Implicit Differential Equations	p. 167
Degeneration of the Contact Structure	p. 168
The Asymptotics of Relaxation Oscillations	p. 170
Degenerate Systems	p. 170
Systems of First Approximation	p. 171
Normalizations of Fast-Slow Systems with Two Slow Variables for ¿ > 0	p. 173
Derivation of the Systems of First Approximation	p. 175
Investigation of the Systems of First Approximation	p. 175
Funnels	p. 177
Periodic Relaxation Oscillations in the Plane	p. 177
Delayed Loss of Stability as a Pair of Eigenvalues Cross the Imaginary Axis	p. 179
Generic Systems	p. 179
Delayed Loss of Stability	p. 180
Hard Loss of Stability in Analytic Systems of Type 2	p. 181
Hysteresis	p. 181
The Mechanism of Delay	p. 182
Computation of the Moment of Jumping in Analytic Systems	p. 182
Delay Upon Loss of Stability by a Cycle	p. 185
Delayed Loss of Stability and "Ducks"	p. 185
Duck Solutions	p. 185
An Example: A Singular Point on the Fold of the Slow Surface	p. 186
Existence of Duck Solutions	p. 188
The Evolution of Simple Degenerate Ducks	p. 189
A Semi-local Phenomenon: Ducks with Relaxation	p. 190
Ducks in <$>{\op R}^3<$> and <$>{\op R}^n<$>	p. 191
Recommended Literature	p. 193
References	p. 195
Additional References	p. 205
Table of Contents provided by Publisher. All Rights Reserved.

Dynamical Systems V

Bifurcation Theory and Catastrophe Theory

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Paperback

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