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Dynamical Systems V : Bifurcation Theory and Catastrophe Theory - V.S. Afrajmovich

Dynamical Systems V

Bifurcation Theory and Catastrophe Theory

Paperback Published: 20th May 1999
ISBN: 9783540653790
Number Of Pages: 274

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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.

Prefacep. 7
Bifurcations of Equilibriap. 10
Families and Deformationsp. 11
Families of Vector Fieldsp. 11
The Space of Jetsp. 11
Sard's Lemma and Transversality Theoremsp. 12
Simplest Applications: Singular Points of Generic Vector Fieldsp. 13
Topologically Versal Deformationsp. 14
The Reduction Theoremp. 15
Generic and Principal Familiesp. 16
Bifurcations of Singular Points in Generic One-Parameter Familiesp. 17
Typical Germs and Principal Familiesp. 17
Soft and Hard Loss of Stabilityp. 19
Bifurcations of Singular Points in Generic Multi-Parameter Families with Simply Degenerate Linear Partsp. 20
Principal Familiesp. 20
Bifurcation Diagrams of the Principal Families (3±) in Table 1p. 21
Bifurcation Diagrams with Respect to Weak Equivalence and Phase Portraits of the Principal Families (4±) in Table 1p. 21
Bifurcations of Singular Points of Vector Fields with a Doubly-Degenerate Linear Partp. 23
A List of Degeneraciesp. 23
Two Zero Eigenvaluesp. 24
Reductions to Two-Dimensional Systemsp. 24
One Zero and a Pair ofPurely Imaginary Eigenvaluesp. 25
Two Purely Imaginary Pairsp. 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 Stabilityp. 35
Definitionsp. 35
Table of Exponentsp. 37
Bifurcations of Limit Cyclesp. 38
Bifurcations of Limit Cycles in Generic One-Parameter Familiesp. 39
Multiplier 1p. 39
Multiplier - 1 and Period-Doubling Bifurcationsp. 41
A Pair of Complex Conjugate Multipliersp. 42
Nonlocal Bifurcations in One-Parameter Families of Diffeomorphismsp. 43
Nonlocal Bifurcations of Periodic Solutionsp. 45
Bifurcations Resulting in Destructions of Invariant Torip. 45
Bifurcations of Cycles in Generic Two-Parameter Families with an Additional Simple Degeneracyp. 48
A List of Degeneraciesp. 48
A Multiplier +1 or -1 with Additional Degeneracy in the Nonlinear Termsp. 49
A Pair of Multipliers on the Unit Circle with Additional Degeneracy in the Nonlinear Termsp. 49
Bifurcations of Cycles in Generic Two-Parameter Families with Strong Resonances of Orders q &neq; 4p. 51
The Normal Form in the Case of Unipotent Jordan Blocksp. 51
Averaging in the Seifert and the Möbius Foliationsp. 52
Principal Vector Fields and their Deformationsp. 53
Versality of Principal Deformationsp. 53
Bifurcations of Stationary Solutions of Periodic Differential Equations with Strong Resonances of Orders q &neq; 4p. 54
Bifurcations of Limit Cycles for a Pair of Multipliers Crossing the Unit Circle at ±ip. 57
Degenerate Familiesp. 57
Degenerate Families Found Analyticallyp. 59
Degenerate Families Found Numericallyp. 59
Bifurcations in Nondegenerate Familiesp. 60
Limit Cycles of Systems with a Fourth Order Symmetryp. 60
Finitely-Smooth Normal Forms of Local Familiesp. 60
A Synopsis of Resultsp. 60
Definitions and Examplesp. 62
General Theorems and Deformations of Nonresonant Germsp. 63
Reduction to Linear Normal Formp. 65
Deformations of Germs of Diffeomorphisms of Poincaré Typep. 66
Deformations of Simply Resonant Hyperbolic Germsp. 66
Deformations of Germs of Vector Fields with One Zero Eigenvalue at a Singular Pointp. 68
Functional Invariants of Diffeomorphisms of the Linep. 69
Functional Invariants of Local Families of Diffeornorphismsp. 70
Functional Invariants of Families of Vector Fieldsp. 71
Functional Invariants of Topological Classifications of Local Families of Diffeomorphisms of the Linep. 71
Feigenbaum Universality for Diffeomorphisms and Flowsp. 73
Period-Doubling Cascadesp. 73
Perestroikas of Fixed Pointsp. 75
Cascades of n-fold Increases of Periodp. 75
Doubling in Hamiltonian Systemsp. 75
The Period-Doubling Operator for One-Dimensional Mappingsp. 75
The Universal Period-Doubling Mechanism for Diffeomorphismsp. 77
Nonlocal Bifurcationsp. 79
Degeneracies of Codimension 1. Summary of Resultsp. 80
Local and Nonlocal Bifurcationsp. 80
Nonhyperbolic Singular Pointsp. 82
Nonhyperbolic Cyclesp. 83
Nontransversal Intersections of Manifoldsp. 84
Contoursp. 85
Bifurcation Surfacesp. 87
Characteristics of Bifurcationsp. 88
Summary of Resultsp. 88
Nonlocal Bifurcations of Flows on Two-Dimensional Surfacesp. 90
Semilocal Bifurcations of Flows on Surfacesp. 90
Nonlocal Bifurcations on a Sphere: The One-Parameter Casep. 91
Generic Families of Vector Fieldsp. 92
Conditions for Genericityp. 94
One-Parameter Families on Surfaces different from the Spherep. 95
Global Bifurcations of Systems with a Global Transversal Section on a Torusp. 96
Some Global Bifurcations on a Klein bottlep. 97
Bifurcations on a Two-Dimensional Sphere: The Multi-Parameter Casep. 98
Some Open Questionsp. 101
Bifurcations of Trajectories Homoclinic to a Nonhyperbolic Singular Pointp. 102
A Node in its Hyperbolic Variablesp. 103
A Saddle in its Hyperbolic Variables: One Homoclinic Trajectoryp. 103
The Topological Bernoulli Automorphismp. 104
A Saddle in its Hyperbolic Variables: Several Homoclinic Trajectoriesp. 105
Principal Familiesp. 106
Bifurcations of Trajectories Homoclinic to a Nonhyperbolic Cyclep. 106
The Structure of a Family of Homoclinic Trajectoriesp. 107
Critical and Noncritical Cyclesp. 107
Creation of a Smooth Two-Dimensional Attractorp. 108
Creation of Complex Invariant Sets (The Noncritical Case)p. 109
The Critical Casep. 109
A Two-Step Transition from Stability to Turbulencep. 111
A Noncompact Set of Homoclinic Trajectoriesp. 112
Intermittencyp. 113
Accessibility and Nonaccessibilityp. 113
Stability of Families of Diffeomorphismsp. 114
Some Open Questionsp. 116
Hyperbolic Singular Points with Homoclinic Trajectoriesp. 116
Preliminary Notions: Leading Directions and Saddle Numbersp. 117
Bifurcations of Homoclinic Trajectories of a Saddle that Take Place on the Boundary of the Set of Morse-Smale Systemsp. 117
Requirements for Genericityp. 118
Principal Families in <$>{\op R}^3<$> and their Propertiesp. 119
Versality of the Principal Familiesp. 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 Systemsp. 126
An Addition: Creation of a Strange Attractor upon Bifurcation of a Trajectory Homoclinic to a Saddlep. 127
Bifurcations Related to Nontransversal Intersectionsp. 129
Vector Fields with No Contours and No Homoclinic Trajectoriesp. 129
A Theorem on Inaccessibilityp. 130
Modulip. 131
Systems with Contoursp. 132
Diffeomorphisms with Nontrivial Basic Setsp. 133
Vector Fields in <$>{\op R}^3<$> with Trajectories Homoclinic to a Cyclep. 133
Symbolic Dynamicsp. 134
Bifurcations of Smale Horseshoesp. 136
Vector Fields on a Bifurcation Surfacep. 138
Diffeomorphisms with an Infinite Set of Stable Periodic Trajectoriesp. 138
Infinite Nonwandering Setsp. 139
Vector Fields on the Two-Dimensional Torusp. 139
Bifurcations of Systems with Two Homoclinic Curves of a Saddlep. 140
Systems with Feigenbaum Attractorsp. 142
Birth of Nonwandering Setsp. 142
Persistence and Smoothness of Invariant Manifoldsp. 143
The Degenerate Family and Its Neighborhood in Function Spacep. 144
Birth of Tori in a Three-Dimensional Phase Spacep. 145
Attractors and their Bifurcationsp. 145
The Likely Limit Set According to Milnor (1985)p. 147
Statistical Limit Setsp. 147
Internal Bifurcations and Crises of Attractorsp. 149
Internal Bifurcations and Crises of Equilibria and Cyclesp. 149
Bifurcations of the Two-Dimensional Torusp. 150
Relaxation Oscillationsp. 154
Fundamental Conceptsp. 155
An Example: van der Pol's Equationp. 155
Fast and Slow Motionsp. 156
The Slow Surface and Slow Equationsp. 157
The Slow Motion as an Approximation to the Perturbed Motionp. 158
The Phenomenon of Jumpingp. 159
Singularities of the Fast and Slow Motionsp. 160
Singularities of Fast Motions at Jump Points of Systems with One Fast Variablep. 160
Singularities of Projections of the Slow Surfacep. 161
The Slow Motion for Systems with One Slow Variablep. 162
The Slow Motion for Systems with Two Slow Variablesp. 163
Normal Forms of Phase Curves of the Slow Motionp. 164
Connection with the Theory of Implicit Differential Equationsp. 167
Degeneration of the Contact Structurep. 168
The Asymptotics of Relaxation Oscillationsp. 170
Degenerate Systemsp. 170
Systems of First Approximationp. 171
Normalizations of Fast-Slow Systems with Two Slow Variables for ¿ > 0p. 173
Derivation of the Systems of First Approximationp. 175
Investigation of the Systems of First Approximationp. 175
Funnelsp. 177
Periodic Relaxation Oscillations in the Planep. 177
Delayed Loss of Stability as a Pair of Eigenvalues Cross the Imaginary Axisp. 179
Generic Systemsp. 179
Delayed Loss of Stabilityp. 180
Hard Loss of Stability in Analytic Systems of Type 2p. 181
Hysteresisp. 181
The Mechanism of Delayp. 182
Computation of the Moment of Jumping in Analytic Systemsp. 182
Delay Upon Loss of Stability by a Cyclep. 185
Delayed Loss of Stability and "Ducks"p. 185
Duck Solutionsp. 185
An Example: A Singular Point on the Fold of the Slow Surfacep. 186
Existence of Duck Solutionsp. 188
The Evolution of Simple Degenerate Ducksp. 189
A Semi-local Phenomenon: Ducks with Relaxationp. 190
Ducks in <$>{\op R}^3<$> and <$>{\op R}^n<$>p. 191
Recommended Literaturep. 193
Referencesp. 195
Additional Referencesp. 205
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9783540653790
ISBN-10: 3540653791
Series: Encyclopaedia of Mathematical Sciences
Audience: General
Format: Paperback
Language: English
Number Of Pages: 274
Published: 20th May 1999
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.55 x 15.55  x 1.73
Weight (kg): 0.44