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9781402087233

Topological Degree Approach to Bifurcation Problems

by
  • ISBN13:

    9781402087233

  • ISBN10:

    1402087233

  • Format: Hardcover
  • Copyright: 2008-12-04
  • Publisher: Springer Verlag

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Summary

"Topological bifurcation theory is one of the most essential topics in mathematics. This book contains original bifurcation results for the existence of oscillations and chaotic behaviour of differential equations and discrete dynamical systems under variation of involved parameters. Using topological degree theory and a perturbation approach in dynamical systems, a broad variety of nonlinear problems are studied, including: non-smooth mechanical systems with dry frictions; weakly coupled oscillators; systems with relay hysteresis; differential equations on infinite lattices of Frenkel-Kontorova and discretized Klein-Gordon types; blue sky catastrophes for reversible dynamical systems; buckling of beams; and discontinuous wave equations." "Precise and complete proofs, together with concrete applications with many stimulating and illustrating examples, make this book valuable to both the applied sciences and mathematical fields, ensuring the book should not only be of interest to mathematicians but to physicists and theoretically inclined engineers interested in bifurcation theory and its applications to dynamical systems and nonlinear analysis."--BOOK JACKET.

Table of Contents

Introductionp. 1
Prefacep. 1
An Illustrative Perturbed Problemp. 1
A Brief Summary of the Bookp. 5
Theoretical Backgroundp. 7
Linear Functional Analysisp. 7
Nonlinear Functional Analysisp. 8
Implicit Function Theoremp. 8
Lyapunov-Schmidt Methodp. 9
Leray-Schauder Degreep. 9
Differential Topologyp. 11
Differentiable Manifoldsp. 11
Symplectic Surfacesp. 12
Intersection Numbers of Manifoldsp. 12
Brouwer Degree on Manifoldsp. 13
Vector Bundlesp. 13
Euler Characteristicp. 14
Multivalued Mappingsp. 14
Upper Semicontinuityp. 14
Measurable Selectionsp. 15
Degree Theory for Set-Valued Mapsp. 15
Dynamical Systemsp. 16
Exponential Dichotomiesp. 16
Chaos in Discrete Dynamical Systemsp. 16
Periodic O.D.Eqnsp. 18
Vector Fieldsp. 18
Center Manifolds for Infinite Dimensionsp. 20
Bifurcation of Periodic Solutionsp. 23
Bifurcation of Periodics from Homoclinics Ip. 23
Discontinuous O.D.Eqnsp. 23
The Linearized Equationp. 25
Subharmonics for Regular Periodic Perturbationsp. 32
Subharmonics for Singular Periodic Perturbationsp. 40
Subharmonics for Regular Autonomous Perturbationsp. 43
Applications to Discontinuous O.D.Eqnsp. 44
Bounded Solutions Close to Homoclinicsp. 53
Bifurcation of Periodics from Homoclinics IIp. 54
Singular Discontinuous O.D.Eqnsp. 54
Linearized Equationsp. 57
Bifurcation of Subharmonicsp. 60
Applications to Singular Discontinuous O.D.Eqnsp. 70
Bifurcation of Periodics from Periodicsp. 75
Discontinuous O.D.Eqnsp. 75
Linearized Problemp. 77
Bifurcation of Periodics in Nonautonomous Systemsp. 82
Bifurcation of Periodics in Autonomous Systemsp. 85
Applications to Discontinuous O.D.Eqnsp. 90
Concluding Remarksp. 96
Bifurcation of Periodics in Relay Systemsp. 97
Systems with Relay Hysteresisp. 97
Bifurcation of Periodicsp. 98
Third-Order O.D.Eqns with Small Relay Hysteresisp. 103
Nonlinear Oscillators with Weak Couplingsp. 107
Weakly Coupled Systemsp. 107
Forced Oscillations from Single Periodicsp. 108
Forced Oscillations from Families of Periodicsp. 111
Applications to Weakly Coupled Nonlinear Oscillatorsp. 113
Bifurcation of Chaotic Solutionsp. 121
Chaotic Differential Inclusionsp. 121
Nonautonomous Discontinuous O.D.Eqnsp. 121
The Linearized Equationp. 122
Bifurcation of Chaotic Solutionsp. 125
Chaos from Homoclinic Manifoldsp. 128
Almost and Quasi Periodic Discontinuous O.D.Eqnsp. 129
Chaos in Periodic Differential Inclusionsp. 135
Regular Periodic Perturbationsp. 135
Singular Differential Inclusionsp. 138
More About Homoclinic Bifurcationsp. 139
Transversal Homoclinic Crossing Discontinuityp. 139
Homoclinic Sliding on Discontinuityp. 140
Topological Transversalityp. 143
Topological Transversality and Chaosp. 143
Topologically Transversal Invariant Setsp. 143
Difference Boundary Value Problemsp. 146
Chaotic Orbitsp. 151
Periodic Points and Extensions on Invariant Compact Subsetsp. 156
Perturbed Topological Transversalityp. 157
Topological Transversality and Reversibilityp. 164
Period Blow-Upp. 164
Period Blow-Up for Reversible Diffeomorphismsp. 165
Perturbed Period Blow-Upp. 168
Perturbed Second Order O.D.Eqnsp. 172
Chains of Reversible Oscillatorsp. 176
Homoclinic Period Blow-Up for Breathersp. 177
Heteroclinic Period Blow-Up for Non-breathersp. 178
Period Blow-Up for Traveling Wavesp. 181
Traveling Waves on Latticesp. 183
Traveling Waves in Discretized P.D.Eqnsp. 183
Center Manifold Reductionp. 185
A Class of Singularly Perturbed O.D.Eqnsp. 189
Bifurcation of Periodic Solutionsp. 189
Traveling Waves in Homoclinic Casesp. 193
Traveling Waves in Heteroclinic Casesp. 194
Traveling Waves in 2 Dimensionsp. 196
Periodic Oscillations of Wave Equationsp. 199
Periodics of Undamped Beam Equationsp. 199
Undamped Forced Nonlinear Beam Equationsp. 199
Existence Results on Periodicsp. 201
Subharmonics from Homoclinicsp. 207
Periodics from Periodicsp. 209
Applications to Forced Nonlinear Beam Equationsp. 210
Weakly Nonlinear Wave Equationsp. 216
Excluding Small Divisorsp. 216
Lebesgue Measures of Nonresonancesp. 220
Forced Periodic Solutionsp. 221
Theory of Numbers and Nonresonancesp. 223
Topological Degree for Wave Equationsp. 227
Discontinuous Undamped Wave Equationsp. 227
Standard Classes of Multi-Mappingsp. 228
M-Regular Multi-Functionsp. 229
Classes of Admissible Mappingsp. 230
Semilinear Wave Equationsp. 231
Construction of Topological Degreep. 232
Local Bifurcationsp. 235
Bifurcations from Infinityp. 237
Bifurcations for Semilinear Wave Equationsp. 238
Chaos for Discontinuous Beam Equationsp. 238
Bibliographyp. 243
Indexp. 257
Table of Contents provided by Ingram. All Rights Reserved.

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