Bifurcation and Chaos in Discontinuous and Continuous Systems

by Feckan, Michal

• ISBN13: 9783642182686
• ISBN10: 3642182682
• Format: Hardcover
• Copyright: 2011-08-02
• Publisher: Springer Verlag

Summary

"Bifurcation and Chaos in Discontinuous and Continuous Systems" provides rigorous mathematical functional-analytical tools for handling chaotic bifurcations along with precise and complete proofs together with concrete applications presented by many stimulating and illustrating examples. A broad variety of nonlinear problems are studied involving difference equations, ordinary and partial differential equations, differential equations with impulses, piecewise smooth differential equations, differential and difference inclusions, and differential equations on infinite lattices as well.This book is intended for mathematicians, physicists, theoretically inclined engineers and postgraduate students either studying oscillations of nonlinear mechanical systems or investigating vibrations of strings and beams, and electrical circuits by applying the modern theory of bifurcation methods in dynamical systems.Dr. Michal Fe#xC4;kan is a Professor at the Department of Mathematical Analysis and Numerical Mathematics on the Faculty of Mathematics, Physics and Informatics at the Comenius University in Bratislava, Slovakia. He is working on nonlinear functional analysis, bifurcation theory and dynamical systems with applications to mechanics and vibrations.

Author Biography

Dr. Michal Feckan is a Professor at the Department of Mathematical Analysis and Numerical Mathematics on the Faculty of Mathematics, Physics and Informatics at the Comenius University in Bratislava, Slovakia. He is working on nonlinear functional analysis, bifurcation theory and dynamical systems with applications to mechanics and vibrations.

Table of Contents

Introduction	p. 1
References	p. 6
Preliminary Results	p. 9
Linear Functional Analysis	p. 9
Nonlinear Functional Analysis	p. 11
Banach Fixed Point Theorem	p. 11
Implicit Function Theorem	p. 11
Lyapunov-Schmidt Method	p. 12
Brouwer Degree	p. 13
Local Invertibility	p. 13
Global Invertibility	p. 14
Multivalued Mappings	p. 14
Differential Topology	p. 15
Differentiable Manifolds	p. 15
Vector Bundles	p. 16
Tubular Neighbourhoods	p. 16
Dynamical Systems	p. 17
Homogenous Linear Equations	p. 17
Chaos in Diffeomorphisms	p. 18
Periodic ODEs	p. 19
Vector Fields	p. 20
Global Center Manifolds	p. 22
Two-Dimensional Flows	p. 22
Averaging Method	p. 23
Carathéodory Type ODEs	p. 24
Singularities of Smooth Maps	p. 24
Jet Bundles	p. 24
Whitney C∞ Topology	p. 25
Transversality	p. 25
Malgrange Preparation Theorem	p. 26
Complex Analysis	p. 26
References	p. 28
Chaos in Discrete Dynamical Systems	p. 29
Transversal Bounded Solutions	p. 29
Difference Equations	p. 29
Variational Equation	p. 30
Perturbation Theory	p. 35
Bifurcation from a Manifold of Homoclinic Solutions	p. 38
Applications to Impulsive Differential Equations	p. 40
Transversal Homoclinic Orbits	p. 44
Higher Dimensional Difference Equations	p. 44
Bifurcation Result	p. 45
Applications to McMillan Type Mappings	p. 51
Planar Integrable Maps with Separatrices	p. 54
Singular Impulsive ODEs	p. 55
Singular ODEs with Impulses	p. 55
Linear Singular ODEs with Impulses	p. 56
Derivation of the Melnikov Function	p. 64
Examples of Singular Impulsive ODEs	p. 68
Singularly Perturbed Impulsive ODEs	p. 70
Singularly Perturbed ODEs with Impulses	p. 70
Melnikov Function	p. 71
Second Order Singularly Perturbed ODEs with Impulses	p. 72
Inflated Deterministic Chaos	p. 73
Inflated Dynamical Systems	p. 73
Inflated Chaos	p. 74
References	p. 83
Chaos in Ordinary Differential Equations	p. 87
Higher Dimensional ODEs	p. 87
Parameterized Higher Dimensional ODEs	p. 87
Variational Equations	p. 88
Melnikov Mappings	p. 90
The Second Order Melnikov Function	p. 93
Application to Periodically Perturbed ODEs	p. 95
ODEs with Nonresonant Center Manifolds	p. 97
Parameterized Coupled Oscillators	p. 97
Chaotic Dynamics on the Hyperbolic Subspace	p. 98
Chaos in the Full Equation	p. 100
Applications to Nonlinear ODEs	p. 105
ODEs with Resonant Center Manifolds	p. 108
ODEs with Saddle-Center Parts	p. 108
Example of Coupled Oscillators at Resonance	p. 109
General Equations	p. 121
Averaging Method	p. 127
Singularly Perturbed and Forced ODEs	p. 131
Forced Singular ODEs	p. 131
Center Manifold Reduction	p. 132
ODEs with Normal and Slow Variables	p. 135
Homoclinic Hopf Bifurcation	p. 135
Bifurcation from Degenerate Homoclinics	p. 136
Periodically Forced ODEs with Degenerate Homoclinics	p. 136
Bifurcation Equation	p. 137
Bifurcation for 2-Parametric Systems	p. 138
Bifurcation for 4-Parametric Systems	p. 144
Autonomous Perturbations	p. 147
Inflated ODEs	p. 150
Inflated Carathéodory Type ODEs	p. 150
Inflated Periodic ODEs	p. 151
Inflated Autonomous ODEs	p. 154
Nonlinear Diatomic Lattices	p. 156
Forced and Coupled Nonlinear Lattices	p. 156
Spatially Localized Chaos	p. 157
References	p. 163
Chaos in Partial Differential Equations	p. 167
Beams on Elastic Bearings	p. 167
Weakly Nonlinear Beam Equation	p. 167
Setting of the Problem	p. 168
Preliminary Results	p. 171
Chaotic Solutions	p. 191
Useful Numerical Estimates	p. 215
Lipschitz Continuity	p. 217
Infinite Dimensional Non-Resonant Systems	p. 220
Buckled Elastic Beam	p. 220
Abstract Problem	p. 224
Chaos on the Hyperbolic Subspace	p. 224
Chaos in the Full Equation	p. 226
Applications to Vibrating Elastic Beams	p. 227
Planer Motion with One Buckled Mode	p. 227
Nonplaner Symmetric Beams	p. 230
Nonplaner Nonsymmetric Beams	p. 235
Multiple Buckled Modes	p. 238
Periodically Forced Compressed Beam	p. 242
Resonant Compressed Equation	p. 242
Formulation of Weak Solutions	p. 242
Chaotic Solutions	p. 243
References	p. 247
Chaos in Discontinuous Differential Equations	p. 249
Transversal Homoclinic Bifurcation	p. 249
Discontinuous Differential Equations	p. 249
Setting of the Problem	p. 250
Geometric Interpretation of Nondegeneracy Condition	p. 255
Orbits Close to the Lower Homoclinic Branches	p. 257
Orbits Close to the Upper Homoclinic Branch	p. 263
Bifurcation Equation	p. 265
Chaotic Behaviour	p. 287
Almost and Quasiperiodic Cases	p. 293
Periodic Case	p. 294
Piecewise Smooth Planar Systems	p. 295
3D Quasiperiodic Piecewise Linear Systems	p. 299
Multiple Transversal Crossings	p. 310
Sliding Homoclinic Bifurcation	p. 312
Higher Dimensional Sliding Homoclinics	p. 312
Planar Sliding Homoclinics	p. 319
Three-Dimensional Sliding Homoclinics	p. 321
Outlook	p. 332
References	p. 332
Concluding Related Topics	p. 335
Notes on Melnikov Function	p. 335
Role of Melnikov Function	p. 335
Melnikov Function and Calculus of Residues	p. 336
Second Order ODEs	p. 340
Applications and Examples	p. 347
Transverse Heteroclinic Cycles	p. 361
Blue Sky Catastrophes	p. 369
Symmetric Systems with First Integrals	p. 370
D'Alembert and Penalized Equations	p. 371
References	p. 373
Index	p. 375
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