Topological Degree Approach to Bifurcation Problems

ISBN-13: 9789048179695 / Angielski / Miękka / 2010 / 261 str.

Michal Fečkan

Opis

This book is devoted to bifurcations of periodic, subharmonic and chaotic oscillations, and travelling waves in nonlinear differential equations and discrete dynamical systems by using the topological degree theory both for single-valued and multi-valued mappings in Banach spaces. Original bifurcation results are proved with applications to a broad variety of nonlinear problems ranging from non-smooth and discontinuous mechanical systems, weakly coupled oscillators, systems with relay hysteresis, through infinite chains of differential equations on lattices involving also spatially discretized partial differential equations, and to string and beam partial differential equations. Next, the chaotic behaviour is also investigated for maps possessing topologically transversally intersecting invariant manifolds. Moreover, periodic orbits with arbitrarily high periods, the so-called blue sky catastrophe, are shown for reversible differential systems and maps. Finally, bifurcations of large amplitude oscillations for discontinuous undamped wave partial differential equations are given as well.

This book is mainly intended for post-graduate students and researchers in mathematics with an interest in applications of topological bifurcation methods to dynamical systems and nonlinear analysis, in particular to differential equations and inclusions, and maps. But, among others, it could also be used either by physicists studying oscillations of nonlinear mechanical systems or by engineers investigating vibrations of strings and beams, and electrical circuits.

Recenzje

From the book reviews:

"This excellent and well-organized book is based on recently published papers of the author using topological degree methods. ... The book should not only be of interest to mathematicians but to physicists and theoretically inclined engineers involved in bifurcation theory and its applications to dynamical systems and nonlinear analysis." (László Hatvani, Acta Scientiarum Mathematicarum (Szeged), Vol. 75 (3-4), 2009)

Spis treści

1. Introduction 1.1. Preface 1.2. An Illustrative Perturbed Problem 1.3. A Brief Summary of the Book 2. Theoretical Background 2.1. Linear Functional Analysis 2.2. Nonlinear Functional Analysis 2.2.1. Implicit Function Theorem 2.2.2. Lyapunov-Schmidt Method 2.2.3. Leray-Schauder Degree 2.3. Differential Topology 2.3.1. Differentiable Manifolds 2.3.2. Symplectic Surfaces 2.3.3. Intersection Numbers of Manifolds 2.3.4. Brouwer Degree on Manifolds 2.3.5. Vector Bundles 2.3.6. Euler Characteristic 2.4. Multivalued Mappings 2.4.1. Upper Semicontinuity 2.4.2. Measurable Selections 2.4.3. Degree Theory for Set-Valued Maps 2.5. Dynamical Systems 2.5.1. Exponential Dichotomies 2.5.2. Chaos in Discrete Dynamical Systems 2.5.3. Periodic O.D.Eqns 2.5.4. Vector Fields 2.6. Center Manifolds For Infinite Dimensions 3. Bifurcation of Periodic Solutions 3.1. Bifurcation of Periodics from Homoclinics I 3.1.1. Discontinuous O.D.Eqns 3.1.2. The Linearized Equation 3.1.3. Subharmonics for Regular Periodic Perturbations 3.1.4. Subharmonics for Singular Periodic Perturbations 3.1.5. Subharmonics for Regular Autonomous Perturbations 3.1.6. Applications to Discontinuous O.D.Eqns 3.1.7. Bounded Solutions Close to Homoclinics 3.2. Bifurcation of Periodics from Homoclinics II 3.2.1. Singular Discontinuous O.D.Eqns 3.2.2. Linearized Equations 3.2.3. Bifurcation of Subharmonics 3.2.4. Applications to Singular Discontinuous O.D.Eqns 3.3. Bifurcation of Periodics from Periodics 3.3.1. Discontinuous O.D.Eqns 3.3.2. Linearized Problem 3.3.3. Bifurcation of Periodics in Nonautonomous Systems 3.3.4. Bifurcation of Periodics in Autonomous Systems 3.3.5. Applications to Discontinuous O.D.Eqns 3.3.6. Concluding Remarks 3.4. Bifurcation of Periodics in Relay Systems 3.4.1. Systems with Relay Hysteresis 3.4.2. Bifurcation of Periodics 3.4.3. Third-Order O.D.Eqns with Small Relay Hysteresis 3.5. Nonlinear Oscillators with Weak Couplings 3.5.1. Weakly Coupled Systems 3.5.2. Forced Oscillations from Single Periodics 3.5.3. Forced Oscillations from Families of Periodics 3.5.4. Applications to Weakly Coupled Nonlinear Oscillators 4. Bifurcation of Chaotic Solutions 4.1. Chaotic Differential Inclusions 4.1.1. Nonautonomous Discontinuous O.D.Eqns 4.1.2. The Linearized equation 4.1.3. Bifurcation of Chaotic Solutions 4.1.4. Chaos from Homoclinic Manifolds 4.1.5. Almost and Quasi Periodic Discontinuous O.D.Eqns 4.2. Chaos in Periodic Differential Inclusions 4.2.1. Regular Periodic Perturbations 4.2.2. Singular Differential Inclusions 4.3. More about Homoclinic Bifurcations 4.3.1. Transversal Homoclinic Crossing Discontinuity 4.3.2. Homoclinic Sliding on Discontinuity 5. Topological Transversality 5.1. Topological Transversality and Chaos 5.1.1. Topologically Transversal Invariant Sets 5.1.2. Difference Boundary Value Problems 5.1.3. Chaotic Orbits 5.1.4. Periodic Points and Extensions on Invariant Compact Subsets 5.1.5. Perturbed Topological Transversality 5.2. Topological Transversality and Reversibility 5.2.1. Period Blow-up 5.2.2. Period Blow-up for Reversible Diffeomorphisms 5.2.3. Perturbed Period Blow-up 5.2.4. Perturbed Second Order O.D.Eqns 5.3. Chains of Reversible Oscillators 5.3.1. Homoclinic Period Blow-up for Breathers 5.

Blurb

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.