Topological Degree Approach to Bifurcation Problems

ISBN-13: 9781402087233 / Angielski / Twarda / 2008 / 261 str.

Michal Feckan

Opis

1. 1 Preface Many phenomena from physics, biology, chemistry and economics are modeled by di?erential equations with parameters. When a nonlinear equation is est- lished, its behavior/dynamics should be understood. In general, it is impossible to ?nd a complete dynamics of a nonlinear di?erential equation. Hence at least, either periodic or irregular/chaotic solutions are tried to be shown. So a pr- erty of a desired solution of a nonlinear equation is given as a parameterized boundary value problem. Consequently, the task is transformed to a solvability of an abstract nonlinear equation with parameters on a certain functional space. When a family of solutions of the abstract equation is known for some para- ters, the persistence or bifurcations of solutions from that family is studied as parameters are changing. There are several approaches to handle such nonl- ear bifurcation problems. One of them is a topological degree method, which is rather powerful in cases when nonlinearities are not enough smooth. The aim of this book is to present several original bifurcation results achieved by the author using the topological degree theory. The scope of the results is rather broad from showing periodic and chaotic behavior of non-smooth mechanical systems through the existence of traveling waves for ordinary di?erential eq- tions on in?nite lattices up to study periodic oscillations of undamped abstract waveequationsonHilbertspaceswithapplicationstononlinearbeamandstring partial di?erential equations. 1.

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.