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Topological Degree Approach to Bifurcation Problems

  • Book
  • © 2008

Overview

Authors:
  1. Michal Fečkan

    1. Department of Mathematical Analysis and Numerical Mathematics Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava, Slovakia

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  • Includes (1) Rigorous proofs of chaotic solutions for discontinuous differential equations and differential inclusions (2) Bifurcations of periodic solutions in differential inclusions and systems with relay hysteresis (3)
  • The persistence of traveling waves under spatial discretization of sine-Gordon and Klein-Gordon partial differential equations (4) Topological degree theory for discontinuous wave partial differential equations (5) Chaotic behavior of maps possessing topologically transversally intersecting invariant manifolds

Part of the book series: Topological Fixed Point Theory and Its Applications (TFPT, volume 5)

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Table of contents (8 chapters)

  1. Front Matter

    Pages i-ix
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  2. Introduction

    Pages 1-6

  3. Theoretical Background

    Pages 7-21

  4. Bifurcation of Periodic Solutions

    Pages 23-119

  5. Bifurcation of Chaotic Solutions

    Pages 121-142

  6. Topological Transversality

    Pages 143-181

  7. Traveling Waves on Lattices

    Pages 183-198

  8. Periodic Oscillations of Wave Equations

    Pages 199-226

  9. Topological Degree for Wave Equations

    Pages 227-241

  10. Back Matter

    Pages 243-261
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Keywords

About this book

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.

Reviews

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)

Authors and Affiliations

  • Department of Mathematical Analysis and Numerical Mathematics Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava, Slovakia

    Michal Fečkan

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