Topological Degree Approach to Bifurcation Problems by Michal Feckan

Topological Degree Approach to Bifurcation Problems

Michal Feckan
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  • ISBN
    9789048179695 / 9048179696
  • Title Topological Degree Approach to Bifurcation Problems
  • Author Michal Feckan
  • Category Topology
  • Format
  • Year 2010
  • Pages 270
  • Publisher
  • Imprint Springer
  • Edition
  • Language English
  • Dimensions 156mm x 234mm x 14mm


This book contains original bifurcation results for the existence of oscillations and chaotic behavior of differential equations and discrete dynamical systems under variation of involved parameters. It studies a broad variety of nonlinear problems.

Publisher Description

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.

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.” (Laszlo Hatvani, Acta Scientiarum Mathematicarum (Szeged), Vol. 75 (3-4), 2009)

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Topological Degree Approach to Bifurcation Problems