Topological Degree Approach to Bifurcation Problems / Edition 1

by Michal Feckan

Overview

1. 1 Preface Many phenomena from physics, biology, chemistry and economics are modeled by differential equations with parameters. When a nonlinear equation is est- lished, its behavior/dynamics should be understood. In general, it is impossible to find a complete dynamics of a nonlinear differential 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 differential eq- tions on inflnite lattices up to study periodic oscillations of undamped abstract wave equations on Hilberts paces with applications to nonlinear beam and string partial differential equations. 1.

Product Details

ISBN-13:	9781402087233
Publisher:	Springer Netherlands
Publication date:	10/10/2008
Series:	Topological Fixed Point Theory and Its Applications , #5
Edition description:	2008
Pages:	261
Product dimensions:	6.20(w) x 9.40(h) x 0.80(d)

Table of Contents

Theoretical Background.- Bifurcation of Periodic Solutions.- Bifurcation of Chaotic Solutions.- Topological Transversality.- Traveling Waves on Lattices.- Periodic Oscillations of Wave Equations.- Topological Degree for Wave Equations.

Editorial 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)

From the Publisher