Bifurcation Theory

An Introduction with Applications to Partial Differential Equations

verfasst von: Hansjörg Kielhöfer

Verlag: Springer New York

Buchreihe : Applied Mathematical Sciences

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

In the past three decades, bifurcation theory has matured into a well-established and vibrant branch of mathematics. This book gives a unified presentation in an abstract setting of the main theorems in bifurcation theory, as well as more recent and lesser known results. It covers both the local and global theory of one-parameter bifurcations for operators acting in infinite-dimensional Banach spaces, and shows how to apply the theory to problems involving partial differential equations. In addition to existence, qualitative properties such as stability and nodal structure of bifurcating solutions are treated in depth. This volume will serve as an important reference for mathematicians, physicists, and theoretically-inclined engineers working in bifurcation theory and its applications to partial differential equations.

The second edition is substantially and formally revised and new material is added. Among this is bifurcation with a two-dimensional kernel with applications, the buckling of the Euler rod, the appearance of Taylor vortices, the singular limit process of the Cahn-Hilliard model, and an application of this method to more complicated nonconvex variational problems.

Inhaltsverzeichnis

Frontmatter

Chapter 0. Introduction

Abstract

Bifurcation Theory attempts to explain various phenomena that have been discovered and described in the natural sciences over the centuries. The buckling of the Euler rod, the appearance of Taylor vortices, and the onset of oscillations in an electric circuit, for instance, all have a common cause: A specific physical parameter crosses a threshold, and that event forces the system to the organization of a new state that differs considerably from that observed before.

Hansjörg Kielhöfer

Chapter I. Local Theory

Abstract

One of the most important analytic tools for the solution of a nonlinear problem

$$F(x, y)=0$$

Hansjörg Kielhöfer

Chapter II. Global Theory

Abstract

Comparable to the importance of the Implicit Function Theorem for the local analysis is the degree of a mapping for any global analysis. Although the theory was originally invented and defined in topology we present an analytical theory developed later. For finite-dimensional continuous mappings it is the Brouwer degree; its extension to infinite dimensions is the Leray–Schauder degree.

Hansjörg Kielhöfer

Chapter III. Applications

Abstract

The Fredholm property is a leitmotiv, since it was assumed in all sections of Chapters I and II. Clearly, all finite-dimensional linear operators have that property, but we have infinite-dimensional applications in mind. A prototype of a Fredholm operator, playing also a central role in applications, is an elliptic operator over a bounded domain. We confine ourselves to operators of second order acting on scalar functions.

Hansjörg Kielhöfer

Backmatter

Titel: Bifurcation Theory
verfasst von: Hansjörg Kielhöfer
Verlag: Springer New York
Electronic ISBN: 978-1-4614-0502-3
Print ISBN: 978-1-4614-0501-6
DOI: https://doi.org/10.1007/978-1-4614-0502-3