Analytic Theory of Global Bifurcation

An Introduction

Boris Buffoni, John Toland

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English

Princeton University Press
02 February 2003

Differential calculus & equations; Analytic topology

Series: Princeton Series in Applied Mathematics

Rabinowitz's classical global bifurcation theory, which concerns the study in-the-large of parameter-dependent families of nonlinear equations, uses topological methods that address the problem of continuous parameter dependence of solutions by showing that there are connected sets of solutions of global extent. Even when the operators are infinitely differentiable in all the variables and parameters, connectedness here cannot in general be replaced by path-connectedness. However, in the context of real-analyticity there is an alternative theory of global bifurcation due to Dancer, which offers a much stronger notion of parameter dependence. This book aims to develop from first principles Dancer's global bifurcation theory for one-parameter families of real-analytic operators in Banach spaces. It shows that there are globally defined continuous and locally real-analytic curves of solutions. In particular, in the real-analytic setting, local analysis can lead to global consequences--for example, as explained in detail here, those resulting from bifurcation from a simple eigenvalue.

Included are accounts of analyticity and implicit function theorems in Banach spaces, classical results from the theory of finite-dimensional analytic varieties, and the links between these two and global existence theory. Laying the foundations for more extensive studies of real-analyticity in infinite-dimensional problems and illustrating the theory with examples, Analytic Theory of Global Bifurcation is intended for graduate students and researchers in pure and applied analysis.

By: Boris Buffoni, John Toland
Imprint: Princeton University Press
Country of Publication: United States
Volume: 9
Dimensions: Height: 235mm, Width: 152mm, Spine: 19mm
Weight: 397g
ISBN: 9780691112985
ISBN 10: 0691112983
Series: Princeton Series in Applied Mathematics
Pages: 184
Publication Date: 02 February 2003
Audience: Professional and scholarly , College/higher education , Undergraduate , Primary
Format: Hardback
Publisher's Status: Active

Boris Buffoni holds a Swiss National Science Foundation Professorship in Mathematics at the Swiss Federal Institute of Technology-Lausanne. John Toland is Professor of Mathematical Sciences at the University of Bath and a Senior Research Fellow of the UK's Engineering and Physical Sciences Research Council