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Topics in Dynamic Bifurcation Theory
Jack K. Hale, Georgia Institute of Technology, Atlanta, GA
A co-publication of the AMS and CBMS.
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CBMS Regional Conference Series in Mathematics
1981; 84 pp; softcover
Number: 47
Reprint/Revision History:
third printing 2001
ISBN-10: 0-8218-1698-5
ISBN-13: 978-0-8218-1698-1
List Price: US$15
Member Price: US$12
All Individuals: US$12
Order Code: CBMS/47
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This set of lectures has two primary objectives. The first one is to present the general theory of first order bifurcation that occur for vector fields in finite dimensional space. Illustrations are given of higher order bifurcations. The second objective, and probably the most important one, is to set up a framework for the discussion of similar problems in infinite dimensions. Parabolic systems and retarded functional differential equations are considered as illustrations and motivations for the general theory.

Readers familiar with ordinary differential equations and basic elements of nonlinear functional analysis will find that the material is accessible and the fundamental results in bifurcation theory are presented in a way to be relevant to direct application. Most of the expository material consists of a concise presentation of basic results and problems in structural stability.

The most significant contribution of the book is the formulation of structural stability and bifurcation in infinite dimensions. Much research should come from this--indeed some have already picked up the ideas in their work.

Readership

Reviews

"[This book] surveys some important aspects of bifurcation theory for ordinary differential equations, including infinite-dimensional cases."

-- Mathematical Reviews

Table of Contents

  • On the definition of bifurcation
  • Structural stability and generic properties in \(\mathbf R^n\)
  • Stability and bifurcation at a zero eigenvalue
  • Stability and bifurcation from a focus
  • First order bifurcation in the plane
  • Two dimensional periodic systems
  • Higher order bifurcation near equilibrium
  • A framework for infinite dimensions
  • Bifurcation in infinite dimensions
  • References
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