# Topics in Dynamic Bifurcation Theory

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*Jack K. Hale*

A co-publication of the AMS and CBMS

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 & Endorsements

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

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Topics in Dynamic Bifurcation Theory

- Cover Cover11 free
- Title i2 free
- Copyright ii3 free
- Contents iii4 free
- 1. Introduction 16 free
- 2. On the definition of bifurcation 38 free
- 3. Structural stability and generic properties in R[sup(n)] 510
- 4. Stability and bifurcation at a zero eigenvalue 1318
- 5. Stability and bifurcation from a focus 2126
- 6. First order bifurcation in the plane 3136
- 7. Two dimensional periodic systems 3742
- 8. Higher order bifurcation near equilibrium 4550
- 9. A framework for infinite dimensions 5358
- 10. Bifurcation in infinite dimensions 6974
- References 7782
- Back Cover Back Cover190