Bifurcations of Dynamical Systems

Abstract

The subject of these lectures is the bifurcation theory of dynamical systems. They are not comprehensive, as we take up some facets of bifurcation theory and largely ignore others. In particular, we focus our attention on finite dimensional systems of difference and differential equations and say almost nothing about infinite dimensional systems. The reader interested in the infinite dimensional theory and its applications should consult the recent survey of Marsden [66] and the conference proceedings edited by Rabinowitz [89]. We also neglect much of the multidimensional bifurcation theory of singular points of differential equations. The systematic exposition of this theory is much more algebraic than the more geometric questions considered here, and Arnold [7,9] provides a good survey of work in this area. We confine our interest to questions which involve the geometric orbit structure of dynamical systems. We do make an effort to consider applications of the mathematical phenomena illustrated. For general background about the theory of dynamical systems consult [102]. Our style is informal and our intent is pedagogic. The current state of bifurcation theory is a mixture of mathematical fact and conjecture. The demarcation between the proved and unproved is small [11]. Rather than attempting to sort out this confused state of affairs for the reader, we hope to provide the geometric insight which will allow him to explore further.