INTRODUCTION

Symmetries abound In nature, in technology, and - especially - in the simplified mathematical models that we study so assiduously. Symmetries complicate things and simplify them. They complicate them by introducing exceptional types of behaviour. Increasing the number of variables Involved, and making vanish things that usually do not vanish. They simplify them by introducing exceptional types of behaviour, increasing the number of variables involved, and making vanish things that usually do not vanish. They violate all the hypotheses of our favourite theorems, yet lead to natural generalizations of those theorems. It is now standard to study the ‘generic’ behaviour of dynamical systems. Symmetry is not generic. The answer is to work within the world of symmetric systems and to examine a suitably restricted Idea of genericity.

The pioneering work of Sattinger [1979, 1983], Vanderbauwhede [1982] and others opened up the possibility of a systematic theory, and during the past decade understanding of the bifurcation of dynamical systems with symmetry has developed into a recognizable subject with Its own distinctive identity: Equivariant Bifurcation Theory. It is not Just a tactical development: it embodies a general strategy for tackling the bifurcations of symmetric nonlinear systems. The technical machinery is extensive - Lie theory, representation theory, invariant theory, dynamical systems, and topology, for example - and the literature has grown to the point where the details can obscure the broader principles of the subject.

Symmetries are often exploited without being made explicit. For example, symmetry often forces multiple eigenvalues; but In any computation of those eigenvalues their multiplicity will emerge in due course.