This book is an expanded version of a Master Class on the symmetric bifurcation theory of differential equations given by the author at the University of Twente in 1995. The notes cover a wide range of recent results in the subject, and focus on the dynamics that can appear in the generic bifurcation theory of symmetric differential equations. Many of the results and examples in the book are new and have not been previously published. The first four chapters contain an accessible presentation of the fundamental work by Field and Richardson on symmetry breaking and the Maximal Isotropy Subgroup Conjecture. The remainder of the book focuses on recent research of the author and includes chapters on the invariant sphere theorem, coupled cell systems, heteroclinic cycles , equivariant transversality, and an Appendix (with Xiaolin Peng) giving a new low dimensional counterexample to the converse of the Maximal Isotropy Subgroup Conjecture. The chapter on coupled cell systems includes a weath of new examples of 'cycling chaos'. The chapter on equivariant transversality is introductory and centres on an extended discussion of an explicit system of four coupled nonlinear oscillators. The style and format of the original lectures has largely been maintained and the notes include over seventy exercises *with hints for solutions and suggestions kfor further reading). In general terms, the notes are directed at mathematicians and aplied scientists working in the field of bifurcation theory who wish to learn about some of the latest developments and techniques in equivariant bifurcation theory. The notes are relatively self-contained and are structured so that they can form the basis for a graduate level course in equivariant bifurcation theory.
Introduction & preliminaries Hyperoctahedral groups A zoo of bifurcations Stability and determinacy The invariant sphere theorem Hetroclinic cycles in equivariant bifurcations Symmetrically coupled cell systems An example of Z2-transversality in a system of four coupled oscillators Geometric methods Converse to the MISC (appendix) Hints and Solutions to selected exercises Bibliography Index