Methods in Equivariant Bifurcations and Dynamical Systems P. Chossat and R. Lauterbach, Advanced Series in Nonlinear Dynamics, Vol. 15. World Scientific, 2000. xv + 404 pages. Price £46.00.
Reviewer: André Vanderbauwhede, Department of Pure Mathematics and Computer Algebra, University of Gent, Belgium.

Equivariant bifurcation theory (i.e. the study of bifurcations in systems which are invariant under the action of a symmetry group) started as a separate subdomain within the theory of dynamical systems in the late seventies, and got into full bloom in the eighties. This was the period during which the basic Equivariant Branching Lemmas for both steady-state and Hopf bifurcation were developed and applied to a large selection of problems. By combining an equivariant version of the Liapunov-Schmidt reduction with singularity theory classifications were obtained of possible bifurcations under most of the symmetries commonly appearing in applications. A particularly rich and successful field of application was in hydrodynamics where for example the Bénard and Taylor-Couette problems got massive attention. A standard reference for these results is the book [1] of Golubitsky, Stewart and Schaeffer which appeared in 1988.

The aim of the book by Chossat and Lauterbach under review is to be a kind of follow-up of [1], reporting on and explaining some of the further developments which took place in equivariant bifurcation theory since the publication of [1]. These include (among others) the invariant sphere theorem, a number of results on bifurcations with maximal and non-maximal isotropy, bifurcations of and from relative equilibria and relative periodic orbits, pattern formation in problems with non-compact symmetry, the robust appearance of heteroclinic cycles, and some unexpected dynamics resulting from forced symmetry breaking (imperfections which destroy the perfect model symmetry). At the same time the book intends to introduce the main tools of equivariant bifurcation theory (such as Liapunov-Schmidt reduction, normal forms, center manifolds, group representations, invariant theory and orbit spaces reductions) to a general audience of applied mathematicians. With the same audience in mind the emphasis is on applications to infinite-dimensional systems described by partial differential equations of parabolic type.

The authors have to a large extend succeeded in their goal; although not all recent developments are covered (due to the extent of the subject this is not feasible) the selection which was made gives a fairly complete overview of the recent accomplishments in this still very active field. The theoretical background which is provided should be a welcome help for many applied mathematicians who want to get a better insight in equivariant bifurcation theory or who want to apply the results and methods of this theory to their own problems.

[1] M. Golubitsky, I. Stewart and D. Schaeffer. Singularities and Groups in Bifurcation Theory, Vol. 2. Springer Verlag, 1988.