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BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY

Attempts to apply mathematics to questions in science or technology often suffer from the problem that an analytical solution of such practical questions is usually either trivial (at least for anyone sufficiently well trained) or impossible (even for the most expert practitioners of mathematics of any given age). In applications of nonlinear dynamical systems this manifests itself as the unfortunate reality that typical dynamical systems, whether expressed as maps or differential equations, usually cannot be solved or even usefully approximated in any analytical form whatsoever. In the absence of explicit solutions, one needs to apply a range of qualitative and asymptotic techniques. More recently, the growth in speed and availability of computers for approximation and simulation has allowed researchers to gain an understanding of the typical behaviour of a wide range of dynamical systems that would have previously been inaccessible to study by either explicit or qualitative methods. Since mathematical models from applications often include symmetries, a natural question is how symmetries of the model manifest themselves in the asymptotic dynamics and bifurcations thereof. This naturally leads to the study of equivariant dynamical systems: flows and maps that commute with an action of a symmetry group. Several areas of applied mathematics, theoretical physics and chemistry have successfully used this approach, going back to Wigner’s work on atomic spectra. In particular, stabilities of equilibria are generically determined by linearizations; these linearizations must commute with the symmetry group, and one can readily split the spectral properties of the linearization into parts according to a decomposition of the action of the group into copies of irreducible representations. However, the usefulness of the group action goes beyond that; it constrains the normal forms that describe the bifurcation of solutions at a center manifold. Some terms are forced to be zero, while others are generically nonzero. It also forces the existence of invariant sets purely by virtue of their symmetries. The main mathematical tools used in this approach consist of Lie group representation theory, equivariant singularity theory for smooth germs of vector fields and adaptation of other methods from dissipative dynamics and bifurcation theory [4], [1] to an equivariant setting. The book presently under review can be seen as a continuation of the research programme of the authors that was gathered together previously into two volumes [2], [3]. However, the present book is not really intended as an updated exposition or extension of the theory in those volumes. Instead it brings in developments of the theory as necessary and as inspired by a number of applications that are discussed throughout the book. The applications could be classified loosely into problems of pattern formation and problems of nonlinear oscillation. For example, pattern formation problems