Dynamics, Patterns and Symmetry

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9704980 Golubitsky Bifurcations to spiral waves and to cellular patterns in partial differential equations posed on a circular disk will be investigated. In unbounded domains, the bifurcation and meandering of spirals and scroll waves and the Ginzburg-Landau theory of spatially extended systems will be considered -- - with application to spatially aperiodic solutions in spatially extended systems. In addition, part of the effort will be devoted to studying the dynamics present in ordinary differential equations with symmetry. Stable ergodicity of chaotic attractors in problems with continuous symmetry, and the existence, stability and bifurcations of robust heteroclinic cycles will be studied. Some of these ideas are relevant to intermittent magnetic dynamos in rotating convection. Patterns appear in physical, chemical, and biological systems and are characteristically striking and reproducible. Consequently, scientists and mathematicians have developed theories to explain the origins of these patterns. There are several approaches to the study of pattern; this one is based on symmetry and bifurcation. It is proposed to investigate the theory and application of symmetric dynamical systems and its relation to pattern formation. There are fundamental differences in the analyses depending on whether the patterns being studied fit neatly into a bounded domain or whether boundaries are unimportant and the equations are posed on infinite domains. Aspects of both cases will be studied.