Mathematical Sciences: Non-Linear Dynamics of Cylindrically Symmetric Systems

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The presence of a nontrivial symmetry group can profoundly affect the dynamics of a physical system. In particular, it has been recently appreciated that the instabilities or bifurcations observed in such systems typically exhibit highly nongeneric features when symmetry breaking occurs. This project is to study the nonlinear mathematical models with detailed investigation of specific physical systems. Particular emphasis is placed on the symmetry groups 0(2) and S0(2) because of their relevance to applications, and on the dynamics arising from multiple instabilities of mode interactions. The technique to be applied are those of equivariant bifurcation theory, center manifold reduction, and normal form analysis. The goal is to obtain an understanding of symmetry breaking transitions between pre- chaotic states as well as the impact of symmetry on the transition to chaos. For specific fluid systems with circular symmetry one expects to obtain results which may be tested in numerical and laboratory experiments.