Mathematical Sciences: Bifurcation and Symmetry

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The investigators continue studies of the effect that symmetry has on the solutions of differential equations. Three particular topics are the stability of heteroclinic cycles (which is related to the existence of intermittency and bursting phenomena), the symmetry of chaotic attractors (which may be related to some of the distinctive patterns observed in turbulent fluid flow), and the coupling of gauge and spatial symmetries (such as occurs in the Ginzburg-Landau model for superconductivity). They will also attempt to relate their results to specific experiments. They believe that both the bursting phenomena associated with heteroclinic cycles and the symmetry-increasing bifurcations of chaotic attractors have been observed in the Couette-Taylor experiment; part of their work will be to make these connections more precise. Many models of physical phenomena including topics as diverse as the transition from conduction to convection in fluid dynamics and the ways in which animals walk contain symmetries in their formulation. These symmetries are known to influence the kinds of solutions that are found in such models. Most previous work has focused on the kinds of equilibria and periodic solutions that appear. This project undertakes to broaden the class of solutions that are studied to include such topics as chaos and intermittency in symmetric systems. Recent research and experiments indicate that this is both reasonable and tractable.