Mathematical Sciences: Differential Dynamical Systems

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The three principal investigators will apply various geometric, symbolic dynamic, and ergodic methods to solve problems concerning finite-dimensional flows and mappings. One investigator will concentrate on fixed-point theorems for surface homeomorphisms, the conjugacy problem for subshifts of finite type, and knots in three-dimensional flows. A second investigator will study the creation of strange attractors through bifurcation at the onset of chaos. The third investigator will use ergodic theory to study the geodesic flow for manifolds with nonpositive curvature. "Dynamical systems" is the mathematics of movement. Two problems of intense interest in this field are the topological nature of closed orbits and bifurcations at the onset of chaos. One investigator will continue his analysis of the knot types of closed orbits in three dimensions. He seeks to use the Jones polynomial which catergorizes knot types to better understand orbits of such systems as the Lorenz and Henon attractors. A second investigator will improve upon our understanding of Smale "horseshoes" created at the onset of chaos. For a system of differential equations describing for example water flow, the existence of these bent topological objects would indicate turbulent behavior.