Mathematical Sciences: Structure of Attractors

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Abstract Barge The main goal of this project is to extend, in parameter space, a recently discovered form of self-similarity displayed by a one-dimensional attractor at the instant of homoclinic bifurcation. The PI has proved that when a one- dimensional unstable manifold has a nondegenerate homoclinic tangency (and the eigenvalues satisfy a nonresonance condition) then in every neighborhood of every point of the closure of the unstable manifold, there is (in the closure of the unstable manifold) a homeomorphic copy of every member of a certain uncountable collection of continua. The goal is to prove that, in typical one-parameter families undergoing homoclinic bifurcation, this extreme local wildness occurs for a set of parameters of positive measure. The ubiquity and diversity of the subcontinua of the closure of the unstable manifold is a result of the recurrence of arbitrary patterns of small folds in the unstable manifold. The main approach to extending the occurence of these patterns in parameter space will be to selectively excise small open sets of parameters for which the "critical orbit" does not recur appropriately, leaving a Cantor set of positive measure. This approach has been used successfully by a number of researchers to establish the existence of transitive attractors with nice measures. The goal here is more topological, less analytical, and the estimates should be less exacting. This project also includes ongoing work on three other topics: algebraic invariants associated with one- and two-dimensional Markov spaces; expansive homeomorphisms on plane continua; and rotational dynamics on invariant plane continua. In mathematical models of physical processes, there frequently occur "attractors" that contain the discription of the long term behavior of the model. The structure (topology) of the attractor reflects qualitative properties of this behavior. In case the model is chaotic, the structure of the attractor is extremely complicated. Chaotic attractors are of two general types: hyperbolic and non-hyperbolic. The hyperbolic attractors are relatively well understood. The developement of techniques for understanding non-hyperbolic systems is perhaps the biggest challenge in dynamics today. Success in this project would provide the first coherent glimpse into the topological structure of non-hyperbolic attractors.