Dynamical Systems: Theory and Applications

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Research topics within the theory of dynamical systems and on its applications are proposed. Four projects are described. The first proposes to leverage techniques developed for low dimensional systems to analyze (infinite dimensional) systems defined by evolutionary partial differential equations. Expected results include methods for detecting strange attractors in physical and mechanical systems. The second project is about networks of dynamical systems. It seeks to systematically relate the aggregate behaviors of such systems to those of their individual components and to the coupling. The third topic lies in the interface between dynamical systems and nonequilibrium statistical physics. One of the objectives here is to shed light on the notion of local equilibrium for systems with deterministic microscopic dynamics. The fourth and final topic is about large deviations, an important statistical property for dynamical systems. A scheme expected to resolve the issue for a large class of nonuniformly hyperbolic systems is proposed.This proposal addresses several topics at the frontier of research in dynamical systems, a branch of modern mathematics concerned with time evolutions of natural processes. Its main focus is the analysis of systems with high complexity, due either to multiple degrees of freedom or to chaos within the system. It is systems of this kind that are most often encountered in applications. The proposed methods of investigation include geometric analysis, probabilistic techniques and numerical simulations. A larger aim of the proposed work is to integrate dynamical systems ideas into other core areas of mathematics such as partial differential equations, and to build connections with other scientific disciplines such as statistical physics and the biological sciences. The resulting cross-fertilization is expected to be beneficial to all. In terms of educational value, the proposed research will provide ample training for emerging mathematicians, as it will involve directly the postdoctoral associates and Ph.D. students of the Principle Investigator.