Frontiers in Dynamical Systems

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Dynamical systems is a branch of modern mathematics concerned with time evolutions of natural and iterative processes. It is both an area of pure mathematics and a subject that lies at the crossroads of multiple scientific disciplines. As an area of pure mathematics, it seeks to develop rigorous theories and techniques that are applicable to general processes, addressing foundational issues of stability and chaos. As a partner with the sciences, it offers qualitative theories and viewpoints, and can be a powerful tool when used in conjunction with numerical computations. While both aspects of dynamical systems are worthy in their own right, there often is a gap that lies between theory and applications. The proposed research has both purely theoretical and application oriented components, one of its goals being the cross-fertilization of ideas. This research will: (1) lead to significant broadening of the scope of dynamical systems theory, (2) strengthen connections between dynamical systems and other areas of mathematics, such as probability and partial differential equations, and (3) build connections between mathematics and statistical physics and theoretical neuroscience. In this proposal there are four specific research projects are presented. One proposes to extend hyperbolic theory (or the theory of chaotic dynamical systems) from finite to infinite dimensions, so that an enlarged theory will be applicable not just to ordinary differential equations but also to large classes of partial differential equations. A second project proposes to extend current techniques for analyzing chaotic dynamics, which often exhibit Gaussian type fluctuations, to systems that produce heavy-tailed statistics. Though well recognized as a hallmark of complex processes, few analytical studies of such dynamical processes have been carried out. A third project seeks to provide rigorous justification for certain fundamental phenomenology in nonequilibrium statistical mechanics, such as the idea of local thermal equilibrium in steady states dynamics. The fourth and final project proposes numerical and analytical studies of phenomena identified in previous computational models of neuroscience, specifically in models of visual cortex. Providing scientific training for young researchers is a very important part of the proposed activity. The principal investigator has a strong and well documented history of mentoring students and junior colleagues, and she fully expects to continue to do so using projects in the current grant.