Five research projects in dynamical systems theory are undertaken. The first three are focused on advances within the subject.The goal of Project 1 is to extend nonuniform hyperbolic theory (or the mathematical theory of chaos), from finite to infinite dimensions, enlarging its range of applicability to include certain classes of evolutionary partial differential equations. Project 2 is about chaotic behavior in randomly perturbed dynamical systems, and Project 3 seeks to extend certain theories of autonomous dynamical systems to time-dependent systems with slowly drifting coefficients. The last two projects export dynamical systems ideas to two disciplines outside of mathematics: nonequilibrium statistical mechanics and theoretical neuroscience. Project 4 addresses issues related to the analog of local temperature in dynamical models that are driven out of equilibrium. Project 5 pertains to neuronal networks, seeking dynamical explanations for various phenomena observed by experimentalists.
The Principal Investigator works with systems that are large, random, or out of equilibrium, three characteristics shared by many real-world systems. Scientifically, the proposed research will (1) lead to significant advances in the theory of dynamical systems, (2) build connections between dynamical systems and other branches of mathematics, such as partial differential equations and probability, and (3) broaden the use mathematics, the applicability of dynamical systems ideas in particular, in engineering, theoretical physics and neuroscience. The cross-fertilization in (2) and (3) will be beneficial to all the areas involved. On the educational front, providing scientific training and guidance for young researchers is an important part of the proposed activity. The Principal Investigator expects to continue this activity using projects in the current proposal, and to continue her public speaking and writing of expository articles as part of the process for dissemination of ideas.
