Mathematical Sciences: Dynamical Systems and Smooth Ergodic Theory

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The principal investigators will continue research in the chaotic behavior of dynamical systems. In particular they will study the Henon and standard maps which are the prototypical examples of nonuniformly elliptic hyperbolic systems in dimension two. The will attempt to prove the existence of chaotic behavior by proving the positivity of Lyapunov exponents. They will also study the Julia sets in complex dynamics, particularly in the nonuniform hyperbolic case. This award will support research in the general area of dynamical systems. A process which is very simple and easy to understand locally can become extremely complicated, nonlinear, and difficult to analyze globally. Dynamical systems is the study of this local to global relationship. Many physical systems can best be modeled using this area of mathematics including fluid flow and turbulence, complex biological systems, mechanical systems, and chemical reactions.