Mathematical Sciences: Dynamical Systems and Smooth Ergodic Theory

主要研究人员将继续研究动力系统的混沌行为。特别是，他们将研究Henon映射和标准映射，它们是二维非一致椭圆型双曲组的典型例子。本文试图通过证明Lyapunov指数的正性来证明混沌行为的存在性。他们还将研究复杂动力学中的Julia集，特别是非均匀双曲情况下的Julia集。该奖项将支持动力系统一般领域的研究。一个在当地非常简单和容易理解的过程可能会变得极其复杂、非线性和难以在全球范围内分析。动力系统是研究这种局部到全球关系的学科。许多物理系统都可以使用这一数学领域进行建模，包括流体流动和湍流、复杂的生物系统、机械系统和化学反应。

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.