Dynamical Systems: Theory and Applications

提出了动力系统理论及其应用方面的研究课题。描述了四个项目。第一个建议利用为低维系统开发的技术来分析由进化偏微分方程式定义的(无限维)系统。预期的结果包括检测物理和机械系统中奇怪吸引子的方法。第二个项目是关于动力系统的网络。它试图系统地将此类系统的聚合行为与其单个组件的聚合行为以及耦合联系起来。第三个主题是动力系统和非平衡统计物理之间的界面。这里的目标之一是阐明具有确定性微观动力学的系统的局部均衡的概念。第四个也是最后一个主题是关于大偏差，这是动力系统的一个重要统计性质。提出了一种解决一大类非一致双曲系统问题的方案，该方案解决了动力系统研究前沿的几个问题，动力系统是现代数学的一个分支，涉及自然过程的时间演化。它的主要焦点是对具有高度复杂性的系统的分析，这些系统要么是由于多个自由度，要么是由于系统中的混沌。在应用中最常遇到的就是这类系统。提出的研究方法包括几何分析、概率技术和数值模拟。拟议工作的一个更大目标是将动力系统的概念整合到偏微分方程式等数学的其他核心领域，并与统计物理和生物科学等其他科学学科建立联系。由此产生的交叉受精预计对所有人都有利。在教育价值方面，拟议的研究将为初出茅庐的数学家提供充分的培训，因为它将直接涉及原理研究员的博士后助理和博士后学生。

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.