Dynamical Systems and Smooth Ergodic Theory

对混沌动力系统的数学理论进行了研究。更具体地说，该项目侧重于以双曲行为为主的动力系统的几何和统计理论。对混沌系统的混合速率和相关统计性质（如中心极限定理）的系统研究继续了早期nsf支持的项目的工作。提出了另外三个研究领域：小随机扰动下的稳定性；某螺线管型奇异吸引子及其可能的应用以及在数学物理中的两个问题——一维薛定谔算子和磁流体力学中的快速发电机问题——中使用动力系统技术。物理现象，如天气和大气扰动，随着时间而演变。动力系统是数学的一个分支，研究这些过程和其他过程的时间演变背后的理论。该领域的研究人员分析物理科学家提出的模型，为解释观测到的现象形成理论基础，最终目的是预测未来的事件。本世纪非线性科学的进步之一是认识到相对简单的物理定律可以而且确实具有极其复杂的表现。看起来不遵循逻辑模式的行为通常被归类为“混乱”。混沌系统是本研究的对象。在这个项目中，将采用各种途径将动力系统的外部混沌表现与其潜在秩序联系起来，同时考虑到由于偶然而发生的随机波动。

Research on the mathematical theory of chaotic dynamical systems is proposed. More specifically, this project focuses on the geometric and statistical theories of dynamical systems with predominantly hyperbolic behavior. A systematic study of the rates of mixing and related statistical properties (such as the Central Limit Theorem) for chaotic systems continues work of an earlier NSF-supported project. Three other areas of investigation are proposed: stability under small random perturbations; strange attractors of a certain solenoidal type and possible applications; and the use of dynamical systems techniques in two problems from mathematical physics, the one-dimensional Schroedinger operator and the fast dynamo problem in magneto-hydrodynamics. Physical phenomena, such as weather and atmospheric disturbances, evolve with time. Dynamical Systems is the branch of mathematics that studies the theories that lie behind the time evolutions of these and other processes. Researchers in this field analyze models proposed by physical scientists, form theoretical bases for explaining observed phenomena, and aim ultimately to predict future occurrences. One of the advances in the nonlinear sciences in this century is the recognition that relatively simple physical laws can and do have extremely complicated manifestations. Behaviors that appear to follow no logical patterns are often grouped under the term "chaos." Chaotic systems are the objects of this research. In this project, various avenues will be pursued to relate the outward chaotic manifestations of a dynamical system to its underlying order, taking into account random fluctuations that occur as a result of chance.