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，这个项目的重点是几何和统计理论的动力系统与占主导地位的双曲行为。 混沌系统的混合速率和相关统计特性（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。其他三个领域的调查建议：稳定性下的小随机扰动;奇怪的吸引力的某种螺线管型和可能的应用;和使用动力系统技术的两个问题，从数学物理，一维薛定谔算子和快速发电机问题的磁流体力学。 物理现象，如天气和大气扰动，随着时间的推移而演变。动力系统是数学的一个分支，它研究这些和其他过程的时间演化背后的理论。该领域的研究人员分析物理科学家提出的模型，形成解释观察到的现象的理论基础，并最终预测未来的发生。非线性科学在本世纪的进步之一是认识到相对简单的物理定律可以而且确实有极其复杂的表现形式。那些看起来不符合逻辑模式的行为通常被归为“混乱”一类。“混沌系统是这项研究的对象。在这个项目中，将寻求各种途径，将动力系统的外在混沌表现与其内在秩序联系起来，同时考虑到由于偶然性而发生的随机波动。

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.