Dynamical Systems

提出了动力系统数学理论的四个课题。每个项目都包含一系列具有共同主题的问题。第一类是具有强耗散和单一不稳定方向的奇异吸引子。文中还提出了在以前的资助下发展的一般理论在任意相维上的推广，以及在诸如非线性振子等具体问题上的应用。第二个项目涉及以双曲行为为主的动力系统的统计行为。所提出的研究重点是在离散和连续时间内导致不同相关衰减率的机制。第三个项目的主题是格子动力系统。对大量耦合在一起的动力系统的聚集行为进行了系统的分析。最后一个项目提出了解决薛定谔算子、运动学快速发电机和Navier-Stokes方程三个互不相关问题的动力系统方法。作为数学的一个分支，动力系统关注受某些潜在规律支配的过程的时间演化。本课题的一个主要目标是发展统一的数学理论来解释观察到的现象和预测未来的发生。在这项提案中，提出了对一些适合数学分析并可能应用于物理和生物科学的模型的调查。相对简单的定律可以导致复杂的动力学，这一点已经有一段时间了。该提案的第一部分重点关注具有混沌行为的系统。提出了两个主题：奇异吸引子的分析和混合的统计理论。(奇怪的吸引子是高度复杂的物体，可以捕捉耗散动力系统的长期行为；它们在自然界和模拟中经常被观察到，但到目前为止一直拒绝进行严格的分析。)其他项目包括大动力系统的聚集性质与其单个组件的聚集性质之间的关系，以及主要研究者认为可以用动力系统方法解决的几个物理和流体力学问题。每个项目都包含一系列具有共同主题的问题。第一类是具有强耗散和单一不稳定方向的奇异吸引子。文中还提出了在以前的资助下发展的一般理论在任意相维上的推广，以及在诸如非线性振子等具体问题上的应用。第二个项目涉及以双曲行为为主的动力系统的统计行为。所提出的研究重点是在离散和连续时间内导致不同相关衰减率的机制。第三个项目的主题是格子动力系统。对大量耦合在一起的动力系统的聚集行为进行了系统的分析。最后一个项目提出了解决薛定谔算子、运动学快速发电机和Navier-Stokes方程三个互不相关问题的动力系统方法。作为数学的一个分支，动力系统关注受某些潜在规律支配的过程的时间演化。本课题的一个主要目标是发展统一的数学理论来解释观察到的现象和预测未来的发生。在这项提案中，提出了对一些适合数学分析并可能应用于物理和生物科学的模型的调查。相对简单的定律可以导致复杂的动力学，这一点已经有一段时间了。该提案的第一部分重点关注具有混沌行为的系统。提出了两个主题：奇异吸引子的分析和混合的统计理论。(奇怪的吸引子是高度复杂的物体，可以捕捉耗散动力系统的长期行为；它们在自然界和模拟中经常被观察到，但到目前为止一直拒绝进行严格的分析。)提出的其他项目包括大动力系统的聚集性质与其单个组件的聚集性质之间的关系，以及主要研究者认为可以用动力系统方法解决的一些来自物理和流体力学的问题。

Four projects on the mathematical theory of dynamical systems areproposed. Each project contains a cluster of problems with a common theme. The first pertains to strange attractors with strong dissipation and a single direction of instability. Extensions of a general theory developed under a previous grant to arbitrary phase-dimensions are proposed, as are applications to concrete problems such as nonlinear oscillators. The second project concerns the statistical behavior of dynamical systems with predominantly hyperbolic behavior. The focus ofthe proposed research is on mechanisms leading to various rates of correlation decay in both discrete and continuous times. The topic of the third project is lattice dynamical systems. A systematic analysis of the aggregate behavior of large numbers of dynamical systems coupled together is proposed. The final project proposes dynamical systems methods of solution for three unrelated problems on the Schrodinger operator, kinematic fast dynamo and Navier-Stokes equations.As a branch of mathematics, dynamical systems is concerned with the time evolutions of processes governed by certain underlying laws.A primary goal of the subject is to develop unifying mathematicaltheories to explain observed phenomena and predict future occurrences. In this proposal, the investigation of a number of models amenable to mathematical analysis and with potential applications to the physical and biological sciences is proposed. It has been known for some time that relatively simple laws can lead to complicated dynamics. The first part of this proposal focuses on systems with chaotic behavior. Two topics are proposed: an analysis of strange attractors and a statistical theory of mixing. (Strange attractors are highly complex objects which capture the long term behaviors of dissipative dynamical systems; they have been observed frequently in nature and in simulations but have thus far resisted rigorous analysis.) Other projects proposed include the relations between aggregate properties of large dynamical systems and those of their individual components, and a few problems from physics and hydrodynamics which the principal investigator believes can be solved by the methods of dynamical systems.Four projects on the mathematical theory of dynamical systems areproposed. Each project contains a cluster of problems with a common theme. The first pertains to strange attractors with strong dissipation and a single direction of instability. Extensions of a general theory developed under a previous grant to arbitrary phase-dimensions are proposed, as are applications to concrete problems such as nonlinear oscillators. The second project concerns the statistical behavior of dynamical systems with predominantly hyperbolic behavior. The focus ofthe proposed research is on mechanisms leading to various rates of correlation decay in both discrete and continuous times. The topic of the third project is lattice dynamical systems. A systematic analysis of the aggregate behavior of large numbers of dynamical systems coupled together is proposed. The final project proposes dynamical systems methods of solution for three unrelated problems on the Schrodinger operator, kinematic fast dynamo and Navier-Stokes equations.As a branch of mathematics, dynamical systems is concerned with the time evolutions of processes governed by certain underlying laws.A primary goal of the subject is to develop unifying mathematicaltheories to explain observed phenomena and predict future occurrences. In this proposal, the investigation of a number of models amenable to mathematical analysis and with potential applications to the physical and biological sciences is proposed. It has been known for some time that relatively simple laws can lead to complicated dynamics. The first part of this proposal focuses on systems with chaotic behavior. Two topics are proposed: an analysis of strange attractors and a statistical theory of mixing. (Strange attractors are highly complex objects which capture the long term behaviors of dissipative dynamical systems; they have been observed frequently in nature and in simulations but have thus far resisted rigorous analysis.) Other projects proposed include the relations between aggregate properties of large dynamical systems and those of their individual components, and a few problems from physics and hydrodynamics which the principal investigator believes can be solved by the methods of dynamical systems.