Dynamical Systems

提出了关于动力系统数学理论的四个项目。每个项目都包含一组具有共同主题的问题。第一类是具有强耗散和单方向不稳定的奇异吸引子。在以前的拨款下发展的一般理论扩展到任意相位尺寸，以及应用到具体问题，如非线性振荡器。第二个项目涉及以双曲行为为主的动力系统的统计行为。提出的研究重点是在离散和连续时间导致各种相关衰减率的机制。第三个专题是晶格动力系统。对耦合在一起的大量动力系统的总体行为进行了系统的分析。最后的项目提出了求解薛定谔算子、运动学快速发电机和Navier-Stokes方程三个不相关问题的动力系统方法。作为数学的一个分支，动力系统关注的是受某些基本规律支配的过程的时间演变。这门学科的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。在本建议中，提出了一些适合数学分析的模型的研究，并提出了在物理和生物科学中具有潜在应用的模型。一段时间以来，人们已经知道，相对简单的定律可以导致复杂的动力学。本建议的第一部分着重于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。（奇异吸引子是高度复杂的物体，它捕捉了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，但迄今为止还没有得到严格的分析。）提出的其他项目包括大型动力系统的总体性质与其单个组成部分的性质之间的关系，以及主要研究者认为可以用动力系统方法解决的一些物理和流体动力学问题。提出了关于动力系统数学理论的四个项目。每个项目都包含一组具有共同主题的问题。第一类是具有强耗散和单方向不稳定的奇异吸引子。在以前的拨款下发展的一般理论扩展到任意相位尺寸，以及应用到具体问题，如非线性振荡器。第二个项目涉及以双曲行为为主的动力系统的统计行为。提出的研究重点是在离散和连续时间导致各种相关衰减率的机制。第三个专题是晶格动力系统。对耦合在一起的大量动力系统的总体行为进行了系统的分析。最后的项目提出了求解薛定谔算子、运动学快速发电机和Navier-Stokes方程三个不相关问题的动力系统方法。作为数学的一个分支，动力系统关注的是受某些基本规律支配的过程的时间演变。这门学科的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。在本建议中，提出了一些适合数学分析的模型的研究，并提出了在物理和生物科学中具有潜在应用的模型。一段时间以来，人们已经知道，相对简单的定律可以导致复杂的动力学。本建议的第一部分着重于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。（奇异吸引子是高度复杂的物体，它捕捉了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，但迄今为止还没有得到严格的分析。）提出的其他项目包括大型动力系统的总体性质与其单个组成部分的性质之间的关系，以及主要研究者认为可以用动力系统方法解决的一些物理和流体动力学问题。

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.