Dynamical Systems

动力系统

• 批准号: 0100538
• 负责人: Lai-Sang Young
• 金额: $ 50万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-06-01 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0100538&HistoricalAwards=false
• 关键词: 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.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.

提出了动力系统数学理论的四个方案。每个项目都包含一组具有共同主题的问题。第一种是具有强耗散和单方向不稳定性的奇异吸引子。根据以前的补助金开发的一般理论的扩展到任意相尺寸的建议，是具体的问题，如非线性振荡器的应用。第二个项目涉及的统计行为的动力系统与占主导地位的双曲行为。拟议研究的重点是导致离散和连续时间内不同相关性衰减率的机制。第三个项目的主题是晶格动力系统。本文提出了一种系统地分析大量耦合在一起的动力系统的聚集行为的方法。期末计画提出动力系统方法来解决薛定谔算子、运动快速发电机和Navier-Stokes方程三个不相关的问题。作为数学的一个分支，动力系统是研究受某些基本定律支配的过程的时间演化。本学科的主要目标是发展统一的理论来解释观测到的现象和预测未来的事件。在这一建议中，提出了一些模型的调查服从数学分析，并与潜在的应用物理和生物科学。一段时间以来，人们已经知道相对简单的定律可能会导致复杂的动力学。该提案的第一部分侧重于具有混沌行为的系统。提出了两个主题：奇怪吸引子的分析和混合的统计理论。（奇异吸引子是捕捉耗散动力学系统长期行为的高度复杂的对象;它们在自然界和模拟中经常被观察到，但迄今为止一直无法进行严格的分析。提出的其他项目包括大型动力系统的集合性质与其单个组成部分的集合性质之间的关系，以及主要研究者认为可以用动力系统方法解决的物理学和流体力学中的一些问题。每个项目都包含一组具有共同主题的问题。第一种是具有强耗散和单方向不稳定性的奇异吸引子。根据以前的补助金开发的一般理论的扩展到任意相尺寸的建议，是具体的问题，如非线性振荡器的应用。第二个项目涉及的统计行为的动力系统与占主导地位的双曲行为。建议的研究重点是在离散和连续时间的相关性衰减的各种速率的机制。第三个项目的主题是晶格动力系统。本文提出了一种系统地分析大量耦合在一起的动力系统的聚集行为的方法。期末计画提出动力系统方法来解决薛定谔算子、运动快速发电机和Navier-Stokes方程三个不相关的问题。作为数学的一个分支，动力系统是研究受某些基本定律支配的过程的时间演化。本学科的主要目标是发展统一的理论来解释观测到的现象和预测未来的事件。在这一建议中，提出了一些模型的调查服从数学分析，并与潜在的应用物理和生物科学。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。该提案的第一部分侧重于具有混沌行为的系统。提出了两个主题：奇怪吸引子的分析和混合的统计理论。（奇异吸引子是捕捉耗散动力学系统长期行为的高度复杂的对象;它们在自然界和模拟中经常被观察到，但迄今为止一直无法进行严格的分析。提出的其他项目包括大型动力系统的总体性质与其各个组成部分之间的关系，以及主要研究者认为可以通过动力系统方法解决的物理学和流体力学的一些问题。