Topics in Finite and Infinite Dimensional Random Dynamical Systems

有限和无限维随机动力系统主题

• 批准号: 0401708
• 负责人: Peter Bates
• 金额: --
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-10-01 至 2010-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0401708&HistoricalAwards=false
• 关键词: Topics; Finite; Infinite; Dimensional; Random

Random dynamical systems arise in the modeling of many phenomena in physics, biology, economics, etc. when uncertainties or random influences, called noises, are taken intoaccount. These random effects are not only introduced to compensate for the defects in some deterministic models, but also are often rather intrinsic phenomena. A fundamental problem is to study the dynamical behavior of orbits of random dynamical systems. This project seeks to establish some of the basic geometric framework for infinite dimensional random dynamical systems. By proving existence and robustness of random invariant manifolds and foliations, results on structural stability of deterministic systems under random perturbations will be sought. Floquet theory containing the multiplicative ergodic theorem will be developed, statistical properties of random attractors, and pattern formation in random media will be explored. The theory of smooth conjugacy for finite dimensional random dynamical systems will also be developed.The broader impacts of this project include the training of graduate students and junior faculty members in this emerging area of dynamics. Applications to the many fields mentioned above will create opportunities to interact on many levels with students and faculty members from other disciplines and will lead to advances in technology. One such example are the possible applications to the study of nano-devices, for instance, random thermal fluctuations or quantum effects must be accounted for in order to accurately predict performance. That field is one of many that is completely open to analysis using tools of the type to be developed here.

随机动力学系统出现在物理，生物学，经济学等许多现象的建模时。这些随机效应不仅是为了补偿某些确定性模型中的缺陷，而且通常是固有的现象。一个基本问题是研究随机动力系统的轨道的动力学行为。该项目旨在为无限尺寸随机动力系统建立一些基本的几何框架。通过证明随机不变的流形和叶子的存在和鲁棒性，将寻求确定性系统在随机扰动下的结构稳定性的结果。将开发包含乘级千古定理的Floquet理论，随机吸引子的统计特性，并将探索随机介质中的模式形成。还将开发出有限维度随机动力学系统的平稳结合理论。该项目的更广泛影响包括在这个新兴动力学领域对研究生和初级教职员工进行培训。上面提到的许多领域的申请将创造机会与其他学科的学生和教职员工在许多层面上进行互动，并将导致技术进步。一个例子是对纳米驱动器的研究的可能应用，例如，必须考虑随机的热波动或量子效应，以便准确预测性能。该领域是使用此处要开发的类型的工具完全开放分析的众多领域之一。