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.

在物理、生物、经济等领域的许多现象的建模中，当考虑到不确定性或称为噪声的随机影响时，随机动力系统就会出现。这些随机效应的引入不仅是为了弥补某些确定性模型的缺陷，而且往往是一种相当固有的现象。一个基本问题是研究随机动力系统轨道的动力学行为。本项目旨在建立无限维随机动力系统的一些基本几何框架。通过证明随机不变流形和随机不变流形的存在性和稳健性，可以得到确定性系统在随机扰动下的结构稳定性的结果。将发展包含乘法遍历定理的Floquet理论，探索随机吸引子的统计性质，以及随机介质中的图案形成。有限维随机动力系统的光滑共轭理论也将得到发展。这个项目的更广泛的影响包括在这个新兴的动力学领域对研究生和初级教师的培训。在上述许多领域的应用将创造与其他学科的学生和教职员工在多个层面上互动的机会，并将导致技术的进步。一个这样的例子是在纳米器件研究中的可能应用，例如，为了准确地预测性能，必须考虑随机热波动或量子效应。该领域是许多完全开放的分析领域之一，可以使用这里将要开发的这种类型的工具进行分析。