Study of global behavior in Dynamical Systems and PDE's

动力系统和偏微分方程中全局行为的研究

• 批准号: 0354567
• 负责人: Rafael de la Llave
• 金额: --
• 依托单位: University of Texas at Austin
• 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: 2004-06-01 至 2010-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0354567&HistoricalAwards=false
• 关键词: Study; global; behavior; Dynamical; Systems

AbstractDe la LlaveThe goal of this project is to develop a variety of methods that allowto understand the long term behavior of dynamical systems and partialdifferential equations. The methods are designed to work together asa toolkit. We also plan to use this to study to several problems thathave been posed in the literature. Among the tools to be developed are: variational methods and their qualitative consequences, KAM theory, normal hyperbolic manifolds, slow manifolds, regularity results for cohomology equations, numerical computations. Among the problems to be considered are the equilibrium configurations of models from statistical mechanics, the problem of PDE describing periodic media and the problem of instability in Hamiltonian systems subject to periodic perturbations.Often one has to make predictions of systems over a large period oftime or over a large spatial extension. Even if a completely detailedstudy is often impossible, one can hope to identify special features inthe system which allow to make significant predictions, at least in some cases. Ideally, these features do not change much if the system is altered and can be computed efficiently. In this proposal, we will develop an array of tools that allow to identify and to compute several of these organizing features, both in the context of systems extended in space and in the context of long term prediction. The more theoretical work will be developed hand in hand with some concrete applications that have been proposed in the natural sciences literature. Among those, the problem of effective properties of periodic (in space) media and the problem of response of a non-linear mechanical system to external forcing which is periodic in time.

该项目的摘要目标是开发多种方法，以了解动态系统和partialDifferenten方程的长期行为。 这些方法旨在将ASA工具包一起使用。 我们还计划使用它来研究文献中遇到的几个问题。 在要开发的工具中有：变分方法及其定性后果，KAM理论，正常双曲线歧管，缓慢的歧管，共同体方程的规律性结果，数值计算。 在要考虑的问题中，有统计力学的模型的平衡配置，描述周期性媒体的PDE问题以及受周期性扰动的哈密顿系统中不稳定性的问题。通常，必须在很大的时间内或在很大的空间扩展过程中对系统进行预测。即使通常不可能进行完全详细的研究，人们也可以希望在系统中识别允许做出重大预测的特殊功能，至少在某些情况下。理想情况下，如果更改系统并可以有效地计算系统，则这些功能不会发生太大变化。在此提案中，我们将开发一系列工具，允许在空间扩展和长期预测的背景下识别和计算其中几个组织功能。 与自然科学文献中提出的一些具体应用有关的理论工作将越来越多。 其中，周期性介质的有效特性的问题以及非线性机械系统对外部强迫的响应问题，这是时间周期性的。