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.

这个项目的目标是发展各种方法来理解动力系统和偏微分方程的长期行为。 这些方法被设计为阿萨工具包一起工作。 我们还计划利用这一点来研究文献中提出的几个问题。 其中要开发的工具是：变分方法及其定性后果，KAM理论，正常双曲流形，慢流形，正则性结果的上同调方程，数值计算。 其中要考虑的问题是从统计力学模型的平衡配置，PDE描述周期性介质的问题和问题的不稳定性的哈密顿系统受到周期性扰动。经常有一个作出预测的系统在一个大的时期oftime或在一个大的空间扩展。即使一个完全详细的研究往往是不可能的，人们可以希望确定系统中的特殊功能，允许作出重大的预测，至少在某些情况下。理想情况下，如果系统发生变化，这些特征不会发生太大变化，并且可以有效地计算。在这个建议中，我们将开发一系列工具，允许识别和计算这些组织功能中的几个，无论是在空间中扩展的系统的背景下，在长期预测的背景下。 更多的理论工作将与自然科学文献中提出的一些具体应用携手发展。 其中，周期性（在空间）介质的有效性质的问题和一个非线性力学系统的响应问题，这是周期性的时间外强迫。