Study of global behavior in Dynamical Systems and PDE's

摘要 De la Llave 该项目的目标是开发各种方法来理解动力系统和偏微分方程的长期行为。 这些方法被设计为作为工具包一起工作。 我们还计划用它来研究文献中提出的几个问题。 待开发的工具包括：变分方法及其定性结果、KAM 理论、正态双曲流形、慢流形、上同调方程的正则性结果、数值计算。 要考虑的问题包括统计力学模型的平衡配置、描述周期性介质的偏微分方程问题以及受周期性扰动影响的哈密顿系统的不稳定性问题。人们通常必须对较长时间段或较大空间范围内的系统进行预测。即使通常不可能进行完全详细的研究，人们也可以希望识别系统中的特殊功能，从而至少在某些情况下做出重要的预测。理想情况下，如果系统发生改变，这些特征不会发生太大变化，并且可以有效地进行计算。在本提案中，我们将开发一系列工具，无论是在空间扩展的系统背景下还是在长期预测背景下，这些工具都可以识别和计算其中的一些组织特征。 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一起发展。 其中，周期性（空间）介质的有效特性问题和非线性机械系统对时间上周期性的外力的响应问题。

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.