Study of global behavior in Dynamical Systems and PDE's

这个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式的长期行为。这些方法被设计为作为工具包一起使用。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、正规双曲流形、慢流形、上同调方程的正则性结果、数值计算。需要考虑的问题包括统计力学模型的平衡构型，描述周期介质的偏微分方程问题，以及受周期扰动的哈密顿系统的不稳定性问题。经常需要对系统在大时间段或大空间范围内进行预测。即使完全详细的研究通常是不可能的，人们也可以希望识别出系统中的特殊特征，这些特征可以做出重要的预测，至少在某些情况下是这样。理想情况下，如果系统发生变化，这些特征不会有太大变化，并且可以高效地进行计算。在这项提案中，我们将开发一系列工具，以便在空间扩展的系统和长期预测的背景下识别和计算这些组织特征中的几个。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(空间)介质的有效性质问题和非线性力学系统对时间周期外力的响应问题。

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.