Analytical, topological and numerical methods in the study of long range behavior in dynamical systems and differential equations

摘要本文的主要目标是设计出能够预测动力系统或PDE的长期（或长范围）行为的方法。我们计划开发一系列广泛的工具（不变流形、变分方法、数值分析），使它们能够协同工作。我们对动态系统的不稳定性和椭圆偏微分方程和耦合网络的全局行为的应用特别感兴趣。许多自然法则被表述为局部的相互作用。空间和时间上的一个点只影响它的近邻。有可能这些局部的相互作用相互抵消这样整体影响就很小系统就不会受到影响也有可能局部的相互作用相互加强导致大规模的影响。这两种选择确实会发生，它们依赖于非常微妙的效果（例如，相当深刻和抽象的数论是非常可测量效果的关键）。尽管数世纪以来，应用数学家们就已经认识到它的重要性，但直到最近，许多人才开发出足够丰富的工具包，开始着手解决它。不同的人，一直在用不同的技术一起工作，他们已经开始产生结果。作为兴趣的见证，该提案的PI已成为CRM（巴塞罗那）2008年秋季和Fields institute（2011年春季）特别学期的共同组织者。

Abstractde la LlaveThe main goal of this proposal is to devise methods that allow to make predictions of the long term (or the long range) behavior of dynamical systems or PDE. We plan to develop a broad array of tools (invariant manifolds, variational methods, numerical analysis) in such a way that they can work together. We are particularly interested in applications to instability in dynamical systems and to global behavior in ellipticpartial differential equations and in coupled networks. Many of the laws of nature are formulated as local interactions. One point in space and time affects only its close neighborhood. It can happen that these local interactions cancel each other out so that the global effect is small and that the systems remain kind of unaffected or it can happen that the local interactions reinforce each other andlead to large scale effects. The two alternatives do happen and they depend on very subtle effects (e.g. ratherdeep and abstract number theory is the key to very measurable effects). Even if the importance of the has been recognized by applied mathematicians for centuries, it is only very recently that a rich enough toolkit has been developed by many start tackling it. Different people, have been making different techniques to work together, and they have started producing results. As a witness to the interest, the PI of this proposal hasbeen co-organizer of special semesters in CRM (Barcelona) Fall 2008 and Fields institute (Spring 2011).