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

这项提议的主要目标是设计出能够对动力系统或PDE的长期(或长期)行为进行预测的方法。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。我们对动力系统的不稳定性、椭圆型偏微分方程组和耦合网络的整体行为的应用特别感兴趣。许多自然规律都是以局部相互作用的形式表述的。空间和时间中的一个点只影响它的近邻。可能发生的情况是，这些局部相互作用相互抵消，因此全球效应很小，系统保持某种程度的不受影响，也可能发生局部相互作用相互加强，导致大规模效应的情况。这两种选择确实发生了，它们依赖于非常微妙的效应(例如，深度和抽象的数论是非常可测量的效应的关键)。尽管应用数学家早在几个世纪前就认识到了它的重要性，但直到最近，许多人才开始研究它，开发了一个足够丰富的工具包。不同的人，一直在创造不同的技术来共同工作，他们已经开始产生效果。作为这一兴趣的见证者，这项提议的PI一直是CRM(巴塞罗那)2008年秋季和菲尔兹学院(2011年春季)特别学期的联合组织者。

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).