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

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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).