Study of global behavior in Dynamical Systems and PDE's

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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.