Analytic and numerical studies of long term behavior in dynamical systems and differential equations

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The goal of this project is to use computational and analytic methods to understand the long term dynamics of several problems that are specified by the Physics etc. The main tools used are analysis, topology and computation. One computes invariant manifolds by studying (numerically, analytically) the invariance equations satisfied by them and uses this as the skeleton. This requires to develop constructive versions of KAM theory, normal hyperbolicity as well as some novel objects such as the scattering map. This strategy can be applied also to renormalization maps (maps in the spaces of maps) and analyze some very chaotic systems.The unifying principle is to find landmarks that organize the long term behavior in concrete systems. This is done using systematically numerical calculations and rigorous mathematics. The use of rigorous mathematics allows to work with confidence close to the breakdown of the objects considered, when the numerics becomes less clear.It is also well suited for applications when one needs to consider concrete mappings, for example in Astrodynamics or in chemical reactions. This involves projects of different levels and we plan to involve students in some of them. The students will get training both in numerics and in rigorous mathematics as well as be able to consider concrete problems.