Analytical and Numerical Studies of Long Term Behavior

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This project involves the development of rigorous and computational methods in the study of dynamics. Specifically, the following will be studied: 1) The theory and computation of invariant manifolds and invariant foliations (specially non-resonant manifolds, manifolds of quasiperiodic motions) for dynamical systems and PDE; 2) Rigidity phenomena in dynamical systems (non-commutative smooth Livsic theory, quasiconformal methods, inducing and renormalization); 3) Variational methods (mechanisms of transport and applications to PDEs); 4) Hamiltonian formulation of water waves and numerical implementations.The long term behavior of a system may be significantly more complicated than the laws governing it. Even very simple laws of evolution, when applied many times, may lead to a variety of behavior which is difficult to predict. A prime example is the behavior of a fluid, whose fundamental equations are known and which is accurately predictable over a short time, but which can develop, after a time, swirls or turbulence that are not easy to predict. In this project, a broad based approach that combines analytical methods and numerical calculations will be used to classify and study quantitatively possible long term behaviors of concrete systems. Both widely applicable-but not quantitative-general theory and quantitative calculations (some of them numerical) in concrete models appearing in applied problems will be developed. The study of fluid motion and the motion of satellites will be given particular attention.