Analytical and Numerical Studies of Long Term Behavior

这个项目涉及动力学研究中严格和计算方法的发展。具体地说，将研究以下内容：1)动力系统和偏微分方程组的不变流形和不变流形(特别是非共振流形、准周期运动流形)的理论和计算；2)动力系统中的刚性现象(非交换光滑Livsic理论、拟共形方法、诱导和重整化)；3)变分方法(传输机制和在偏微分方程组中的应用)；4)水波的哈密顿表述和数值实现。系统的长期行为可能比支配它的定律复杂得多。即使是非常简单的进化定律，当被多次应用时，也可能导致难以预测的各种行为。一个最好的例子是流体的行为，它的基本方程是已知的，在短时间内可以准确地预测，但经过一段时间后，它可能会形成难以预测的漩涡或湍流。在这个项目中，将使用分析方法和数值计算相结合的广泛方法来分类和定量研究混凝土系统可能的长期行为。在应用问题中出现的具体模型中，既有广泛适用但不是定量的一般理论和定量计算(其中一些是数值的)，都将得到发展。流体运动和卫星运动的研究将受到特别关注。

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.