Analytical and Numerical Studies of Long Term Behavior

这个项目涉及到动力学研究中严谨和计算方法的发展。具体而言，将研究以下内容：1)动力系统和PDE的不变流形和不变叶形（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lisic理论、拟共形方法、归纳和重整化）；3)变分方法（输运机制及其在pde中的应用）；4)水波的哈密顿公式及其数值实现。一个系统的长期行为可能比控制它的定律要复杂得多。即使是非常简单的进化规律，如果多次应用，也可能导致各种难以预测的行为。一个典型的例子是流体的行为，它的基本方程是已知的，并且在短时间内可以准确地预测，但在一段时间后，它可能会发展成不容易预测的漩涡或湍流。在这个项目中，结合分析方法和数值计算的基础广泛的方法将用于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，将发展广泛适用的——但不是定量的——一般理论和定量计算（其中一些是数值计算）。流体运动和卫星运动的研究将得到特别的重视。

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.