Analytical and Numerical Studies of Long Term Behavior

该项目涉及在动力学研究中发展严格的计算方法。 具体而言，将研究以下内容：1）不变流形和不变叶理的理论与计算（特别是非共振流形，拟周期运动流形）和偏微分方程; 2）动力系统中的刚性现象（非对易光滑Livsic理论，拟共形方法，诱导和重整化）; 3）变分方法（迁移机制和对偏微分方程的应用）; 4）水波的哈密顿公式和数值实现。系统的长期行为可能比控制它的定律复杂得多。即使是非常简单的演化定律，当多次应用时，可能导致各种难以预测的行为。 一个最好的例子是流体的行为，其基本方程是已知的，并且在短时间内可以准确预测，但是经过一段时间后，它可以发展出不容易预测的漩涡或湍流。 在这个项目中，一个基础广泛的方法，结合分析方法和数值计算将被用来分类和定量研究混凝土系统可能的长期行为。 这两个广泛适用的，但不是定量的一般理论和定量计算（其中一些数值），在具体的模型中出现的应用问题将得到发展。 将特别注意对流体运动和卫星运动的研究。

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.