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

这个项目的目标是使用计算和分析方法来理解由物理学等指定的几个问题的长期动态。使用的主要工具是分析、拓扑学和计算。人们通过研究不变流形所满足的不变方程(数值地、解析地)来计算不变流形，并将其用作骨架。这就需要发展KAM理论的构造性版本、法双曲性以及一些新的物体，如散射图。这一策略也可以应用于重整化映射(映射空间中的映射)和分析一些非常混沌的系统。统一的原则是找到组织具体系统中长期行为的标志点。这是使用系统的数值计算和严格的数学来完成的。当数值变得不那么清晰时，严格数学的使用允许在接近所考虑的对象的分解的情况下自信地工作。当需要考虑具体映射时，例如在天体动力学或化学反应中，它也非常适合应用。这涉及不同层次的项目，我们计划让学生参与其中的一些项目。学生将接受数学和严谨数学方面的培训，并能够考虑具体问题。

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.