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

动力系统和微分方程中长期行为的分析和数值研究

• 批准号: 1162544
• 负责人: Rafael de la Llave
• 金额: $ 27.29万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1162544&HistoricalAwards=false
• 关键词: Analytic; numerical; studies; long; term

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.

该项目的目标是使用计算和分析方法来理解由物理等指定的几个问题的长期动力学。主要使用的工具有分析、拓扑和计算。人们通过研究流形所满足的不变性方程来计算不变性流形，并以此作为骨架。这就需要发展KAM理论的建设性版本，正常双曲以及一些新的对象，如散射图。这种策略也可以应用于重整化映射（映射空间中的映射），并分析一些非常混乱的系统。统一的原则是找到在具体系统中组织长期行为的标志。这是通过系统的数值计算和严格的数学来完成的。当数字变得不太清楚时，使用严格的数学可以更有信心地接近所考虑的对象的分解。它也非常适合于需要考虑具体映射的应用，例如在天体动力学或化学反应中。这涉及到不同层次的项目，我们计划让学生参与其中。学生们将得到数值和严格数学的训练，并能够考虑具体问题。