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.

该项目的目的是使用计算和分析方法来了解物理学指定的几个问题的长期动态。所使用的主要工具是分析，拓扑和计算。一个人通过研究（分析性地）来计算不变的歧管，它们满足并将其用作骨架。这需要开发KAM理论的建设性版本，正常的双曲线以及一些新的对象，例如散射图。该策略也可以应用于重新归一化的地图（地图中的地图）并分析一些非常混乱的系统。统一的原则是找到在混凝土系统中组织长期行为的地标。这是使用系统数值计算和严格数学的。严格的数学使用可以在数字崩溃的情况下，在数字变得越来越清晰时，可以自信地工作。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。这涉及不同级别的项目，我们计划让学生参与其中一些。学生将接受数字和严格数学的培训，并能够考虑具体问题。