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Analytical, topological and numerical methods in the study of long range behavior in dynamical systems and differential equations

研究动力系统和微分方程中长程行为的分析、拓扑和数值方法

基本信息

批准号：
0901389
负责人：
Rafael de la Llave
金额：
$ 30万
依托单位：
University of Texas at Austin
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2009
资助国家：
美国
起止时间：
2009-09-01 至 2012-04-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901389&HistoricalAwards=false
关键词：
Analytical topological numerical methods study

项目摘要

Abstractde la LlaveThe main goal of this proposal is to devise methods that allow to make predictions of the long term (or the long range) behavior of dynamical systems or PDE. We plan to develop a broad array of tools (invariant manifolds, variational methods, numerical analysis) in such a way that they can work together. We are particularly interested in applications to instability in dynamical systems and to global behavior in ellipticpartial differential equations and in coupled networks. Many of the laws of nature are formulated as local interactions. One point in space and time affects only its close neighborhood. It can happen that these local interactions cancel each other out so that the global effect is small and that the systems remain kind of unaffected or it can happen that the local interactions reinforce each other andlead to large scale effects. The two alternatives do happen and they depend on very subtle effects (e.g. ratherdeep and abstract number theory is the key to very measurable effects). Even if the importance of the has been recognized by applied mathematicians for centuries, it is only very recently that a rich enough toolkit has been developed by many start tackling it. Different people, have been making different techniques to work together, and they have started producing results. As a witness to the interest, the PI of this proposal hasbeen co-organizer of special semesters in CRM (Barcelona) Fall 2008 and Fields institute (Spring 2011).

这个建议的主要目标是设计方法，允许作出预测的长期（或长期）的动力系统或PDE的行为。我们计划开发一系列广泛的工具（不变流形，变分方法，数值分析），以便它们可以一起工作。我们特别感兴趣的应用程序的不稳定性动力系统和全球行为的椭圆偏微分方程和耦合网络。自然界的许多法则都是以局部相互作用的形式表述的。空间和时间中的一个点只影响它的近邻。这些局部的相互作用可能会相互抵消，这样整体效应就很小，系统就不会受到影响，或者局部的相互作用可能会相互加强，导致大尺度效应。这两种选择确实发生了，而且它们依赖于非常微妙的效应（例如，相当深刻和抽象的数论是非常可测量的效应的关键）。尽管应用数学家们几个世纪以来就认识到了这一问题的重要性，但直到最近，才有足够丰富的工具箱被开发出来，许多人开始着手解决这一问题。不同的人一直在使用不同的技术来协同工作，他们已经开始产生结果。作为兴趣的见证，这个建议的PI已经在CRM（巴塞罗那）2008年秋季和菲尔兹研究所（2011年春季）的特别学期的共同组织者。