Theory and Applications for Infinite Dimensional Dynamical Systems

无限维动力系统的理论与应用

• 批准号: 0200961
• 负责人: Peter Bates
• 金额: $ 16.8万
• 依托单位: Brigham Young University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0200961&HistoricalAwards=false
• 关键词: Theory; Applications; Infinite; Dimensional; Dynamical

PI: Peter Bates, Brigham Young UniversityCo-PI: Keining Lu, Brigham Young UniversityDMS-0200961Abstract:This project will develop the geometrical theory of invariant manifolds and foliations for semiflows in Banach space with emphasis on how this can be applied to important areas of science. Of particular interest are how one may establish the existence of these structures knowing that good approximations exist in some sense. Applications which will be addressed include analysis of lattice dynamical systems, the existence of modulated waves in fluid models, the form and dynamics of vortices in superconducting materials, the form and dynamics of material phase boundaries, including those with random perturbations in the driving forces, and the development of good computational algorithms.Since most physical systems are highly complex, it is usually impossible to solve the equations which are proposed to describe and predict the relevant physical phenomena. Even if it were possible to solve one of these equations, the imprecision in measurement and our imperfect understanding of the actual physical laws may call into question the meaning of the solutions. This project will develop a theory whereby, under certain conditions, one can describe the nature of solutions and guarantee that the physical system behaves according to this description to a reasonable degree of accuracy. Special attention will be given to equations describing phenomena in material science, including phase transitions and superconductivity.

主要研究者：Peter Bates，杨百翰大学合作伙伴：Keining Lu，杨百翰大学DMS-0200961摘要：本项目将发展Banach空间中半流的不变流形和叶理的几何理论，重点是如何将其应用于重要的科学领域。特别感兴趣的是如何建立这些结构的存在，知道在某种意义上存在良好的近似。将涉及的应用包括晶格动力学系统的分析，流体模型中调制波的存在，超导材料中涡旋的形式和动力学，材料相边界的形式和动力学，包括驱动力中随机扰动的那些，以及良好的计算算法的发展。通常不可能求解提出来描述和预测相关物理现象的方程。即使有可能解出这些方程中的一个，测量的不精确性和我们对实际物理定律的不完全理解也可能使解的意义受到质疑。该项目将开发一种理论，在某些条件下，人们可以描述解决方案的性质，并保证物理系统的行为根据这种描述达到合理的准确度。将特别关注描述材料科学中的现象的方程，包括相变和超导性。