Topics in Infinite Dimensional Dynamical Systems

无限维动力系统主题

• 批准号: 9970894
• 负责人: Peter Bates
• 金额: $ 16.2万
• 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: 1999
• 资助国家: 美国
• 起止时间: 1999-06-01 至 2003-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970894&HistoricalAwards=false
• 关键词: Topics; Infinite; Dimensional; Dynamical; Systems

AbstractBates/LuGeometrical theory of infinite-dimensional dynamical systems will be developed. Included here are conditions giving the existence and persistence of invariant manifolds and foliations. These will be used to describe global qualitative behavior of trajectories and to establish conditions which guarantee structural stability. Applications of this theory to topics such as lattice dynamics, cellular neural networks, nonlinear parabolic partial differential equations and some nonlinear hyperbolic partial differential equations will be considered. Efficient computational approaches to simulate some of these will be sought, based upon the geometrical theory developed.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.

摘要Bates/Lu型无限维动力系统几何理论将得到发展。这里包括给出不变流形和叶的存在和持久性的条件。这些将被用来描述轨迹的全球质量行为，并建立保证结构稳定性的条件。这一理论在格子动力学、细胞神经网络、非线性抛物型偏微分方程组和一些非线性双曲型偏微分方程中的应用将被考虑。基于发展起来的几何理论，人们将寻求有效的计算方法来模拟其中的一些现象。由于大多数物理系统是高度复杂的，通常不可能求解提出的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，测量的不精确度和我们对实际物理定律的不完全理解也可能会让人质疑这些解的意义。这个项目将开发一种理论，在某些条件下，一个人可以描述解决方案的性质，并保证物理系统按照这种描述以合理的精度运行。将特别关注描述材料科学现象的方程，包括相变和超导。