Topics in Infinite Dimensional Dynamical Systems

无限维动力系统的几何理论将得到发展。 包括在这里的条件，使存在性和持久性不变的流形和叶状。 这些将用于描述轨迹的全局定性行为，并建立保证结构稳定性的条件。 这个理论的应用主题，如晶格动力学，细胞神经网络，非线性抛物型偏微分方程和一些非线性双曲型偏微分方程将被考虑。 基于几何理论的发展，我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非常复杂，通常不可能求解用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。

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.