Topics in Infinite Dimensional Dynamical Systems

[摘要]将发展无限维动力系统的几何理论。这里包括给出不变流形和叶形存在和持续的条件。这些将用于描述轨迹的整体定性行为，并建立保证结构稳定性的条件。该理论在晶格动力学、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等主题中的应用将被考虑。基于已发展的几何理论，将寻求有效的计算方法来模拟其中的一些。由于大多数物理系统是高度复杂的，通常不可能解出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，测量的不精确和我们对实际物理定律的不完善的理解也可能使解的意义受到质疑。这个项目将发展一种理论，据此，在某些条件下，人们可以描述解的性质，并保证物理系统按照这种描述以合理的精度运行。将特别注意描述材料科学现象的方程，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.