Topics in Infinite Dimensional Dynamical Systems

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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.