Theory and Applications for Infinite Dimensional Dynamical Systems

Brigham Young University, Provo UT

Investigators

Abstract

PI: Peter Bates, Brigham Young University Co-PI: Keining Lu, Brigham Young University DMS-0200961 Abstract: 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.