The low-dimensional dynamic behavior of infinite dimensional systems is a field of intense mathematical research, and the theoretical part of this work explores and builds on a number of such directions. The investigator and his colleagues study several new aspects of the theory of inertial and approximate inertial manifolds, and explore and establish new connections between them and the method of empirical eigenfunctions. At the same time, a goal of the proposal is the computational implementation of the new methods and the study of certain numerical analysis aspects associated with them (stability, interaction with symmetries, and preservation of dissipation). Dissipative time-dependent partial differential equations arise from descriptions of a variety of important physical, chemical, and biological phenomena. The purpose of this project is to develop, analyze, computationally implement, and test methods for the reduction of such equations to accurate, small dynamical systems. This is a subject of intense research interest in its own right; at the same time, it provides a bridge to the study of many complex physicochemical and engineering systems, with important implications for the simulation, analysis, prediction and control of their dynamic behavior. An integral part of this work is the application of the techniques to models of realistic systems of current research interest, such as pattern formation in catalytic and electrochemical reactions. The investigator and his colleagues believe that the attempt to tackle realistic problems strengthens the theoretical and computational aspects of the research, and provides new ideas and directions.
