In many areas of science and engineering differential equations are employed as models of complex phenomena. The focus in this proposal is on the approximation of solutions of infinite dimensional dynamical systems. In particular, the investigator and his colleagues are interested in understanding the dynamics of the finite dimensional approximations and how they relate to the dynamics of the original dynamical system. The specific issues to be investigated are related to stability of solutions of spatially discrete reaction-diffusion equations, the dynamics of so-called anti-diffusion lattice differential equations, dynamics and computation of traveling waves for neutral mixed type functional differential equations and rigorous computation of periodic orbits for retarded delay equations, robust stability for time dependent linear Hamiltonian systems, numerical techniques for efficient computation of Lyapunov exponent like quantities based upon nonlinear flows, and approximation techniques for stability spectra of delay equations and partial differential equations. Techniques to investigate these systems combine numerical analysis and dynamical systems ideas to gain a better understanding of approximation dynamics.
Often in complex systems the differential equations used in modeling are infinite dimensional, but necessarily these models are approximated by finite dimensional systems in order to perform numerical simulations. In order to understand the behavior of these models, the investigator and his colleagues are interested in similarities and differences in the behavior of the original infinite dimensional system and finite dimensional approximations, for example those obtained through discretization. Understanding the impact of discretization leads to greater confidence when making inferences from simulation results. In addition, stability analysis is useful in understanding the robustness of complex biological phenomena that occur in the environment and in identifying instabilities, for example, in models of weather prediction. Discrete models of both biological and physical processes are becoming more important as the need for detailed, microscopic models increases and should benefit from improved analysis and computational capabilities.