Topics in Computational Dynamics

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Models of complex systems often involve nonlinear differential equations that must be approximated numerically to gain detailed information. To understand the behavior of such approximations, this project explores similarities and differences in the dynamic behavior of the original model and what is obtained through numerical simulations. Understanding when the original model and numerical simulations yield similar dynamic behavior leads to greater confidence when making inferences from simulation results. In addition, stability analysis is useful in understanding the robustness of complex biological and physical phenomena. Of particular emphasis in this project is the application of these results to the dynamic behavior of models of climate dynamics. The techniques to be developed include approaches to dimension reduction and uncertainty quantification based upon Lyapunov exponent theory. The understanding of biological and physical processes will increase due to the computational techniques developed to address the dynamic behavior of detailed, microscopic models. In many areas of science and engineering dynamical systems are employed as models of complex phenomena. The focus of this project is on approximation of solutions of nonlinear dynamical systems. In particular, the investigator and 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 main research topics to be investigated are related to the application of time dependent orthogonal change of variables and in the dynamics of traveling waves under discretization. Specific interests include the use of Lyapunov vectors (analogues of eigenvectors in time dependent stability analysis) for dimensional reduction of dissipative nonlinear differential equations, robust stability for time dependent linear Hamiltonian systems, and techniques for analyzing time dependent stability of numerical time stepping techniques, stability of plane wave solutions to bistable reaction-diffusion equations in periodic media, the impact of time and space discretization on traveling waves, and non-planar traveling waves in discrete media. Techniques to investigate these systems combine numerical analysis and dynamical systems ideas to gain a better understanding of computational dynamics both qualitatively and quantitatively.