Grant: Discovery Grants Program - Individual

My main research interest is the development, analysis, implementation, and application of numerical methods for analyzing the solution behavior of dynamical systems. My work is motivated by the fascinating changes of behavior exhibited by solutions to such equations as parameters are varied: there is tremendous satisfaction in devising new numerical algorithms that help increase our understanding of these phenomena. My recent research includes the determination of so-called invariant manifolds in the famous Lorenz equations, and especially the topological changes of these manifolds as parameter values enter the chaotic regime. Another study concerns the determination of stable and unstable manifolds of periodic orbits in the circular restricted 3-body problem, as these are of great current interest in the design of low-cost orbit transfers in space mission design. Also of practical interest is the analysis of laser models, several of which predict the qualitative and even quantitative behavior of actual physical experiments very well.I try to make my numerical methods available for use in widely varying research areas. A notable success has been the development of the influential and widely used research software AUTO, which continues to evolve in cooperation with researchers in a world-wide network. AUTO ""demos"", including those for several of the above-mentioned research projects, are freely available on-line for use by others in related research.I will further develop highly accurate continuation and bifurcation algorithms and powerful software for dynamical systems, with emphasis on nonlinear elliptic and parabolic partial differential equations in general 2-dimensional spatial domains, and subject to boundary and integral constraints. This work has important applications in many research areas, such as pattern formation in biology, fluid flow problems, and chemical reactions. In the longer term generalization to 3-dimensional spatial domains is anticipated.