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 nonlinear equations. I have worked on dynamical systems, especially those modeled by ordinary differential equations, and I am also interested in nonlinear elliptic and parabolic partial differential equations. My work is motivated by the fascinating changes of behavior exhibited by solutions to nonlinear equations as parameters are varied: There is tremendous satisfaction in devising new numerical algorithms that help increase our understanding of these phenomena. Recent applications range from a classification of elemental periodic orbits of the Circular Restricted 3-Body Problem, which is relevant to space mission design, to the computation of invariant manifolds, which explains an apparent discontinuous response in a cardiac pacemaker cell model. I try to make the numerical methods available for use in widely varying research areas. A notable success has been the development of the research software AUTO, which continues to evolve in cooperation with researchers in a world-wide network. I will further develop high-order accurate, adaptive discretizations of elliptic PDEs, resulting in software to track solutions, and determine their bifurcations, even in near-singular applications. Another focus is continuation techniques for periodic orbits (including infinite period orbits) and invariant manifolds in dissipative as well as conservative systems. Important applications will be studied.