This project will investigate the theory and applications of equivariant dynamical systems. In particular, using the symmetries of differential equations and models, we will study patterns that occur both in space and in time. We will explore locomotor central pattern generators for bipeds and quadrupeds, a classification of bursting types, hypermeander of spiral waves, and magnetic dynamos. We will also study a combination of Euclidean symmetry and chaotic dynamics that leads to a deterministic mechanism for stochastic Brownian motion, and Ginzburg-Landau theory for spatially-extended systems.
Patterns appear in physical, chemical, and biological systems in both space and time and are striking and reproducible. Often these patterns result from symmetries, and our research will study pattern formation by further developing the theory and application of symmetric dynamical systems. Applications that will be investigated range from the study of muscle rhythms in animal gaits and intermittency in magnetic dynamos to the complicated ways in which bursting signals are produced and spiral waves move.
