Dynamics near Turing patterns: modulations, bifurcations, and defects

University Of Minnesota-Twin Cities, Minneapolis MN

Investigators

Abstract

This project aims at a description of dynamics near Turing patterns in spatially extended systems that do not necessarily possess a free energy. Turing-like patterns are spatially periodic and stationary structures that emerge in a self-organized fashion. Typical contexts are reaction-diffusion settings, but also various convection experiments, or nonlinear optical feedback systems. We propose a systematic investigation of perturbations of Turing patterns, bifurcations from Turing patterns, and defects embedded in Turing patterns. We will aim at model-independent descriptions whenever possible, emphasizing topological and geometric intersection theory rather than explicit solutions. Some specific problems are the existence of dislocations in Turing patterns, symmetry-breaking bifurcations from Turing patterns, inhomogeneities and parameter ramps, and connections with Liesegang structures. Self-organized, regular patterns are striking when they appear in nature, in sand dunes, and on animal coats. They also carry great technological potential when exploited in manufacturing processes at nanoscales. In both cases, one would like to understand when and how regular structures form, and at which characteristic scale. We plan to study those questions using geometric and topological tools in the analysis. Those tools can shed light on the role of inhomogeneities and defects in the selection of patterns in situations where precise models and parameter values are unavailable. Ultimately, we would like to understand those self-organized pattern formation processes from macroscopic, universal principles that are largely independent of the underlying detailed microscopic processes.