Collaborative Research: Absolute and essential instabilities in spatially extended systems

NSF Award Abstract - DMS-0203301

Mathematical Sciences: Collaborative Research: Absolute and essential instabilities in spatially extended systems

Abstract

0203301 Scheel

This project explores several instability mechanisms of coherent states, such as fronts, pulses and spiral waves, that occur in spatially extended systems far from equilibrium. The common theme of these mechanisms is that they involve transport phenomena caused by diffusion and nonlinearities. Examples of such transitions are backfiring instabilities of fronts, period-doubling bifurcations of spiral waves, and the effect of inhomogeneities on the dynamics of spiral waves. While transport is most conveniently modeled and described using an idealized unbounded domain, boundaries may well enhance or inhibit the instability through partial reflection or generation of waves. It is the aim of this project to develop techniques that can help to investigate instability mechanisms simultaneously on bounded and unbounded domains. One of the expected outcomes will be a description of instability mechanisms that is robust with respect to typical boundary conditions in large reactors. Technically, our approach is based on methods from spatial dynamics that allow us to derive sharp pointwise estimates which capture explicitly the effects of boundaries.

Complicated spatio-temporal patterns that arise due to the interplay of chemical reactions and diffusion have been observed experimentally in a number of specific reactions (among them the Belousov-Zhabotinsky reaction and the chlorite-iodide-malonic acid reaction). Similar phenomena have been observed during fibrillation of cardiac tissue where spiral waves act as organizing centers for the complex spatio-temporal dynamics. Other examples where irregular patterns occur are the interaction of pulses in oscillatory media, backfiring of excitation pulses in catalytic reactions, and optical bistability in nonlinear photonic gratings. The focus of this project is to analyze some of these instability mechanisms by analytical means. This will not only further our understanding of pattern formation in chemical and biological systems, but will eventually allow for a systematic control of patterns, for instance, in the catalytic oxidation of carbon-monoxide and in the propagation of calcium waves in intracellular tissue.