Patterns in Continuous Systems

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

NSF Award Abstract - DMS-0072444 Mathematical Sciences: Patterns in Continuous Systems Abstract 0072444 Knobloch This project studies bifurcations and pattern formation in systems with symmetry. Such systems arise naturally in a variety of applications. The project focuses on the dynamical consequences of weak forced symmetry-breaking perturbations, and seeks thereby to identify those aspects of equivariant dynamics that are (i) insensitive to such perturbations, and (ii) due to them. In many cases such perturbations are responsible for the creation of global bifurcations, and these in turn may introduce chaotic dynamics into the system. Applications to a number of continuum systems will be worked out, focusing on the generation of burst-like behavior in hydrodynamical systems such as convection and the Faraday instability, and the dynamics of acoustically driven shape oscillations of drops and bubbles. This project investigates how the patterns set up in various physical systems change as the parameters specifying the system vary, with emphasis on systems having various spatial and temporal symmetries. The approach can identify new types of complex behavior and the mechanisms responsible for it. Applications include, for example, the behavior of water waves in regions of different shapes. The project investigates the effects of slight changes in the shape of the region, and seeks to identify which aspects of the behavior are insensitive to such perturbations, and which are due to them. In many cases such perturbations are responsible for the introduction of chaotic dynamics into systems that would otherwise behave in a simple manner. A number of additional applications will be worked out, including burst-like behavior in hydrodynamical systems and oscillations of drops and bubbles.