Hopping Behavior and Effects of Noise in Cellular Pattern-Forming Systems

We study the effects of multiplicative noise on a spatio-temporal pattern forming nonlinear Partial Differential Equation (PDE) model for premixed flame instability, known as the Kuramoto-Sivashinsky equation, in a circular domain. Modifications of a previously developed numerical integration scheme allow for longer time integration in the presence of noise. In order to gain additional insight, we focus on a region of parameter space where hopping patterns of the deterministic system arise as well as the region of parameter space where the transition between a single ring to multiple rings of cells appears. We discuss the numerical challenges in the integration of the Kuramoto-Sivashinsky equation in polar coordinates with the addition of the noise term. We also study the effects of additive and multiplicative noise on the normal forms or amplitude equations that describe the dynamics of hopping patterns. Finally we show some results of the implementation of the numerical scheme to solve both the PDE and the normal form equations and discuss preliminary findings.