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

Abstract: We study the effects of multi-plicative noise on a spatio-temporal pattern forming nonlinear Partial Differential Equa-tion (PDE) model for premixed flame insta-bility, known as the Kuramoto-Sivashinsky equation, in a circular domain. Modifica-tions of a previously developed numerical in-tegration scheme allow for longer time in-tegration in the presence of noise. In or-der 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 mul-tiple 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 dy-namics 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 pre-liminary findings