Pattern Formation with Two Length Scales: Spatiotemporal Chaos

Abstract

Three-wave interactions (or resonant triads) are the lowest-order nonlinear interaction in pattern formation and arise between waves with different orientations when the sum of two wavevectors equals a third one. When a pattern has only one length scale, stripe patterns are possible but three-wave interactions are responsible for the prevalence of hexagons close to onset. In problems with two length scales, there is a much wider range of possible three-wave interactions, leading to more complex structures such as superhexagons, stars, quasipatterns and even spatiotemporal chaos (STC). We investigate the role that nonlinear wave interactions play in the formation of STC in a model partial differential equation (PDE) in the case that the length scale ratio is $\sqrt{7}$, relevant to superlattice patterns in the Faraday wave experiment. The simpler aspects of the dynamics can be represented by a system of ordinary differential equations (ODEs) derived from the PDE using weakly nonlinear theory. We analyze the equilibrium patterns in these ODEs and evaluate their stability, comparing the results with direct numerical simulations of the model PDE. The ODEs predict parameter regimes where there are no stable simple equilibria, which is where we typically find complex behavior in the PDE. We have conducted a careful study of the transition from simple patterns to patterns that include modes beyond the finite-dimensional subspace imposed in the reduction to the ODEs, to time-dependent competition between different triads, ending up with fully developed STC. For our choice of length scale ratio, we show that four-wave interactions also play an important role. Our analysis is relevant to any pattern-forming system with three-wave interactions involving two length scales, such as the Faraday wave experiment, coupled reaction-diffusion systems, and pattern formation in dryland vegetation.