Pattern selection in extended periodically forced systems: a continuum coupled map approach.

Abstract

We propose that a useful approach to the modeling of periodically forced extended systems is through continuum coupled map (CCM) models. CCM models are discrete time, continuous space models, mapping a continuous spatially varying field xi(n)(x) from time n to time n+1. The efficacy of CCM models is illustrated by an application to experiments of Umbanhowar, Melo, and Swinney [Nature 382, 793 (1996)] on vertically vibrated granular layers. Using a simple CCM model incorporating temporal period doubling and spatial patterning at a preferred length scale, we obtain results that bear remarkable similarities to the experimental observations. The fact that the model does not make use of physics specific to granular layers suggests that similar phenomena may be observed in other (nongranular) periodically forced, strongly dissipative systems. We also present a framework for the analysis of pattern selection in CCM models using a truncated modal expansion. Through the analysis, we predict scaling laws of various quantities, and these laws may be verifiable experimentally.