Resonant Pattern Formation in a Spatially Extended Chemical System

Abstract

When an oscillatory nonlinear system is driven by a periodic external stimulus, the system can lock at rational multiples p : q of the driving frequency. The frequency range of this resonant locking at a given p : q depends on the amplitude of the stimulus; the frequency width of locking increases from zero as the stimulus amplitude increases from zero, generating an “Arnol’d tongue” in a graph of stimulus amplitude vs stimulus frequency. Physical systems that exhibit frequency locking include electronic circuits [1, 2], Josephson junctions [3], chemical reactions [4], fields of fireflies [5, 6], and forced cardiac systems [7, 8]. Most studies of frequency locking have concerned either maps or systems of a few coupled ODEs. The Arnol’d tongue structure of the sine circle map has been extensively studied, and the theory of periodically driven ODE systems has been well developed [9], but there has been very little analysis of frequency locking phenomena in PDEs, except for a few studies of the parametrically excited Mathieu equation with diffusion and damping [10, 11, 12] and the parametrically excited complex Ginzburg-Landau equation [13, 14]. Our interest here is in the effect of periodic forcing on pattern forming systems such as convecting fluids, liquid crystals, granular media, and reaction-diffusion systems. Such systems are often subject to periodic forcing (e.g., circadian forcing of biological systems), but the effect of forcing on the bifurcations to patterns has not been examined in experiments or analyzed in PDE models of these systems.