Anna L. Lin

Development of Standing-Wave Labyrinthine Patterns ∗

Experiments on a quasi-two-dimensional Belousov-Zhabotinsky (BZ) reaction-diffusion system, pe- riodically forced at approximately twice its natural frequency, exhibit resonant labyrinthine patterns that develop through two distinct mechanisms. In both cases, large amplitude labyrinthine patterns f orm that consist ofinterpenetrating fingers off requency-locked regions differing in phase by π. Analysis of a forced complex Ginzburg-Landau equation captures both mechanisms observed for the f ormation ofthe labyrinths in the BZ experiments: a transverse instability off ront structures and a nucleation of stripes from unlocked oscillations. The labyrinths are found in the experiments and in the model at a similar location in the forcing amplitude and frequency parameter plane.

Resonance in periodically inhibited reaction-diffusion systems

Abstract We have conducted experiments on a periodically inhibited oscillatory Belousov–Zhabotinsky (BZ) reaction–diffusion system in a regime in which the patterns oscillate at half the frequency of the forcing (the 2:1 resonance regime). The periodic perturbations of the photosensitive (ruthenium-catalyzed) reaction are made with light, which inhibits the oscillatory behavior. Increasing the light intensity increases the refractory period (time for recovery from inhibition), which decreases the oscillation frequency of the patterns in the medium. We investigate the behavior for two different levels of inhibitor concentration in the reagent feed to determine the shape of the 2:1 resonant regime as a function of the forcing intensity and forcing frequency. At high forcing intensity, the inhibition leads to a transition from traveling waves (spirals) to standing waves. Simulations of a reaction–diffusion model with FitzHugh–Nagumo kinetics yield behavior similar to that observed in the experiments.

Resonant and nonresonant patterns in forced oscillators

Uniform oscillations in spatially extended systems resonate with temporal periodic forcing within the Arnold tongues of single forced oscillators. The Arnold tongues are wedge-like domains in the parameter space spanned by the forcing amplitude and frequency, within which the oscillators frequency is locked to a fraction of the forcing frequency. Spatial patterning can modify these domains. We describe here two pattern formation mechanisms affecting frequency locking at half the forcing frequency. The mechanisms are associated with phase-front instabilities and a Turing-like instability of the rest state. Our studies combine experiments on the ruthenium catalyzed light-sensitive Belousov-Zhabotinsky reaction forced by periodic illumination, and numerical and analytical studies of two model systems, the FitzHugh-Nagumo model and the complex Ginzburg-Landau equation, with additional terms describing periodic forcing.

Resonant Pattern Formation in a Spatially Extended Chemical System

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.