RESUMEN

We study instabilities and pattern formation in reaction-diffusion layers that are diffusively coupled. For two-layer systems of identical two-component reactions, we analyze the stability of homogeneous steady states by exploiting the block symmetric structure of the linear problem. There are eight possible primary bifurcation scenarios, including a Turing-Turing bifurcation that involves two disparate length scales whose ratio may be tuned via the interlayer coupling. For systems of n-component layers and nonidentical layers, the linear problem's block form allows approximate decomposition into lower-dimensional linear problems if the coupling is sufficiently weak. As an example, we apply these results to a two-layer Brusselator system. The competing length scales engineered within the linear problem are readily apparent in numerical simulations of the full system. Selecting a sqrt[2]:1 length-scale ratio produces an unusual steady square pattern.

RESUMEN

We study time-periodic forcing of spatially extended patterns near a Turing-Hopf bifurcation point. A symmetry-based normal form analysis yields several predictions, including that (i) weak forcing near the intrinsic Hopf frequency enhances or suppresses the Turing amplitude by an amount that scales quadratically with the forcing strength, and (ii) the strongest effect is seen for forcing that is detuned from the Hopf frequency. To apply our results to specific models, we perform a perturbation analysis on general two-component reaction-diffusion systems, which reveals whether the forcing suppresses or enhances the spatial pattern. For the suppressing case, our results are consistent with features of previous experiments on the chlorine dioxide-iodine-malonic acid chemical reaction. However, we also find examples of the enhancing case, which has not yet been observed in experiment. Numerical simulations verify the predicted dependence on the forcing parameters.

Asunto(s)

RESUMEN

Parametrically-excited surface waves, forced by a repeating sequence of delta-function impulses, are considered within the framework of the Zhang-Viñals model [W. Zhang and J. Viñals, J. Fluid Mech. 336, 301 (1997)]. With impulsive forcing, the linear stability analysis can be carried out exactly and leads to an implicit equation for the neutral stability curves. As noted previously [J. Bechhoefer and B. Johnson, Am. J. Phys. 64, 1482 (1996)], in the simplest case of N=2 equally-spaced impulses per period (which alternate up and down) there are only subharmonic modes of instability. The familiar situation of alternating subharmonic and harmonic resonance tongues emerges only if an asymmetry in the spacing between the impulses is introduced. We extend the linear analysis for N=2 impulses per period to the weakly nonlinear regime, where we determine the leading order nonlinear saturation of one-dimensional standing waves as a function of forcing strength. Specifically, an analytic expression for the cubic Landau coefficient in the bifurcation equation is derived as a function of the dimensionless spacing between the two impulses and the fluid parameters that appear in the Zhang-Viñals model. As the capillary parameter is varied, one finds a parameter regime of wave amplitude suppression, which is due to a familiar 1:2 spatiotemporal resonance between the subharmonic mode of instability and a damped harmonic mode. This resonance occurs for impulsive forcing even when harmonic resonance tongues are absent from the neutral stability curves. The strength of this resonance feature can be tuned by varying the spacing between the impulses. This finding is interpreted in terms of a recent symmetry-based analysis of multifrequency forced Faraday waves [J. Porter, C. M. Topaz, and M. Silber, Phys. Lett. 93, 034502 (2004); C. M. Topaz, J. Porter, and M. Silber, Phys. Rev. E 70, 066206 (2004)].