A simple generalization of the Swift-Hohenberg equation is proposed as a model for the patternforming dynamics of a two-dimensional field with two unstable length scales. The equation is used to study the dynamics of surface waves in a fluid driven by a linear combination of two frequencies. The model exhibits steady-state solutions with two-, four-, six-, and twelve-fold symmetric patterns, similar to the periodic and quasiperiodic patterns observed in recent experiments.PACS numbers: 47.54.+r, 47.35.+i, 47.20.Ky, 61.44.Br Parametric excitations of surface waves have been extensively studied since their first discovery by Faraday [1] over a century and a half ago. In the basic experimental setup an open container of fluid is subjected to vertical sinusoidal oscillations, which periodically modulate the effective gravity. When the driving amplitude a exceeds a critical threshold a c a standing-wave instability occurs with temporal frequency ω one half that of the driving frequency. The characteristic spatial wavelength of the standing-wave pattern is selected through the dispersion relation ω(k) of the fluid. One typically observes patterns of stripes or squares in such experiments. It is only in recent years that a variety of additional patterns -some with quasiperiodic rather than periodic long range order -have been observed [2][3][4][5][6]. We shall focus here on a particular set of experiments, performed by Edwards and Fauve [3], in which a fluid was driven by a linear combination of two frequencies, forming periodic patterns with 2-, 4-, and 6-fold symmetry, and quasiperiodic patterns with 12-fold symmetry.Previous theoretical work [6][7][8][9][10][11] has focused mainly on a description through amplitude equations with an angledependent interaction β(θ ij ) between pairs of modes. Such an interaction, which is either postulated or derived from the underlying microscopic dynamics, can be chosen to stabilize N -fold symmetric patterns for arbitrary N . Müller [10] has also used a set of two coupled partial differential equations, where the pattern of a primary field is stabilized by coupling to a secondary field which provides an effective space-dependent forcing. Newell and Pomeau [11] have coupled multiple fields in a similar way. In both cases the coupling between the different fields is achieved through resonant triad interactions, similar to the interactions we shall introduce below.We propose a simple rotationally-invariant modelequation, governing the dynamics of a real field u(x, y, t), which describes the amplitude of the standing-wave pattern. Our approach is different in that it searches for the minimal requirements for reproducing the steady states, which are observed in the experiments of Edwards and Fauve [3]. We incorporate into our model only the two most essential aspects of the system: 1. The dynamics is damped at frequencies away from the two forcing frequencies, and therefore the wavelengths involved in the selected pattern lie in narrow bands about two critical wavelengths.2. The dri...

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In the bistable regime of the FitzHugh-Nagumo model of reaction-diffusion systems, spatially homogeneous patterns may be nonlinearly unstable to the formation of compact "localized states." The formation of space-filling patterns from instabilities of such structures is studied in the context of a nonlocal contour dynamics model for the evolution of boundaries between high and low concentrations of the activator. An earlier heuristic derivation [D.M. Petrich and R.E. Goldstein, Phys. Rev. Lett. 72, 1120Lett. 72, (1994] is made more systematic by an asymptotic analysis appropriate to the limits of fast inhibition, sharp activator interfaces and small asymmetry in the bistable minima. The resulting contour dynamics is temporally local, with the normal component of the velocity involving a local contribution linear in the interface curvature and a nonlocal component having the form of a screened Biot-Savart interaction. The amplitude of the nonlocal interaction is set by the activator-inhibitor coupling and controls the "lateral inhibition" responsible for the destabilization of localized structures such as spots and stripes, and the repulsion of nearby interfaces in the later stages of those instabilities. The phenomenology of pattern formation exhibited by the contour dynamics is consistent with that seen by Lee, McCormick, Ouyang, and Swinney in experiments on the iodide-ferrocyanide-sulfite reaction in a gel reactor. Extensive numerical studies of the underlying partial differential equations are presented and compared in detail with the contour dynamics. The similarity of these phenomena (and their mathematical description) with those observed in amphiphilic monolayers, Type-I superconductors in the intermediate state, and magnetic fluids in Hele-Shaw geometry are emphasized.

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