We review experimental and theoretical work addressing pattern formation on anisotropic and heterogeneous catalytic surfaces. These systems are typically modeled by reaction-diffusion equations reflecting the kinetics and transport of the involved chemical species. Here, we demonstrate the influence of anisotropy and heterogeneity in a simplified model, the FitzHugh-Nagumo equations. Anisotropy causes stratification of labyrinthine patterns and spiral defect chaos in bistable media. For heterogeneous media, we study the situation where the heterogeneity appears on a length scale shorter than the typical pattern length scale. Homogenization, i.e., computation of effective medium properties, is applied to an example and illustrated with simulations in one (fronts) and two dimensions (spirals). We conclude with a discussion of open questions and promising directions that comprise the coupling of the microscopic structure of the surface to the macroscopic concentration patterns and the fabrication of nanostructures with heterogeneous surfaces as templates. (c) 2002 American Institute of Physics.

An extension of the complex Ginzburg-Landau equation describing resonant spatio-temporal forcing of oscillatory media is investigated. Periodic forcing in space and time leads to spatial structures with two different symmetries: harmonic patterns with the same and subharmonic patterns with twice the wavelength of the external forcing. A linear stability analysis of the homogeneous state carried out analytically leads to subharmonic patterns for intermediate forcing strength, while harmonic modes prevail for very weak and strong forcing amplitudes. Numerical simulations confirm the analytical predictions for weak forcing and show coexistence between the two types of patterns beyond threshold. In addition, traveling localized patterns such as phase flips in subharmonic patterns and traveling patches of subharmonic patterns in a harmonic background have been discovered. In the parameter range of Benjamin-Feir turbulence, stable subharmonic patterns occur upon forcing, which undergo a transition scenario back to irregular dynamics for increasing values of the control parameter.

Statistical topic models such as the Latent Dirichlet Allocation (LDA)