Traveling-wave fragments in anisotropic excitable media

Mertens, F.; Gottschalk, Nathalie; Bär, Markus; Eiswirth, M.; Mikhailov, Alexander S.; Imbihl, R.

doi:10.1103/physreve.51.r5193

Cited by 38 publications

(30 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…3 Experimental and theoretical studies of such anisotropic systems showed novel phenomena like ordered arrays of topological defects, 4 anisotropic phase turbulence, 5 reaction-diffusion waves with sharp corners, [6][7][8] as well as traveling wave fragments along a preferred orientation. 9 Anisotropy is also often present in pattern formation processes in biological media, e.g., in cardiac tissue. 10,11 In reaction-diffusion systems, anisotropy usually enters via the diffusion constants.…”

Section: Diffusion Anisotropymentioning

confidence: 99%

Pattern formation on anisotropic and heterogeneous catalytic surfaces

Bär

Meron

Utzny

2002

Chaos: An Interdisciplinary Journal of Nonlinear Science

View full text Add to dashboard Cite

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.

show abstract

Section: Diffusion Anisotropymentioning

confidence: 99%

Pattern formation on anisotropic and heterogeneous catalytic surfaces

Bär

Meron

Utzny

2002

Chaos: An Interdisciplinary Journal of Nonlinear Science

View full text Add to dashboard Cite

show abstract

“…One of these approaches has been based on an ad hoc replacement of some of the system parameters (for example, the diffusion constant, or coupling constants) with spatially varying parameters [21,22,23,24,25,26,27] or species dependent parameters [28]. In another phenomenological approach, nonlinearity has been introduced into the diffusion term by taking the Laplacian of a power of the dependent variable [29,30,31], i.e., ∇u m where m = 1 is standard diffusion, m > 1 is interpreted as slow diffusion, and m < 1 is interpreted as fast diffusion.…”

Section: Introductionmentioning

confidence: 99%

Existence of Turing Instabilities in a Two-Species Fractional Reaction-Diffusion System

Henry¹,

Wearne²

2002

SIAM J. Appl. Math.

186

121

View full text Add to dashboard Cite

Abstract. We introduce a two-species fractional reaction-diffusion system to model activatorinhibitor dynamics with anomalous diffusion such as occurs in spatially inhomogeneous media. Conditions are derived for Turing-instability induced pattern formation in these fractional activatorinhibitor systems whereby the homogeneous steady state solution is stable in the absence of diffusion but becomes unstable over a range of wavenumbers when fractional diffusion is present. The conditions are applied to a variant of the Gierer-Meinhardt reaction kinetics which has been generalized to incorporate anomalous diffusion in one or both of the activator and inhibitor variables. The anomalous diffusion extends the range of diffusion coefficients over which Turing patterns can occur. An intriguing possibility suggested by this analysis, which can arise when the diffusion of the activator is anomalous but the diffusion of the inhibitor is regular, is that Turing instabilities can exist even when the diffusion coefficient of the activator exceeds that of the inhibitor.

show abstract

“…For example, such structures are observed in planar gas-discharge systems with a high ohmic barrier (Fig. 1a) [1][2][3], granular materials [4] and optical systems [5,6], in modified Belousov-Zhabotinsky reactions [7][8][9][10][11], in catalytic reduction of NO with hydrogen on a Rh(110) surface [12] and oxidation of CO on Pt(110) (Fig. 1b) [13].…”

Section: Introductionmentioning

confidence: 96%