Hostname: page-component-68945f75b7-zpsnj Total loading time: 0 Render date: 2024-08-05T17:48:20.537Z Has data issue: false hasContentIssue false
  • English
  • Français

Interaction of Turing and Hopf Modes in the Superdiffusive Brusselator Model Near a Codimension Two Bifurcation Point

Published online by Cambridge University Press:  09 June 2010

J. C. Tzou ,
A. Bayliss ,
B.J. Matkowsky  and
V.A. Volpert
Get access

Abstract

Spatiotemporal patterns near a codimension-2 Turing-Hopf point of the one-dimensional superdiffusive Brusselator model are analyzed. The superdiffusive Brusselator model differs from its regular counterpart in that the Laplacian operator of the regular model is replaced by ∂α/∂|ξ|α, 1 < α < 2, an integro-differential operator that reflects the nonlocal behavior of superdiffusion. The order of the operator, α, is a measure of the rate of superdiffusion, which, in general, can be different for each of the two components. A weakly nonlinear analysis is used to derive two coupled amplitude equations describing the slow time evolution of the Turing and Hopf modes. We seek special solutions of the amplitude equations, namely a pure Turing solution, a pure Hopf solution, and a mixed mode solution, and analyze their stability to long-wave perturbations. We find that the stability criteria of all three solutions depend greatly on the rates of superdiffusion of the two components. In addition, the stability properties of the solutions to the anomalous diffusion model are different from those of the regular diffusion model. Numerical computations in a large spatial domain, using Fourier spectral methods in space and second order Runge-Kutta in time are used to confirm the analysis and also to find solutions not predicted by the weakly nonlinear analysis, in the fully nonlinear regime. Specifically, we find a large number of steady state patterns consisting of a localized region or regions of stationary stripes in a background of time periodic cellular motion, as well as patterns with a localized region or regions of time periodic cells in a background of stationary stripes. Each such pattern lies on a branch of such solutions, is stable and corresponds to a different initial condition. The patterns correspond to the phenomenon of pinning of the front between the stripes and the time periodic cellular motion. While in some cases it is difficult to isolate the effect of the diffusion exponents, we find characteristics in the spatiotemporal patterns for anomalous diffusion that we have not found for regular (Fickian) diffusion.

Keywords

Brusselator modelsuperdiffusionTuring patternHopf bifurcationcodimension-2 bifurcationweakly nonlinear analysisamplitude equationsspatiotemporal patterns
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 6 , Issue 1: Instability and patterns. Issue dedicated to the memory of A. Golovin , 2011 , pp. 87 - 118
Copyright
© EDP Sciences, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This paper is dedicated to the memory of our colleague and friend Alexander (Sasha) Golovin

References

Assemat, P., Bergeon, A., Knobloch, E.. Spatially localized states in Marangoni convection in binary mixtures . Fluid Dynamics Research 40 (2008), 852-876.CrossRefGoogle Scholar
Bensimon, D., Shraiman, B.I., Croquette, V.. Nonadiabatic effects in convection . Phys. Rev. A38 (1988), 5461-5464.CrossRefGoogle Scholar
Burke, J., Knobloch, E.. Localized states in the generalized Swift-Hohenberg equation . Phys. Rev. E73 (2006), art. 56211. Google Scholar
Burke, J., Knobloch, E.. Homoclinic snaking: structure and stability Chaos 17 (2007), art. 037102. CrossRefGoogle ScholarPubMed
Crampin, E.J., Gaffney, E.A., Maini, P.K.. Reaction and diffusion on growing domains: Scenarios for robust pattern formation . Bull. Math. Biol., 61 (1999), 1093-1120.CrossRefGoogle ScholarPubMed
De Wit, A.. Spatial patterns and spatiotemporal dynamics in chemical systems . Adv. Chem. Phys., 109 (1999), 435–513.Google Scholar
De Wit, A., Lima, D., Dewel, G., Borckmans, P.. Spatiotemporal dynamics near a codimension-two point . Phys. Rev. E54 (1996), 261–271.Google Scholar
Golovin, A.A., Matkowsky, B.J., Volpert, V.A.. Turing pattern formation in the Brusselator model with superdiffusion . SIAM J. Appl. Math., 69 (2008), 251–272.CrossRefGoogle Scholar
Malomed, B.A., Nepomnyashchy, A.A., Tribelsky, M.I.. Domain boundaries in convection patterns . Phys. Rev. A42 (1990), 7244-7263.CrossRefGoogle Scholar
Metzler, R., Klafter, J.. The random walk’s guide to anomalous diffusion: a fractional dynamics approach . Phys. Rep. 339 (2000), No. 1, 1–77.CrossRefGoogle Scholar
Nec, Y., Nepomnyashchy, A.A., Golovin, A.A.. Oscillatory instability in super-diffusive reaction-diffusion systems: Fractional amplitude and phase diffusion equations . EPL, 82 (2008), 58003.CrossRefGoogle Scholar
K.B. Oldham, J. Spanier. The fractional calculus. Academic Press, New York, 1974.
Pomeau, Y.. Front motion, metastability and subcritical bifurcations in hydrodynamics . Physica D23 (1986), 3-11.Google Scholar
Tzou, J.C., Matkowsky, B.J., Volpert, V.A.. Interaction of Turing and Hopf modes in the superdiffusive Brusselator model . Appl. Math. Lett. 22 (2009), 1432–1437.CrossRefGoogle Scholar
Yan, L., Dolnik, M., Zhabotinsky, A.M., Epstein, I.R.. Pattern formation arising from interaction between Turing and wave instabilities . J. Chem. Phys., 117 (2002), 7259–7265.Google Scholar