Interaction of Turing and Hopf modes in the superdiffusive Brusselator model near a codimension two bifurcation point

J. C. Tzou, Alvin Bayliss, Bernard J Matkowsky*, Vladimir Volpert

*Corresponding author for this work

Research output: Contribution to journalArticle

11 Citations (Scopus)

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.

Original languageEnglish (US)
Pages (from-to)87-118
Number of pages32
JournalMathematical Modelling of Natural Phenomena
Volume6
Issue number1
DOIs
StatePublished - Jan 1 2011

Fingerprint

Turing
Bifurcation Point
Codimension
Superdiffusion
Interaction
Amplitude Equations
Spatio-temporal Patterns
Anomalous Diffusion
Nonlinear analysis
Diffusion Model
Nonlinear Analysis
Integro-differential Operators
Three Solutions
Model
Fourier Method
Motion
Mixed Mode
Fully Nonlinear
Stability criteria
Runge-Kutta

Keywords

  • Brusselator model
  • Hopf bifurcation
  • amplitude equations
  • codimension-2 bifurcation
  • spatiotemporal patterns
  • superdiffusion
  • turing pattern
  • weakly nonlinear analysis

ASJC Scopus subject areas

  • Modeling and Simulation

Cite this

@article{a8a709cf4f3041409fb366f60c24fa8e,
title = "Interaction of Turing and Hopf modes in the superdiffusive Brusselator model near a codimension two bifurcation point",
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 model, Hopf bifurcation, amplitude equations, codimension-2 bifurcation, spatiotemporal patterns, superdiffusion, turing pattern, weakly nonlinear analysis",
author = "Tzou, {J. C.} and Alvin Bayliss and Matkowsky, {Bernard J} and Vladimir Volpert",
year = "2011",
month = "1",
day = "1",
doi = "10.1051/mmnp/20116105",
language = "English (US)",
volume = "6",
pages = "87--118",
journal = "Mathematical Modelling of Natural Phenomena",
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}

TY - JOUR

T1 - Interaction of Turing and Hopf modes in the superdiffusive Brusselator model near a codimension two bifurcation point

AU - Tzou, J. C.

AU - Bayliss, Alvin

AU - Matkowsky, Bernard J

AU - Volpert, Vladimir

PY - 2011/1/1

Y1 - 2011/1/1

N2 - 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.

AB - 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.

KW - Brusselator model

KW - Hopf bifurcation

KW - amplitude equations

KW - codimension-2 bifurcation

KW - spatiotemporal patterns

KW - superdiffusion

KW - turing pattern

KW - weakly nonlinear analysis

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U2 - 10.1051/mmnp/20116105

DO - 10.1051/mmnp/20116105

M3 - Article

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JO - Mathematical Modelling of Natural Phenomena

JF - Mathematical Modelling of Natural Phenomena

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