Spatiotemporal patterns and diffusion-induced chaos in a chemical system with equal diffusion coefficients

Arnéodo, A.; Elezgaray, Juan

doi:10.1016/0375-9601(90)90792-m

Cited by 24 publications

(6 citation statements)

References 23 publications

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“…In condensed phase combustion, the development of relaxation oscillations does not inhibit the onset of chaotic regimes (Bayliss & Matkowsky 1990). The destabilizing effects of unequal fixed boundary conditions in reaction-diffusion systems has recently been explored (Arneodo & Elezgaray 1990). The destabilizing effects of unequal fixed boundary conditions in reaction-diffusion systems has recently been explored (Arneodo & Elezgaray 1990).…”

Section: Discussionmentioning

confidence: 99%

Asymptotic and numerical study of Brusselator chaos

1991

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We investigate the Brusselator reaction-diffusion equations with periodic boundary conditions. We consider the range of values of the parameters used by Kuramoto in his study of chaotic concentration waves. We determine numerically the bifurcation diagram of the long-time travelling and standing wave solutions using a highly accurate Fourier pseudo-spectral method.For moderate values of the bifurcation parameter, we have found a sequence of instabilities leading either to periodic and quasiperiodic standing waves, or to chaotic regimes. However, for large values of the control parameter, we have found only uniform time-periodic solutions or time-periodic travelling wave solutions. Our numerical study motivates a new asymptotic analysis of the Brusselator equations for large values of the control parameter and small diffusion coefficients. This analysis explains the numerical predictions. The chaotic regime is limited to moderate values of the control parameter and periodic solutions are the only solutions for large values of the control parameter. We identify the stabilizing mechanism as the relaxation oscillations which appear when the control parameter is large. Our asymptotic result on the stability of periodic solutions is then generalized to a class of two-variable reaction-diffusion equations.

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Section: Discussionmentioning

confidence: 99%

Asymptotic and numerical study of Brusselator chaos

1991

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“…With the slow manifolds (7), we have found conditions where the oscillating single-front pattern undergoes secondary instabilities leading to more complicated spatia-temporal behavior. 85 In Fig. 5(a), we show a chaotically oscillating front structure computed with 0 = 10 -2 in Eq.…”

Section: Diffusion-induced Spatio-temporal Chaosmentioning

confidence: 98%

“…In the present numerical study, we mainly focus on the early bifurcations of front patterns observed on the way to this "extended system" limit and discuss the existence of up to three-front pattern solutions. 85 The numerical patterns shown in Figs. 2 and 3 have been obtained with the following form of the slow manifold: feu) = u 2 -u 3 + us.…”

Section: Asymmetric Feedingmentioning

confidence: 99%

Modeling reaction–diffusion pattern formation in the Couette flow reactor

Elezgaray

Arnéodo

1991

The Journal of Chemical Physics

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Slow passage through a supercritical Hopf bifurcation: Timedelayed response in the Belousov-Zhabotinsky reaction in a batch reactorWe report on a numerical and theoretical study of spatio-temporal pattern forming phenomena in a one-dimensional reaction-diffusion system with equal diffusion coefficients. When imposing a concentration gradient through the system, this model mimics the sustained stationary and periodically oscillating "front structures" observed in a recent experiment conducted in the Couette flow reactor. Conditions are also found under which oscillations of the nontrivial spatial patterns become chaotic. Singular perturbation techniques are used to study the existence and the linear stability of single-front and multi-front patterns. A nonlinear analysis of bifurcating patterns is carried out using a center manifold/normal form approach. The theoretical predictions of the normal form calculations are found in quantitative agreement with direct simulations of the Hopfbifurcation from steady to oscillating front patterns. The remarkable feature of these sustained spatio-temporal phenomena is the fact that they organize due to the interaction of the diffusion process with a chemical reaction which itself would proceed in a stationary manner if diffusion was negligible. This study clearly demonstrates that complex spatia-temporal patterns do not necessarily result from the coupling of oscillators or nonlinear transport.

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“…These equations are an extension of the FitzHugh-Nagumo equations [FitzHugh, 1962;Nagumo et al, 1962] and have been chosen to model chemical and physical systems [Purwins et al, 1988;Arneodo & Elezgaray, 1990]. Different parts of the domain are set to different homogeneous states, the interfaces between these homogeneous regions will be smoothed in the course of time due to diffusion and so-called trigger-fronts are formed.…”

Section: Reaction-diffusion Models Of Voronoi Diagram Constructionmentioning

confidence: 99%

The Formation of Voronoi Diagrams in Chemical and Physical Systems: Experimental Findings and Theoretical Models

Costello

Ratcliffe

Adamatzky

et al. 2004

Int. J. Bifurcation Chaos

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The work discusses the formation of Voronoi diagrams in spatially extended nonlinear systems taking experimental and theoretical results into account. Concerning experimental systems a number of chemical systems used previously as prototype chemical processors and a barrier gas-discharge system are investigated. Although the underlying microscopic processes are very different, both types of systems show self-organized Voronoi diagrams for suitable parameters. Indeed certain chemical systems exhibit Voronoi diagrams as an output state for two distinct sets of parameters one that corresponds to the interaction of stable forced trigger waves and the other that corresponds to the spontaneous initiation and interaction of waves due to point instabilities in the system. In the case of the chemical systems front initiation, propagation and interaction (annihilation) are the primary mechanisms for Voronoi diagram formation, in the case of the barrier gas-discharge system regions of vanishing electric field define the medial axes of the Voronoi diagram. On the basis of cellular automata models the general concept of the formation of Voronoi diagrams has been explained, and related mechanisms have been simulated. Another intuitive approach towards the understanding of self-organized Voronoi diagrams has been given on the basis of reaction-diffusion models explaining the formation of Voronoi diagrams as a result of the mutual interactions of trigger fronts. The variety of systems exhibiting Voronoi diagrams as stationary states indicates that Voronoi diagrams are a generic and natural pattern formation phenomenon.

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Spatiotemporal patterns and diffusion-induced chaos in a chemical system with equal diffusion coefficients

Cited by 24 publications

References 23 publications

Asymptotic and numerical study of Brusselator chaos

Asymptotic and numerical study of Brusselator chaos

Modeling reaction–diffusion pattern formation in the Couette flow reactor

The Formation of Voronoi Diagrams in Chemical and Physical Systems: Experimental Findings and Theoretical Models

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