Stationary patterns of chemical concentration in the Belousov-Zhabotinskii reaction

doi:10.1016/0167-2789(90)90022-H

Physica D: Nonlinear Phenomena

Volume 43, Issue 1, May 1990, Pages 140-152

https://doi.org/10.1016/0167-2789(90)90022-H Get rights and content

Abstract

The Belousov-Zhabotinskii (BZ) reaction is a complicated chemical reaction which has the property that, instead of progressing smoothly from reactants to products or perhaps reaching equilibrium, it may instead exhibit periodic oscillations in the concentrations of certain intermediate products. In addition, spatially periodic patterns may occur when diffusion of the chemical species is taken into account. This paper investigates these patterns in the one-dimensional case in which a time-independent stripy pattern would be observed, corresponding to standing waves of chemical concentration being formed across the containing vessel. The Oregonator model of the reaction is assumed, and the resulting differential equations are solved using Fourier-series expansions. Each Fourier coefficient is then further expressed as a high-order perturbation series in an appropriate small parameter. The series are summed using various acceleration techniques, and are also analyzed to provide some insight into the mathematical structure of the solution. Large-amplitude concentration patterns have been generated in this way, and are discussed. The possibility of pattern generation when all three diffusion coefficients are equal is discussed in detail.

References (22)

J.D. Dockery et al.
Dispersion of traveling waves in the Belousov-Zhabotinskii reaction
Physica D
(1988)
J.A. Vastano et al.
Chemical pattern formation with equal diffusion coefficients
Phys. Lett. A
(1987)
A.M. Zhabotinsky et al.
Autowave processes in a distributed chemical system
J. Theor. Biol.
(1973)
P.K. Becker et al.
Stationary concentration patterns in the Oregonator model of the Belousov-Zhabotinskii reaction
J. Phys. Chem.
(1985)
C. Domb et al.
On the susceptibility of a ferromagnetic above the Curie point
R.J. Field
Limit cycle oscillations in the reversible Oregonator
J. Chem. Phys.
(1975)
R.J. Field et al.
Oscillations on chemical systems II. Thorough analysis of temporal oscillations in the bromate-cerium-malonic acid system
J. Am. Chem. Soc.
(1972)
R.J. Field et al.
Oscillations in chemical systems IV. Limit cycle behavior in a model of a real chemical reaction
J. Chem. Phys.
(1974)
L.K. Forbes
Periodic solutions of high accuracy to the forced Duffing equation: Perturbation series in the forcing amplitude
J. Austral. Math. Soc. Ser. B
(1987)
L.K. Forbes
A series analysis of forced transverse oscillations in a spring-mass system
SIAM J. Appl. Math.
(1989)

J. Grasman

Asymptotic Methods for Relaxation Oscillations and Applications

Cited by (3)

On stability and uniqueness of stationary one-dimensional patterns in the Belousov-Zhabotinsky reaction
1991, Physica D: Nonlinear Phenomena
In this paper, a numerical solution technique is presented for obtaining time-independent spatial patterns of chemical concentration in the Belousov-Zhabotinsky reaction. The Oregonator model of the reaction is assumed, and the concentrations of each of the three principal intermediate species are expressed as Fourier series in the spatial variable, and the coefficients are then obtained by a straightforward Galerkin technique. Results of extensive numerical calculation are discussed, and it is demonstrated that, in many instances, the spatial pattern obtained is not unique, in the sense that at least two different patterns could be produced at the same values of the chemical rate constants and diffusion coefficients.
The problem of whether the steady patterns are stable in time to small perturbations is discussed in detail. A criterion for stability of patterns of infinitesimal amplitude is established, and it is shown that, when all the diffusion coefficients are equal, there can be no stable small-amplitude solutions. However in the non-linear case, large-amplitude “quasi-stable” patterns have been discovered and these are discussed.
Comment on stationary patterns of chemical concentration in the Belousov-Zhabotinskii reaction
1990, Physica D: Nonlinear Phenomena
Solutions to the reaction-diffusion equations corresponding to the three-variable Oregonator model of the oscillatory Belousov-Zhabotinskii reaction were computed recently by Forbes using a series expansion method. In particular, solutions were obtained which correspond to stationary, relatively large-amplitude, spatial patterns in the concentrations of the three intermediate species assuming all their diffusion coefficients to be equal. We show that these solutions are unstable to numerical integration and so are not expected to be physically realizable. However, the increased rate at which the Forbes patterns decay when a critical parameter is perturbed slightly shows that there is some marginal stability associated with them.
Non-smooth feedback control for Belousov–Zhabotinskii reaction–diffusion equations: semi-analytical solutions
2016, Journal of Mathematical Chemistry

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Stationary patterns of chemical concentration in the Belousov-Zhabotinskii reaction

Abstract

Physica D

Phys. Lett. A

J. Theor. Biol.

Stationary concentration patterns in the Oregonator model of the Belousov-Zhabotinskii reaction

J. Phys. Chem.

On the susceptibility of a ferromagnetic above the Curie point

Limit cycle oscillations in the reversible Oregonator

J. Chem. Phys.

Oscillations on chemical systems II. Thorough analysis of temporal oscillations in the bromate-cerium-malonic acid system

J. Am. Chem. Soc.

Oscillations in chemical systems IV. Limit cycle behavior in a model of a real chemical reaction

J. Chem. Phys.

Periodic solutions of high accuracy to the forced Duffing equation: Perturbation series in the forcing amplitude

J. Austral. Math. Soc. Ser. B

A series analysis of forced transverse oscillations in a spring-mass system

SIAM J. Appl. Math.

Asymptotic Methods for Relaxation Oscillations and Applications