Guy Dewel

Absolute and convective instabilities in a one-dimensional Brusselator flow model

The paper considers a one-dimensional Brusselator model with a uniform flow of the mixture of reaction components. An absolute as well as a convective instability can arise for both the Hopf and the Turing modes. The corresponding linear stability analysis is presented and supported by the results of computer simulations of the nonlinear equations. Finally, the condition for spatially undamped tails (the Cherenkov condition) is obtained. This represents a new mechanism for pattern formation in chemical reaction-diffusion systems.

Competition in ramped Turing structures

Abstract Stationary pattern selection and competition in the uniform Brusselator in two (2D) and three (3D) dimensions are reviewed, including reentrant hexagonal and striped zig-zag phases. Influences of linear or chain-like profiles of the pool chemicals on this selection are presented in the form of numerical experiments. The relation with the recent experimental patterns obtained with the CIMA reaction is discussed.

Pattern formation in the bistable Gray-Scott model

This paper presents a computer simulation study of a variety of far-from-equilibrium phenomena that can arise in a bistable chemical reaction-diffusion system which also displays Turing and Hopf instabilities. The Turing bifurcation curve and the wave number for the patterns of maximum linear growth rate are obtained from a linear stability analysis. The distribution in parameter space of a wide variety of different spatio-temporal attractors that can be reached through a strong, local perturbation of the linearly stable homogeneous steady state is mapped out. These include global Turing structures, stable localized structures, interacting fronts, mixed Turing-Hopf modes, and spatio-temporal chaos. Special emphasis is given to the newly discovered spot multiplication process in which cell-like structures replicate themselves until they occupy the entire system. We also present results on the formation of lace-like patterns.

Three-dimensional dissipative structures in reaction-diffusion systems

Abstract We study analytically and numerically the tridimensional pattern selection problem for reaction-diffusion systems. Qualitative agreement is found with the recent experimental results.

Modeling of the kinetic oscillations in the CO oxidation on Pt(100)

We analyze a model recently introduced by Imbihl et al. to describe the kinetic oscillations in the catalytic oxidation of CO on Pt(100). However we describe the surface reconstruction by a time dependent Ginzburg–Landau equation. With realistic values of the rate constants the region of oscillations in the PCO×PO2 diagram extends to very low values of the partial pressures (PCO=1.5×10−6 Torr, PO2=2.7×10−5 Torr) in agreement with the experimental results. In the investigation of the critical points and of the dynamics it has been possible to identify both saddle‐node infinite period (SNIPER) and Hopf bifurcations.

DIFFUSIVE INSTABILITIES AND CHEMICAL REACTIONS

Diffusive instabilities provide the engine for an ever increasing number of dissipative structures. In this class autocatalytic chemical systems are prone to generate temporal and spatial self-organization phenomena. The development of open spatial reactors and the subsequent discovery in 1989 of the stationary reaction–diffusion patterns predicted by Turing [1952] have triggered a large amount of research. This review aims at a comparison between theoretical predictions and experimental results obtained with various type of reactors in use. The differences arising from the use of reactions exhibiting either bistability of homogeneous steady states or a single one in a CSTR are emphasized.

Subcritical transitions to Turing structures

Abstract The existence of localized structures is discussed within the framework of the pattern selection problem for a model for the chlorine dioxide-iodine-malonic acid reaction that represents a key to the understanding of the recently obtained Turing structures.

Reentrant hexagonal Turing structures

Abstract A new structural transition from stripes to hexagons is reported for the two-dimensional Brusselator model. Its existence is explained by the properties of the quadratic nonlinear coupling.

Pattern selection and localized structures in reaction-diffusion systems

We present some theoretical concepts that have been used in the study of chemical disspative structures together with a brief description of recent experimental work on Turing patterns and their interaction with travelling waves.

Turing Bifurcations and Pattern Selection

Pattern forming instabilities in spatially extended dissipative systems driven away from equilibrium have been the focus of a large activity for many years. The goal of this chapter is to present some theoretical concepts that have been developed to understand and describe these dissipative structures [1] from a macroscopic point of view. Although these methods present generic features we shall only be concerned with chemical patterning and shall not discuss here instabilities in hydrodynamics, liquid crystals and nonlinear optics that all present similar types of organization because the latter have been the subject of recent well-documented reviews [2–5]. Moreover, we essentially consider the self-organization of structures discarding the spatial patterning resulting from boundary conditions.