Sustained patterns in chlorite–iodide reactions in a one‐dimensional reactor

Temporally quasiperiodic data, propagating in the laboratory frame, can be rendered periodic by Galilean transformation

For a broad class of distributions of temperature, concentration, or another quantity propagating rectilinearly, we show that temporally quasiperiodic behavior in the laboratory frame can be rendered periodic by Galilean transformation. The approach is illustrated analytically and numerically using as an example a closed-form model distribution generated from a one-dimensional partial differential equation, and a detailed process is developed to determine frame speed from more general quasiperiodic, one-dimensional, temporally- and spatially-discretized data. The approach is extended to two- and three-dimensional rectilinear propagation, and its application to nonrectilinear propagation, along with implications for interpreting noise-corrupted data, are also discussed.

Chemical computing based on Turing patterns in two coupled cells with equal transport coefficients

Two diffusively coupled reaction cells with a nonlinear reaction are used to perform chemical computing based on targeted perturbations switching between two Turing patterns defining two states of a logical device.

Solitary Marangoni-driven convective structures in bistable chemical systems

Bistable chemical fronts can be deformed by Marangoni-driven convective flows induced by gradients of surface tension across the front. We investigate here the nonlinear dynamics of such a system by simulations of two-dimensional Navier-Stokes equations coupled to a reaction-diffusion-convection equation for a surface-active chemical species present in the bulk of the solution and involved in a bistable kinetics. We show that Marangoni flows cannot only alter the shape and speed of the front but also change the relative stability of the two stable steady states, reversing in some cases the direction of propagation of the front with regard to the pure reaction-diffusion situation. A detailed parametric study discusses the properties of the asymptotic dynamics as a function of the Marangoni number M and of a kinetic parameter d.

Spatial bistability: a source of complex dynamics. From spatiotemporal reaction-diffusion patterns to chemomechanical structures

We show experimentally and theoretically that reaction systems characterized by a slow induction period followed by a fast evolution to equilibrium can readily generate "spatial bistability" when operated in thin gel reactors diffusively fed from one side. This phenomenon which corresponds to the coexistence of two different stable steady states, not breaking the symmetry of the boundary conditions, can be at the origin of diverse reaction-diffusion instabilities. Using different chemical reactions, we show how stationary pulses, labyrinthine patterns or spatiotemporal oscillations can be generated. Beyond simple reaction-diffusion instabilities, we also demonstrate that the cross coupling of spatial bistability with the size responsiveness of a chemosensitive gel can give rise to autonomous spatiotemporal shape patterns, referred to as chemomechanical structures.

Introduction: Self-organization in nonequilibrium chemical systems

The field of self-organization in nonequilibrium chemical systems comprises the study of dynamical phenomena in chemically reacting systems far from equilibrium. Systematic exploration of this area began with investigations of the temporal behavior of the Belousov-Zhabotinsky oscillating reaction, discovered accidentally in the former Soviet Union in the 1950s. The field soon advanced into chemical waves in excitable media and propagating fronts. With the systematic design of oscillating reactions in the 1980s and the discovery of Turing patterns in the 1990s, the scope of these studies expanded dramatically. The articles in this Focus Issue provide an overview of the development and current state of the field.

Noise-covered drift bifurcation of dissipative solitons in a planar gas-discharge system

The trajectories of propagating self-organized, well-localized solitary patterns (dissipative solitons) in the form of electrical current filaments are experimentally investigated in a planar quasi-two-dimensional dc gas-discharge system with high Ohmic semiconductor barrier. Earlier phenomenological models qualitatively describing the experimental observations in terms of a particle model predict a transition from stationary filaments to filaments traveling with constant finite speed due to an appropriate change of the system parameters. This prediction motivates a search for a drift bifurcation in the experimental system, but a direct comparison of experimentally recorded trajectories with theoretical predictions is impossible due to the strong influence of noise. To solve this problem, the filament dynamics is modeled using an appropriate Langevin equation, allowing for the application of a stochastic data analysis technique to separate deterministic and stochastic parts of the dynamics. Simulations carried out with the particle model demonstrate the efficiency of the method. Applying the technique to the experimentally recorded trajectories yields good agreement with the predictions of the model equations. Finally, the predicted drift bifurcation is found using the semiconductor resistivity as control parameter. In the resulting bifurcation diagram, the square of the equilibrium velocity scales linearly with the control parameter.

Dynamical effects induced by long range activation in a nonequilibrium reaction-diffusion system

We show both experimentally and numerically that the time scales separation introduced by long range activation can induce oscillations and excitability in nonequilibrium reaction-diffusion systems that would otherwise only exhibit bistability. Namely, we show that in the chlorite-tetrathionate reaction, where the autocatalytic species H+ diffuses faster than the substrates, the spatial bistability domain in the nonequilibrium phase diagram is extended with oscillatory and excitability domains. A simple model and a more realistic model qualitatively account for the observed dynamical behavior. The latter model provides quantitative agreement with the experiments.

Two-dimensional model of a reaction-diffusion system as a typewriter

Pattern formation is a common phenomenon, which appears in biological systems, especially in cell differentiation processes. The proper level for understanding the creation of patterns seems to be a physicochemical description. The most fundamental models should be based on systems, in which only chemical reactions and diffusion transport occur (reaction-diffusion systems). In order to present a richness of patterns, we show here the asymptotic patterns in the form of capital letters obtained in two-dimensional reaction-diffusion systems with zero-flux boundary conditions. All capital letters are obtained in the same model, but initial conditions and sizes of the systems are different for each letter. The chemical model consists of elementary reactions and is realistic. It can be realized experimentally in continuous-flow unstirred reactor with an enzymatic reaction allosterically inhibited by an excess of its reactant and product.

Coexistence of large amplitude stationary structures in a model of reaction-diffusion system

The two-variable reaction-diffusion model of a chemical system describing the spatiotemporal evolution to large amplitude stationary periodical structures in a one-dimensional open, continuous-flow, unstirred reactor is investigated. Numerical solutions show that the structures are generated by divisions of the traveling impulse and its stopping at the boundary of the system. Analyses of projections of numerical solutions on the phase plane of two variables elaborated in the present paper allow qualitative explanation of the results. The coexistence of the large amplitude stationary periodical structures is shown. A number of coexisting structures grows strongly with increasing length of the reactor and may be as large as one wishes. The relationship of these results to biological systems is stressed.

Spatial multistability and nonvariational effects

We investigate the phenomenon of spatial multistability of fronts in thin bistable systems and stress the important role played by the absence of a variational principle. Nonvariational effects allow, for instance, two different immobilized fronts to coexist. The morphological instability of the corresponding nucleating solution can then lead, even in the absence of any diffusive instability, to nontrivial patterns in the depth of one-side-fed reactors.

On the global orbits in a bistable CML

In an infinite one-dimensional coupled map lattice (CML) for which the local map is piecewise affine and bistable, we study the global orbits using a spatiotemporal coding introduced in a previous work. The set of all the fixed points is first considered. It is shown that, under some restrictions on the parameters, the latter is a Cantor set, and we introduce an order to study the fixed points' existence. This also involves the proof of the coexistence of propagating fronts and stationary structures. In the second part, we analyze the global orbits which occur for strong coupling using the splitting of the dynamics into two independent (sub-)lattices, and emphasize the description of various traveling structures. (c) 1997 American Institute of Physics.

Chaos and intermittent bursting in a reaction-diffusion process

Karhunen-Loeve decomposition is done on a chaotic spatio-temporal solution obtained from a nonlinear reaction-diffusion model of a chemical system simulating a chemical process in an open Couette-flow reactor. Using a Galerkin projection of the dominant Karhunen-Loeve modes back onto the nonlinear partial differential system, we obtain an ordinary differential equation model of the same process. Major features such as intermittent and chaotic bursting of the nonlinear process as well as the mechanism of transition to chaos are shown to exist in the low-dimensional model as well as the PDE model. From the low-dimensional model the onset of intermittent bursts followed by small amplitude oscillations is shown to arise due to a sequence of saddle-node bifurcations.

Transition to chemical turbulence

Experiments have been conducted on Turing-type chemical spatial patterns and their variants in a quasi-two-dimensional open spatial reactor with a chlorite-iodide-malonic acid reaction. A variety of stationary spatial structures-hexagons, stripes, and mixed states-were observed, and transitions to these states were studied. For conditions beyond those corresponding to the emergence of patterns, a transition was observed from stationary spatial patterns to chemical turbulence, which is marked by a continuous motion of the pattern within a domain and of the grain boundaries between domains. The transition to chemical turbulence was analyzed by measuring the correlation length, the average pattern speed, and the total length of the domain boundaries. The emergence of chemical turbulence is accompanied by a large increase in the defects in the pattern, which suggests that this is an example of defect-mediated turbulence.