Two-variable Oregonator Model Research Articles

Information Processing Using Networks of Chemical Oscillators.

I believe the computing potential of systems with chemical reactions has not yet been fully explored. The most common approach to chemical computing is based on implementation of logic gates. However, it does not seem practical because the lifetime of such gates is short, and communication between gates requires precise adjustment. The maximum computational efficiency of a chemical medium is achieved if the information is processed in parallel by different parts of it. In this paper, I review the idea of computing with coupled chemical oscillators and give arguments for the efficiency of such an approach. I discuss how to input information and how to read out the result of network computation. I describe the idea of top-down optimization of computing networks. As an example, I consider a small network of three coupled chemical oscillators designed to differentiate the white from the red points of the Japanese flag. My results are based on computer simulations with the standard two-variable Oregonator model of the oscillatory Belousov–Zhabotinsky reaction. An optimized network of three interacting oscillators can recognize the color of a randomly selected point with >98% accuracy. The presented ideas can be helpful for the experimental realization of fully functional chemical computing networks.

Cross-diffusion in the two-variable Oregonator model

We explore the effect of cross-diffusion on pattern formation in the two-variable Oregonator model of the Belousov-Zhabotinsky reaction. For high negative cross-diffusion of the activator (the activator being attracted towards regions of increased inhibitor concentration) we find, depending on the values of the parameters, Turing patterns, standing waves, oscillatory Turing patterns, and quasi-standing waves. For the inhibitor, we find that positive cross-diffusion (the inhibitor being repelled by increasing concentrations of the activator) can induce Turing patterns, jumping waves and spatially modulated bulk oscillations. We qualitatively explain the formation of these patterns. With one model we can explain Turing patterns, standing waves and jumping waves, which previously was done with three different models.

Turing space in reaction-diffusion systems with density-dependent cross diffusion

Reaction-diffusion systems with cross-diffusion terms that depend linearly on density are studied via linear stability analysis and weakly nonlinear analysis. We obtain and analyze the conditions for the Turing instability and derive a universal form of these conditions. We discuss the features of the pattern-forming regions in parameter space for a cross activator-inhibitor system, the Brusselator model, and for a pure activator-inhibitor system, the two-variable Oregonator model. The supercritical or subcritical character of the Turing bifurcation for the Brusselator is determined by deriving an amplitude equation for patterns near the instability threshold.

Reduction waves in the two-variable Oregonator model for the BZ reaction

Numerical simulations of the two-variable Oregonator in a one-dimensional reaction–diffusion model are undertaken to show the formation of single reduction pulses. These are seen to exist over relatively narrow ranges of the (dimensionless) kinetic parameters ϵ and f arising in the derivation of the Oregonator model, though they are seen for all values of the diffusion coefficient ratio D considered. For the smaller values of D a direct transition from single reduction pulses to wave trains is found. For equal diffusion coefficients, D=1.0, this transition involves a sequence of complex spatio-temporal dynamics, including localized oscillatory behaviour and the successive spreading of a region in the reduced state. The present results are compared with results from the three-variable Oregonator and the Rovinsky–Zhabotinsky models for the BZ reaction, as well as with previous experimental observations.

Spatiotemporal chaos arising from standing waves in a reaction-diffusion system with cross-diffusion

We show that quasi-standing wave patterns appear in the two-variable Oregonator model of the Belousov-Zhabotinsky reaction when a cross-diffusion term is added, no wave instability is required in this case. These standing waves have a frequency that is half the frequency of bulk oscillations displayed in the absence of diffusive coupling. The standing wave patterns show a dependence on the systems size. Regular standing waves can be observed for small systems, when the system size is an integer multiple of half the wavelength. For intermediate sizes, irregular patterns are observed. For large sizes, the system shows an irregular state of spatiotemporal chaos, where standing waves drift, merge, and split, and also phase slips may occur.

On wave trains arising in the two-variable Oregonator model for the BZ reaction

The two-variable Oregonator model for the Belousov–Zhabotinsky reaction is considered over the parameter range when regular wave trains arise at large times. Numerical integrations of the corresponding initial-value problem in a spatially 1D system show the evolution of wave trains over a wide range of parameter values when the kinetics are unstable. An approach using the method of matched asymptotic expansions based on a kinetic parameter ϵ being small is derived and reveals a complex structure to the individual waves that make up the wave train.

Simulating strange attraction of acellular slime mould Physarum polycephaum to herbal tablets

Plasmodium of acellular slime mould Physarum polycephalum exhibits traits of wave-like behaviour. The plasmodium’s behaviour can be finely tuned in laboratory experiments by using herbal tablets. A single tablet acts as a fixed attractor: plasmodium propagates towards the tablet, envelops the tablet with its body and stays around the tablet for several days. Being presented with several tablets the plasmodium executes limit cycle like motions. The plasmodium performs sophisticated routines of movement around tablets: rotation, splitting, and annihilation. We use the two-variable Oregonator model to simulate the plasmodium behaviour in the presence of the herbal tablets. Numerical experiments confirm that using long-distance attracting and short-distance repelling fields we can organise arbitrary movement of plasmodia.

Travelling waves in the Oregonator model for the BZ reaction

Numerical solutions of the travelling wave equations that arise in the two-variable Oregonator model for the BZ reaction are obtained for a wide range of parameter values. The main feature is the saddle-node bifurcation in the solutions, giving upper bounds for the existence of travelling waves and indicating the change from the excitable to the subexcitable regimes. The values of the upper bound fm of the stoichiometry factor f in the model are determined in terms of the excitability parameter e for various values of the kinetic parameter q and D, the ratio of the diffusion coefficients of HBrO2 and the oxidized form of the catalyst.

Formation of Somitogenesis-like Pattern in a Reaction–Diffusion System

The Belousov–Zhabotinsky reaction system showing stationary patterns is realized on the basis of water-in-oil microemulsions with the surfactant sodium bis(2-ethylhexyl)sulfosuccinate. We experimentally demonstrate the formation of somitogenesis-like pattern in which chemical waves arising from spontaneous bulk oscillations are successively arrested and then stacked. The experimental results are qualitatively reproduced by numerical simulations using the two-variable oregonator model. These show that a somitogenesis can be accounted for by the genuine reaction–diffusion model.

Active media under rotational forcing

The bubble-free Belousov-Zhabotinsky reaction has been used to study the effects of centrifugal forces on autowave propagation. The reaction parameters were chosen such that the system oscillates naturally creating target waves. In the present study, the system was forced to rotate with a constant velocity around a central axis. In studying the effects of such a forcing on the system, we focused on target dynamics. The system reacts to this forcing in different ways, the most spectacular being a dramatic increase in the period of the target, the effect growing stronger as we move away from the center of rotation. A numerical study was carried out using the two-variable Oregonator model, modified to include convective effects through the diffusion coefficient. The numerical results showed a good qualitative agreement with those of the experiments.

Chemical Turbulence and Line Defects Induced by Gradient Effects in a Three-Dimensional Reaction−Diffusion System

We report experimental observations of chemical turbulence and line defects in a three-dimensional (3-D) Belousov-Zhabotinsky (BZ) reaction-diffusion system. Transitions from spiral waves to 3-D chemical turbulence to line defects are observed. These transitions are caused by concentration gradients across the third dimension in the 3-D reaction medium, indicating the observed line defects have a 3-D structure. The line defects come out of the 3-D turbulent state, and become smooth with the increase of the control parameter. Simulation with the two-variable Oregonator model in the 3-D system reproduces similar line defects.

Travelling waves in the Oregonator model for the BZ reaction

Numerical solutions of the travelling wave equations that arise in the two-variable Oregonator model for the BZ reaction are obtained for a wide range of parameter values. The main feature is the saddle-node bifurcation in the solutions, giving upper bounds for the existence of travelling waves and indicating the change from the excitable to the subexcitable regimes. The values of the upper bound fm of the stoichiometry factor f in the model are determined in terms of the excitability parameter e for various values of the kinetic parameter q and D, the ratio of the diffusion coefficients of HBrO2 and the oxidized form of the catalyst.

Kinetics of the wavetrain in the two-variable Oregonator model

In this paper we first discuss the asymptotic behaviours of solitary impulses in reaction-diffusion equations of Oregonator under the equal, single and double diffusion conditions, respectively. Then we obtain the asymptotic equations of motion of wavetrains for this model by a superposition of solitary impulses and finally we analyse the stability of wavetrains.

Flow-driven instabilities in the Belousov–Zhabotinsky reaction: Modelling and experiments

The development of propagating patterns arising from the differential flow of reactants through a tubular reactor is investigated. The results from a series of experimental runs, using the BZ reaction, are presented to show how the wavelength and propagation speed of the patterns depend on the imposed flow velocity and the concentration of BrO3− in the inflow. A model for this system, based on a two-variable Oregonator model for the BZ reaction, is considered. A stability analysis of the model indicates that the mechanism for pattern formation is through a convective instability. Numerical simulations confirm the existence of propagating patterns and are in reasonable agreement with the experimental observations.

Onset and Synchronization of Complex Dynamic Behavior in the Light-Sensitive Belousov−Zhabotinsky Reaction with Periodic and Nearly Periodic Switching

In this paper we studied the behavior of a model of a periodically driven photosensitive Belousov−Zhabotinsky reaction. The computations were carried out with a two-variable Oregonator model modified to account for photosensitivity. The external light intensity was periodically switched between two levels. By keeping the total cycle length fixed while varying the duration of the positive and negative perturbations, a variety of dynamical behaviors can be observed, including phase locking, torus oscillations, periodic transitions, and chaos. The results suggest that not only are the forcing frequency and amplitude important, as was already known, but the detailed waveform of the external forcing is also essential in determining the behavior of a driven dynamical system. Two scenarios have been investigated in this study: (a) the system was driven between two limit cycles; (b) the system was driven between excitable and oscillatory states. Random modulation of the durations of the positive and negative perturbations (low and high illumination states) leads to synchronization of complex behavior. Calculating the leading Lyapunov exponent confirms that this form of random driving may be used to suppress chaotic behavior.

Link between the effect of an electric field on wave propagation and the curvature-velocity relation

The link between the effect of an electric field in a 1D medium and the curvature-velocity relation is studied theoretically by means of a two-variable reaction-diffusion system and numerically in the framework of a two-variable Oregonator model. Theoretical results predict the existence of a relationship between critical electric field ( E cr) and critical curvature ( K cr), which is highlighted by means of the so-called V-shaped patterns.

A note on oscillations in a simple model of a chemical reaction

AbstractWe investigate oscillatory behaviour in the famous Belousov-Zhabotinskii chemical reaction, as described by the simple two-variable Oregonator model. It is shown that oscillations are possible only in certain parameter regions. Numerical results are presented, and the presence of fold bifurcations discussed.

Convective chemical-wave propagation in the Belousov-Zhabotinsky reaction.

We investigate the onset of convection for chemical-wave propagation in the Belousov-Zhabotinsky reaction based on the two-variable Oregonator model coupled with the fluid dynamic equations. For chemical waves in a vertical slab, two-dimensional convection occurs only for slab widths greater than a critical threshold width. The convective threshold is different for ascending and descending waves. Convectionless waves are flat and propagate with constant speed. Above the onset of convection, the wave velocity increases and the flat wave deforms due to two counterrotating steady rolls. For a horizontal slab, convection is always present and the wave velocity increases with increasing slab width. Our results are compared with experiments.

Three-dimensional scroll ring dynamics in the Belousov-Zhabotinskii reagent and in the two-variable Oregonator model

Three-dimensional scroll ring dynamics in the Belousov-Zhabotinskii reagent and in the two-variable Oregonator model

Chemical vortex dynamics in the Belousov-Zhabotinskii reaction and in the two-variable oregonator model

Chemical vortex dynamics in the Belousov-Zhabotinskii reaction and in the two-variable oregonator model