Oregonator Model Research Articles

Shape- and size-dependent patterns in self-oscillating polymer gels

Polymeric hydrogels that exhibit autonomous, coupled chemical and mechanical oscillations are a unique example of synthetic, active soft matter. Here, we explore the effects of gel aspect ratio and absolute dimensions on pattern formation in hydrogels undergoing the Belousov-Zhabotinsky (BZ) reaction. We synthesize and analyze N-isopropylacrylamide gels containing covalently bound BZ catalyst and polyacrylamide-silica gel composites containing physically associated BZ catalyst. Through both experiments and computational simulations, we demonstrate that the oscillating chemical patterns within BZ gels can be altered by changing the shape and size of the gel, and that these patterns evolve over long timescales. In our simulations, we utilize an improved Oregonator model, which explicitly accounts for the total concentration of the catalyst grafted onto the polymer network. We find that the three-dimensional simulations of the BZ gels successfully reproduce patterns, oscillation periodicity, and catalyst concentration-dependence observed in experiments. Together, these findings validate our theoretical and computational approaches for modeling chemomechanical coupling in active, chemo-responsive gels, and enable future studies that exploit the shape- and size-confinement effects of self-oscillating reactions.

Kinematics of spiral waves under feedback-related spatial gradients

The kinematics of spiral waves with artificially constructed spatial excitability is numerically investigated in the Oregonator model. On an assumption that the rotation center of spiral's tip drifts at angle δ to the direction of a local gradient, a kinematic formula of motion of spiral's tip is derived. To test the formula, we have presented two forms of feedback-related spatial fields with radial gradients (RGs) and concentric circular gradients (CGs) both centering on a reference point. It is found that both rigidly rotating and meandering spiral waves are attracted to the reference point of an inward RG and a clockwise CG perturbation but moved away from it under an outward RG and a counterclockwise CG. Simulations of the drift-velocity formulas provide a quantitative testing of the numerical results.

Control of noise-induced coherent behaviors in an array of excitable elements by time-delayed feedback

We investigate feedback-controlled coherent dynamics in a two-dimensional array of excitable elements. We demonstrate that one can freely enhance or reduce the spatiotemporal coherence of noise-induced oscillation, such as coherence resonance and phase synchronization, by controlling both the delay time and the feedback gain. Furthermore, we find that noise-induced oscillations are entrained by the feedback force in a certain range of the delay time. Experimental observations are approximately reproduced in a numerical simulation with a forced Oregonator model.

Phase Wave between Two Oscillators in the Photosensitive Belousov−Zhabotinsky Reaction Depending on the Difference in the Illumination Time

To investigate the nature of the phase wave between two connected oscillators, the photosensitive Belousov-Zhabotinsky (BZ) reaction was examined for two connected circular reaction fields, which were drawn by using computer software and then projected on a filter paper soaked with BZ solution by using a liquid-crystal projector. The difference in the time at which illumination was terminated between the two circles (Deltat(0)) was changed to control the time at which the phase wave was induced. When Deltat(0) was small (0-3 s), the phase wave normally propagated on the two circles in one direction. In contrast, when Deltat(0) was large (6-10 s), the velocity of the wave decreased near the intersection of the two circles. These different features are discussed in relation to the excitability of the circles and Deltat(0). The experimental results were qualitatively reproduced by a numerical calculation based on the modified three-variable Oregonator model that included photosensitivity.

Inward propagating chemical waves in Taylor vortices

Advection-reaction-diffusion (ARD) waves in the Belousov-Zhabotinsky reaction in steady Taylor-Couette vortices have been visualized using magnetic-resonance imaging and simulated using an adapted Oregonator model. We show how propagating wave behavior depends on the ratio of advective, chemical and diffusive time scales. In simulations, inward propagating spiral flamelets are observed at high Damköhler number (Da). At low Da, the reaction distributes itself over several vortices and then propagates inwards as contracting ring pulses--also observed experimentally.

Turing pattern of the Oregonator model

The Oregonator model is the mathematical dynamics which describes the Field–Körös–Noyes mechanics of the famous Belousov–Zhabotinskii˘ reaction. In this work, we establish some fundamental analytic properties of this dynamics and its corresponding steady state. Under various conditions on the parameters and the size of the reactor, we examine the existence and non-existence of non-constant steady states. In particular, for some properly chosen parameter ranges, we prove the occurrence of the Turing pattern generated by this Oregonator model. Our results exhibit interesting and very different roles of the diffusion rates and the reactor in the formation of the Turing pattern. Our mathematical analysis mainly relies on a priori estimates and the topological degree argument.

Oscillation in Penetration Distance in a Train of Chemical Pulses Propagating in an Optically Constrained Narrowing Channel

A chemical wave train propagating in a narrowing excitable channel surrounded by a nonexcitable field is investigated by using a photosensitive Belousov-Zhabotinsky (BZ) reaction. The considered geometry is created as a dark triangle surrounded by an illuminated area where the reaction is suppressed by the light-induced generation of bromide ion. For a low illumination level, a pulse train terminates at a constant position. However, as the light intensity increases, the position at which subsequent pulses disappear changes periodically, so that the period-doubling of penetration depth occurs. Two-dimensional simulations based on a modified Oregonator model for the photosensitive BZ reaction reproduce the essential features of the experimental observation.

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.

A simple method for simulating biochemical reaction networks

We propose a simple method of simulating chemical reaction networks by using a multiset rewriting system: abstract rewriting system on multisets (ARMS). We simulate the oregonator model, a mathematical model of the Belousov-Zhabotinskii reaction, and we obtain the same behavior when the differential equations or the Gillespie method are used. We also compose a 1- and 2-dimensional cellular automata by using ARMS, and confirm that it shows the pattern forming that is observed in the BZ reaction.

Simulation of the oregonator model by using deterministic ARMS

We propose a deterministic method for simulating chemical reactions, deterministic abstract rewriting system on multisets (DARMS), which is based on the concept of mass action low. The feasibility and utility of DARMS are demonstrated by applying it to the oregonator, which is a well-known model of the Belousov-Zhabotinskii (BZ) reaction.

Wave Propagation in the Photosensitive Belousov−Zhabotinsky Reaction Across an Asymmetric Gap

The photosensitive Belousov-Zhabotinsky (BZ) reaction was investigated at an asymmetrically illuminated gap, which was drawn using computer software and then projected on a filter paper soaked with BZ solution using a liquid-crystal projector. The probability of the chemical wave passing through the gap with asymmetric illumination was different from that through its mirror image. The location at which the wave disappeared and the time delay of the chemical wave passing through the gap changed depending on the velocity of chemical wave propagation. The experimental results were qualitatively reproduced by a theoretical calculation based on the three-variable Oregonator model that included photosensitivity. These results suggest that the photosensitive BZ reaction may be useful for studying spatiotemporal development that depends on the geometry of excitable fields.

Robust entrainment phenomena of oscillations by delay time in the photosensitive Oregonator model

The influences of delayed feedback on the oscillating behaviors are numerically investigated by using the photosensitive Oregonator model with a Hopf point. We find that the time delay in the robust entrainment phenomena determines the time scale of the system, that is, T_{m}=(tau+delta)N (N=1,2,cdots, three dots, centered) , where T_{m} is the mean period of the oscillation and delta is a small constant compared with the delay time tau . Further, our numerical simulation shows that, when the system has a characteristic period T0 under the feedback with time delay, there exists an asymptotical line delta=delta_{0}T_{0} ( delta_{0} independent of any parameters) in the entrainment region with increasing strength of the feedback control c ; when the system has no characteristic period, the above linear relation is also kept, and delta decreases with increasing c .

If BZ medium did spanning trees these would be the same trees as Physarum built

A sub-excitable Belousov–Zhabotinsky (BZ) medium exhibits self-localized wave-fragments which may travel for relatively long time preserving their shape. Using Oregonator model of the BZ medium we imitate foraging behavior of a true slime mold, Physarum polycephalum, on a nutrient-poor substrate. We show that given erosion post-processing operations the BZ medium can approximate a spanning tree of a planar set and thus is computationally equivalent to Physarum in the domain of proximity graph construction.

Temperature dependence of the Oregonator model for the Belousov-Zhabotinsky reaction

It is well known that temperature plays an important role in the chemical dynamics of Belousov-Zhabotinsky (BZ) reactions. The five-step Oregonator model of the BZ reaction has been elaborated here to investigate the temperature effect. The bifurcation dynamics has been calculated in the phase space spanned by initial reagents concentration ratio, stoichiometric factor and temperature. The combination of activation energies of the individual reaction steps governs the dependence of oscillation period and amplitude with temperature. The analysis reveals the existence of temperature limits beyond which no oscillations occur.

APPLICATIONS OF LOCAL ACTIVITY THEORY OF CNN TO CONTROLLED COUPLED OREGONATOR SYSTEMS

The study of chemical reactions with oscillating kinetics has drawn increasing interest over the last few decades because it also contributes towards a deeper understanding of the complex phenomena of temporal and spatial organizations in biological systems.The Cellular Nonlinear Network (CNN) local activity principle introduced by Chua [1997, 2005] has provided a powerful tool for studying the emergence of complex patterns in a homogeneous lattice formed by coupled cells.Recently, Yang and Epstein proposed a reaction–diffusion Oregonator model with five variables for mimicking the Belousov–Zhabotinskii reaction. The Yang–Epstein model can generate oscillatory Turing patterns, including the twinkling eye, localized spiral and concentric wave structures.In this paper, we first propose a modified Yang–Epstein's Oregonator model by introducing a controller, and then map the revised Oregonator reaction–diffusion system into a reaction–diffusion Oregonator CNN. The Oregonator CNN has two cell equilibrium points Q1= (0, 0, 0, 0, 0) and Q2, representing the "original" equilibrium point and an additional equilibrium point, respectively.The bifurcation diagrams of the Oregonator CNN are calculated using the analytical criteria for local activity. The bifurcation diagrams of the Oregonator CNN at Q1have only locally active and unstable regions; and the ones at Q2have locally passive regions, locally active and unstable regions, and edge of chaos regions. The calculated results show that the parameter groups of the Oregonator CNN which generate complex patterns are located on the edge of chaos regions, or on locally active unstable regions near the edge of chaos boundary.Numerical simulations show also that the Oregonator CNNs can generate similar dynamics patterns if the parameter groups are selected the same as those of the Yang–Epstein model. In particular, the parameters of the Yang–Epstein model which exhibit twinkling-eye patterns, and pinwheel patterns are located on the edges of chaos regions near the boundaries of locally active unstable regions with respect to Q2. The parameters of the Yang–Epstein models which exhibit labyrinthine stripelike patterns are located on the locally active unstable regions near the boundaries of the edge of chaos regions with respect to Q2. However the parameter group of the Yang–Epstein model with isolated spiral patterns is in the locally passive region near the boundary with edge of chaos with respect to Q2, whose trajectories tend to the equilibrium point Q2.Choosing a kind of triggering initial conditions given in [Chua, 1997], and the parameters of the Oregonator equations with the twinkling-eye patterns, pinwheel patterns, labyrinthine stripelike patterns, and isolated spiral patterns, three kinds of new spiral waves generated by the Oregonator CNNs were observed by numerical simulations. They seem to be essentially different patterns to those generated by the Oregonator CNNs with initial conditions consisting of equilibrium points plus small random perturbations.Our results demonstrate once again Chua's assertion that a wide spectrum of complex behaviors may exist if the corresponding CNN cell parameters are chosen in or near the edge of chaos region.

Turing instability in reaction-subdiffusion systems

We determine the conditions for the occurrence of Turing instabilities in activator-inhibitor systems, where one component undergoes subdiffusion and the other normal diffusion. If the subdiffusing species has a nonlinear death rate, then coupling between the nonlinear kinetics and the memory effects of the non-Markovian transport process advances the Turing instability if the inhibitor subdiffuses and delays the Turing instability if the activator subdiffuses. We apply the results of our analysis to the Schnakenberg model, the Gray-Scott model, the Oregonator model of the Belousov-Zhabotinsky reaction, and the Lengyel-Epstein model of the chlorine dioxide-iodine-malonic acid reaction.

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.

Second-order splitting schemes for a class of reactive systems

We consider the numerical time integration of a class of reaction–transport systems that are described by a set of ordinary differential equations for primary variables. In the governing equations, the terms involved may require the knowledge of secondary variables, which are functions of the primary variables. Specifically, we consider the case where, given the primary variables, the evaluation of the secondary variables is computationally expensive. To solve this class of reaction–transport equations, we develop and demonstrate several computationally efficient splitting schemes, wherein the portions of the governing equations containing chemical reaction terms are separated from those parts containing the transport terms. A computationally efficient solution to the transport sub-step is achieved through the use of linearization or predictor–corrector methods. The splitting schemes are applied to the reactive flow in a continuously stirred tank reactor (CSTR) with the Davis–Skodjie reaction model, to the CO + H 2 oxidation in a CSTR with detailed chemical kinetics, and to a reaction–diffusion system with an extension of the Oregonator model of the Belousov–Zhabotinsky reaction. As demonstrated in the test problems, the proposed splitting schemes, which yield efficient solutions to the transport sub-step, achieve second-order accuracy in time.

Complex dynamics in the Oregonator model with linear delayed feedback

The Belousov-Zhabotinsky (BZ) reaction can display a rich dynamics when a delayed feedback is applied. We used the Oregonator model of the oscillating BZ reaction to explore the dynamics brought about by a linear delayed feedback. The time-delayed feedback can generate a succession of complex dynamics: period-doubling bifurcation route to chaos; amplitude death; fat, wrinkled, fractal, and broken tori; and mixed-mode oscillations. We observed that this dynamics arises due to a delay-driven transition, or toggling of the system between large and small amplitude oscillations, through a canard bifurcation. We used a combination of numerical bifurcation continuation techniques and other numerical methods to explore the dynamics in the strength of feedback-delay space. We observed that the period-doubling and quasiperiodic route to chaos span a low-dimensional subspace, perhaps due to the trapping of the trajectories in the small amplitude regime near the canard; and the trapped chaotic trajectories get ejected from the small amplitude regime due to a crowding effect to generate chaotic-excitable spikes. We also qualitatively explained the observed dynamics by projecting a three-dimensional phase portrait of the delayed dynamics on the two-dimensional nullclines. This is the first instance in which it is shown that the interaction of delay and canard can bring about complex dynamics.

Anomalous pulse interaction in dissipative media

We review a number of phenomena occurring in one-dimensional excitable media due to modified decay behind propagating pulses. Those phenomena can be grouped in two categories depending on whether the wake of a solitary pulse is oscillatory or not. Oscillatory decay leads to nonannihilative head-on collision of pulses and oscillatory dispersion relation of periodic pulse trains. Stronger wake oscillations can even result in a bistable dispersion relation. Those effects are illustrated with the help of the Oregonator and FitzHugh-Nagumo models for excitable media. For a monotonic wake, we show that it is possible to induce bound states of solitary pulses and anomalous dispersion of periodic pulse trains by introducing nonlocal spatial coupling to the excitable medium.