Light-sensitive Belousov-Zhabotinsky Research Articles

Effect of inhomogeneities on spiral wave dynamics in the Belousov-Zhabotinsky reaction

We examine the effects of controlled, slowly varying spatial inhomogeneities on spiral wave dynamics in the light sensitive Belousov-Zhabotinsky chemical reaction-diffusion system. We measure the speed of the grain boundary that separates two spirals, the speed of the lower frequency spiral being swept away by the grain boundary, and the speed of the slow drift of the highest frequency spiral. The grain boundary speeds are shown to be related to the frequency of rotation and wave number of the spiral pattern, as predicted from analysis of the complex Ginzburg-Landau equation [M. Hendrey, Phys. Rev. Lett.10.1103/PhysRevLett.82.859 82, 859 (1999); M. Hendrey,, Phys. Rev. E10.1103/PhysRevE.61.4943 61, 4943 (2000)].

Dynamics of the Spiral Tip in a Closed Belousov–ZhabotinskyReaction

Dynamics of spiral tip rotating in a closed system of the lightsensitive Belousov–Zhabotinsky reaction is studied under a homogeneousand steady illumination. The time dependence of the kinematicalparameters of meandering spiral is presented and the experimentalevidence is obtained for self-synchronization of the spiral tip in aclosed B-Z system without external feedback.

Noise-induced wave nucleations in an excitable chemical reaction

We study both experimentally and numerically the temporal coherence of noise-induced wave nucleations in excitable media subjected to external fluctuations with finite correlation time. The experiments are performed with the light-sensitive variant of the Belousov-Zhabotinsky (BZ) reaction forced by an exponentially correlated dichotomous fluctuating illumination. We find that there exists an optimal correlation time for which nucleations coherence reaches a maximum. The same behavior is obtained in numerical simulations with a stochastic Oregonator model, modified to describe the light-sensitive BZ reaction.

Feedback-mediated control of spiral waves

A theoretical approach is developed which allows one to describe the drift of a spiral wave mediated by a feedback signal taken as an integral of a system variable over a certain domain of an excitable medium. The underlying reaction–diffusion equations are reduced to an autonomous system of ordinary differential equations for the coordinates of the core center of the spiral wave. A new method to determine the drift velocity field of the spiral core is proposed based on a superposition principle. This approach is applied to analyze the drift velocity field for a feedback signal generated from a one-point detector and from integration domains of different size and shape, including global feedback. It is shown that a variation of the shape of the integration domain can induce bifurcations that change the attractor structure of the drift velocity field. These theoretical predictions are in good quantitative agreement with numerical simulations performed for the Oregonator model and the complex Ginzburg–Landau equation. The results are also confirmed by experiments with the light-sensitive Belousov–Zhabotinsky reaction.

Resonance tongues and patterns in periodically forced reaction-diffusion systems.

Various resonant and near-resonant patterns form in a light-sensitive Belousov-Zhabotinsky (BZ) reaction in response to a spatially homogeneous time-periodic perturbation with light. The regions (tongues) in the forcing frequency and forcing amplitude parameter plane where resonant patterns form are identified through analysis of the temporal response of the patterns. Resonant and near-resonant responses are distinguished. The unforced BZ reaction shows both spatially uniform oscillations and rotating spiral waves, while the forced system shows patterns such as standing-wave labyrinths and rotating spiral waves. The patterns depend on the amplitude and frequency of the perturbation, and also on whether the system responds to the forcing near the uniform oscillation frequency or the spiral wave frequency. Numerical simulations of a forced FitzHugh-Nagumo reaction-diffusion model show both resonant and near-resonant patterns similar to the BZ chemical system.

Spiral wave dynamics controlled by a square-shaped sensory domain

Spiral waves rotating rigidly in a thin layer of the light-sensitive Belousov–Zhabotinsky (BZ) reaction are subjected to a time-dependent uniform illumination. A non-local feedback algorithm computes the illumination intensity to be proportional to the average wave activity within a square-shaped sensory domain. The investigations show a broad spectrum of dynamical responses which results in square- and cross-shaped trajectories of the spiral tip, including reflections at the virtual walls. The geometry of the sensory domain is crucial in determining size and shape of the tip trajectories. A theoretical approach is proposed to explain the observed phenomena.

Antispirals in an artificial tissue of oscillatory cells.

In a tissue of oscillatory cells the active intracellular medium is surrounded by a membrane and the cells are separated by inactive extracellular medium. The synchronization properties of a system of such coupled oscillatory cells have been emulated using the light-sensitive Belousov-Zhabotinsky reaction. Experimental results for four coupled cells are confirmed by numerical simulations. We have furthermore demonstrated the existence of antispirals and antipacemaker waves with inward propagating waves in larger cell assemblies of this type. Such dynamical structures are extremely rare in homogeneous chemical systems where the generic behavior is normal spirals and target patterns with outward-moving waves.

Periodic forcing and feedback control of nonlinear lumped oscillators and meandering spiral waves.

It is shown that meandering spiral waves rotating in excitable media subjected to periodic external forcing or feedback control resemble many features of nonlinear lumped oscillators. In particular, the period shift function obtained for the Poincaré oscillator is qualitatively identical to that for spiral waves under fixed phase control. On the other hand, under one-channel feedback control, meandering spiral waves exhibit quite different dynamic regimes appearing as specific features of a distributed system. In particular, three types of attractors (resonance, entrainment, and asynchronous) of spiral waves are observed in experiments with the light-sensitive Belousov-Zhabotinsky reaction and in numerical simulations performed for the underlying Oregonator model. A theory of the resonance attractor for meandering spiral waves is developed which predicts the attractor radius and specifies the basins of attraction in good quantitative agreement with the numerical computations and experimental observations.

Dynamics of spiral waves under phase feedback control in a Belousov-Zhabotinsky reaction.

Feedback control of spiral waves by the phases of the spiral tip is investigated experimentally in a light-sensitive Belousov-Zhabotinsky reaction. The phases of rotation (Psi(r)) and meandering (Psi(m)) of the spiral tip are obtained in real time during experiments. It is found that, for both meandering and rigid rotating spirals, one can manipulate the spirals to move with any arbitrary paths by the feedback signals derived from Psi(r). Synchronization between meandering and rotation dynamics can be induced when both Psi(m) and Psi(r) are used simultaneously as control signals. Experimental findings are compared well with numerical simulations of the Oregonator model.

Transition from local to global feedback control of spiral wave dynamics

The dynamics of spiral waves rotating in a thin layer of the light-sensitive Belousov–Zhabotinsky reaction are controlled by a time-dependent uniform illumination. The intensity of the illumination is taken to be proportional to the average wave activity observed within a circular domain of the reaction layer. It is shown that an increase of the domain size, corresponding to a transition from local to global feedback control, drastically changes the conditions for stabilization and/or destabilization of the rigidly rotating spiral wave. A theoretical approach is proposed in order to explain the experimentally observed phenomena.

Instabilities of the resonance attractor for spiral waves in an excitable medium.

During an experimental study of the resonance attractor for spiral waves in the light-sensitive Belousov-Zhabotinsky reaction, strong deviations of the attractor trajectories from circular orbits are observed if the time delay in the feedback loop becomes relatively long. A theory is developed that reduces the spiral wave dynamics under a long-delayed control to a higher order iterative map. Then the observed deviations are explained to be a result of instabilities appearing due to the Neimark bifurcation of the map. The theoretical predictions are in good agreement with the experimental data.

Measurements of kinematical parameters of spiral waves in media of low excitability

The dynamics of spiral waves rotating in a thin layer of the light sensitive Belousov–Zhabotinsky reaction mixture are studied under a homogeneous and steady illumination. At a given composition of the excitable medium, the spiral waves meand, when no or low intensity light is applied, or rigidly rotate, when the light intensity is increased sufficiently. There exists, however, a critical value of light intensity above which no wave activity is supported by the medium, since its excitability is too strongly reduced by the illumination. In the vicinity of this critical value the basic kinematical parameters of rigidly rotating spirals (such as the rotation period, wavelength, propagation velocity, and the diameter of the spiral core) are measured as a function of the illumination intensity. The experimental observations are in good agreement with the predictions based on an earlier proposed kinematical theory of spiral waves in media of low excitability.

Spiral wave dynamics under feedback derived from a confined circular domain.

Spiral waves are induced in a thin layer of the light-sensitive Belousov-Zhabotinsky reaction and controlled by a time-dependent uniform illumination. The intensity of the illumination is taken to be proportional to the average wave activity observed within a circular domain of the reaction layer. Stabilization and destabilization of spiral waves, as well as the existence of additional types of attractors, are demonstrated. A simple mathematical description is proposed in order to explain the existence of the attractors and the dependence of their size on the time delay in the feedback loop.

Transitions between orbits of resonance attractors for spiral waves.

Spiral waves rotating in a thin layer of the light-sensitive Belousov-Zhabotinsky reaction can be controlled by the application of short light pulses at instants corresponding to the passage of a wave front through a measuring point. It is shown that such a feedback results in a drift of the spiral wave core along a discrete set of stable circular orbits centered at the measuring point, in good quantitative agreement with the theory of the resonance attractor recently developed. Variations of parameters in the feedback loop initiate transitions between orbits of different size. Thus a spiral wave drift can be induced along a snail-shaped trajectory with permanently growing distance from the measuring point.

Brownian motion of spiral waves driven by spatiotemporal structured noise

Spiral chemical waves subjected to a spatiotemporal random excitability are experimentally and numerically investigated in relation to the light-sensitive Belousov-Zhabotinsky reaction. Brownian motion is identified and characterized by an effective diffusion coefficient which shows a rather complex dependence on the time and length scales of the noise relative to those of the spiral. A kinematically based model is proposed whose results are in good qualitative agreement with experiments and numerics.

EFFECTS OF A QUENCHED DISORDER ON WAVE PROPAGATION IN EXCITABLE MEDIA

The behavior of autowaves under the effect of a quenched disorder is studied in the framework of the light-sensitive Belousov–Zhabotinsky reaction. This allows us to introduce spatial disorder on the excitability by projecting patterns of light transmittance. In particular, we have selected a dichotomic random distribution of levels of transmittance. If the two values of transmittance are equally probable and allows wave propagation without breaking the waves, we find an opposite effect on the wave front velocity and shape depending on the considered dimension. On the other hand, if one of the two values of the transmittance distribution is set on the nonexcitable region, percolation phenomena can arise by changing the number of excitable sites. The different addressed situations are analytically interpreted giving theoretical predictions for the experimental and numerical results.

Structure of the Resonance Attractor for Spiral Waves in Excitable Media

Recent experiments in the light-sensitive Belousov-Zhabotinsky reaction have shown that a pulsatory modulation of excitability controlled by a feedback can force the spiral wave core to execute a circular trajectory around a fixed measuring point of the medium. A theory is developed to explain the existence and stability of this resonance attractor which is based on reducing the spiral wave dynamics to a low-dimensional map. The analysis of this map predicts the existence of a discrete set of stable circular trajectories in good quantitative agreement with numerical simulations of the two-component Oregonator model. Only a small part of this set has so far been observed experimentally.

Percolation thresholds in chemical disordered excitable media

The behavior of chemical waves advancing through a disordered excitable medium is investigated in terms of percolation theory and autowave properties in the framework of the light-sensitive Belousov-Zhabotinsky reaction. By controlling the number of sites with a given illumination, different percolation thresholds for propagation are observed, which depend on the relative wave transmittances of the two-state medium considered.

Wave Propagation in a Medium with Disordered Excitability

The effect of quenched disorder on the propagation of autowaves in excitable media is studied both experimentally and numerically in relation to the light-sensitive Belousov-Zhabotinsky reaction. The spatial disorder is introduced through a random distribution with two different levels of transmittance. In one dimension the (time-averaged) wave speed is smaller than the corresponding to a homogeneous medium with the mean excitability. Contrarily, in two dimensions the velocity increases due to the roughening of the front. Results are interpreted using kinematic and scaling arguments. In particular, for $d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$ we verify a theoretical prediction of a power-law dependence for the relative change of the propagation speed on the disorder amplitude.

Resonant pattern formation in achemical system

A periodic force applied to a nonlinear pendulum can cause the pendulum to become entrained at a frequency that is rationally related to the applied frequency, a phenomenon known as frequency-locking1. A recent theoretical analysis showed that anarray of coupled nonlinear oscillators can exhibit spatial reorganization when subjected to external periodic forcing2. We present here experimental evidence that reaction–diffusion processes, which govern pattern evolution and selection in many chemical and biological systems3, can also exhibit frequency-locking phenomena. For example, periodic optical forcing of the light-sensitive Belousov–Zhabotinsky (BZ) reaction transforms a rotating spiral wave4 to a labyrinthine standing-wave pattern (Fig. 1). As the forcing frequency is varied, we observe a sequence of frequency-locked regimes, analogous to the frequency-locked ‘tongues’ of a driven nonlinear pendulum, except that in the reactor different frequencies correspond to different spatial patterns. Resonant interactions leading to standing-wave patternshave not been observed previously in chemical or biological media, but periodic forcing (such as circadian rhythm) is abundant in nature and may lead to similar pattern-forming phenomena.The reaction occurs in a 0.4-mm-thick porous membrane disk that is placed between two reservoirs (I and II) which are continously refreshed with BZ reagents. The region shown is a 13 × 13 mm section of the 25-mm-diameter reactor. The patterns in the membrane are determined from light absorption measurements at 450 nm. In the perturbed region, light of 0.2 W m−2intensity in the spectral range 430–470 nm is periodically switched on for 6 s and then off for 13 s. Spatially uniform illumination is achieved using a video projector (Sanyo PLC-220N) calibrated by measuring light reflected from a scatter plate and then adjusting each pixel to have the same intensity. The concentrations of thechemicals in reservoirs I and II are: [malonic acid]I = 0.22 M, [BrO3−]I = 0.046 M, [BrI−] = 0.2 M, [H2SO4]I = 0.8 M, [Ru(bpy)32+]II = 1.0 mM, [H2SO4]II = 0.8 M, [BrO3−]II = 0.184 M (here bpy indicates 2,2′-bipyridine). The volume of each reservoir is 10 ml. The flow rate in reservoir I is 20 ml h−1; in reservoir II, 5 ml h−1. The system is maintained at a temperature of 23 °C.