Kenneth Showalter

Chemical waves and patterns

Spiral Waves. Lingering Mysteries about Organizing Centers in the Belousov-Zhabotinsky Medium and its Oregonator Model A. Winfree. Spiral Wave Dynamics S. Muller, T. Plesser. A Theory of Rotating Scroll Waves in Excitable Media J. Tyson, J. Keener. Spiral Waves in Weakly Excitable Media A.S. Mikhailov, V.S. Zykov. Spiral Meandering D. Barkley. Spiral and Target Waves in Finite and Discontinuous Media A.-A. Sepulchre, A. Babloyantz. Turing and Turing-like Patterns. Turing Patterns: from Myth to Reality J. Boissonade, E. Dulos, P. DeKepper. Onset and Beyond Turing Pattern Formation Q. Ouyang, H.L. Swinney. The Chemistry behind the First Experimental Chemical Examples of Turing Patterns I. Lengyel, I.R. Epstein. Turing Bifurcations and Pattern Selection P. Borckmans, G. Dewel, A. De Witt, D. Walgraef. The Differential Flow Instabilities M. Menzinger, A. Rovinsky. Chemical Wave Dynamics. Wave Propagation and Pattern Formation in Nonuniform Reaction-Diffusion Systems A. Zhabotinsky. Chemical Front Propagation: Initiation and Stability E. Mori, X. Chu, J. Ross. Pattern Formation on Catalytic Surfaces M. Eiswirth, G. Ertl. Simple and Complex Reaction-Diffusion Fronts S.K. Scott, K. Showalter. Modeling Front Pattern Formation and Intermittent Bursting Phenomena in the Couette Flow Reactor A. Arneodo, J. Elegaray. Fluctuations and Chemical Waves. Probabilistic Approach to Chemical Instabilities and Chaos G. Nicolis, F. Baras, P. Geysermans, P. Peeters. Internal Noise, Oscillations, Chaos and Chemical Patterns R. Kapral, X.-G. Wu. Index.

Dynamical Quorum Sensing and Synchronization in Large Populations of Chemical Oscillators

Populations of certain unicellular organisms, such as suspensions of yeast in nutrient solutions, undergo transitions to coordinated activity with increasing cell density. The collective behavior is believed to arise through communication by chemical signaling via the extracellular solution. We studied large, heterogeneous populations of discrete chemical oscillators (∼100,000) with well-defined kinetics to characterize two different types of density-dependent transitions to synchronized oscillatory behavior. For different chemical exchange rates between the oscillators and the surrounding solution, increasing oscillator density led to (i) the gradual synchronization of oscillatory activity, or (ii) the sudden “switching on” of synchronized oscillatory activity. We analyze the roles of oscillator density and exchange rate of signaling species in these transitions with a mathematical model of the interacting chemical oscillators.

Controlling chaos in the Belousov—Zhabotinsky reaction

DETERMINISTIC chaos is characterized by long-term unpredictability arising from an extreme sensitivity to initial conditions. Such behaviour may be undesirable, particularly for processes dependent on temporal regulation. On the other hand, a chaotic system can be viewed as a virtually unlimited reservoir of periodic behaviour which may be accessed when appropriate feedback is applied to one of the system parameters1. Feedback algorithms have now been successfully applied to stabilize periodic oscillations in chaotic laser2, diode3, hydrodynamic4 and magnetoelastic5 systems, and more recently in myocardial tissue6. Here we apply a map-based, proportional-feedback algorithm7,8 to stabilize periodic behaviour in the chaotic regime of an oscillatory chemical system: the Belousov–Zhabotinsky reaction.

Navigating Complex Labyrinths: Optimal Paths from Chemical Waves

The properties of excitable media are exploited to find minimum-length paths in complex labyrinths. Optimal pathways are experimentally determined by the collection of time-lapse position information on chemical waves propagating through mazes prepared with the Belousov-Zhabotinsky reaction. The corresponding velocity fields provide maps of optimal paths from every point in an image grid to a particular target point. Collisions of waves that were temporarily separated by obstacles mark boundary lines between Significantly different paths with the same absolute distance. The pathfinding algorithm is tested in very complex mazes with a simple reaction-diffusion model.

Noise-supported travelling waves in sub-excitable media

The detection of weak signals of nonlinear dynamical systems in noisy environments may improve with increasing noise, reaching an optimal level before the signal is overwhelmed by the noise. This phenomenon, known as stochastic resonance,, has been characterized in electronic, laser, magnetoelastic, physical and chemical systems. Studies of stochastic resonance and noise effects in biological, and excitable dynamical systems have attracted particular interest, because of the possibility of noise-supported signal transmission in neuronal tissue and other excitable biological media. Here we report the positive influence of noise on wave propagation in a photosensitive Belousov–Zhabotinsky reaction. The chemical medium, which is sub-excitable and unable to support sustained wave propagation, is illuminated with light that is spatially partitioned into an array of cells in which the intensity is randomly varied. Wave propagation is enhanced with increasing noise amplitude, and sustained propagation is achieved at an optimal level. Above this level, only fragmented waves are observed.

Mixed‐mode oscillations in chemical systems

A prototype model is exploited to reveal the origin of mixed‐mode oscillations. The initial oscillatory solution is born at a supercritical Hopf bifurcation and exhibits subsequent period doubling as some parameter is varied. This period‐2 solution subsequently loses stability, but continues to exist−regaining stability to form the 11 mixed‐mode state (one large plus one small excursion). Other mixed‐mode states lie on isolated branches or ‘‘isolas’’ of limit cycles in the one‐parameter bifurcation diagram and are separated by regions of chaos. As a second parameter is varied, the number of isola solutions increases and the ‘‘gaps’’ between them become narrower, leading to correspondingly more complete Devil’s staircases. An exactly comparable scenario is shown to arise in the three variable model of the Belousov–Zhabotinsky reaction proposed recently by Gyorgyi and Field [Nature 335, 808 (1992)].

Motion analysis of self-propelled Pt-silica particles in hydrogen peroxide solutions.

Silica microspheres that are half-coated with platinum metal undergo self-propulsion in solutions of H(2)O(2), with the average speed increasing with increasing H(2)O(2) concentration. Microscopic observation of the particle motion, with segmentation of the image data, demonstrates that the particles move, on average, with the platinum-coated region oriented opposite to the direction of motion. Velocity autocorrelation and motion direction analyses show that the direction of motion is highly correlated with the particle orientation. The effect of the observation time interval on the measured translational diffusion coefficient and the apparent particle motion is analyzed.

LOGIC GATES IN EXCITABLE MEDIA

The interaction of chemical waves propagating through capillary tubes is studied experimentally and numerically. Certain combinations of two or more tubes give rise to logic gates based on input and output signals in the form of chemical waves and wave initiations. The geometrical configuration, the temporal synchronization of the waves, and the ratio of the tube radius to the critical radius of the excitable medium determine the features of the logic gates.

Excitability, wave reflection, and wave splitting in a cubic autocatalysis reaction-diffusion system

The excitability properties of a two-variable cubic autocatalysis model for chemical oscillations are examined. The reaction-diffusion behaviour of this model is studied in a one-dimensional configuration with differing relative diffusivities of the species. Wave reflection at no-flux boundaries is examined and described in terms of reactant depletion in the wave front and reactant influx in the wave back. Waves are also reflected upon collision with other waves. Wave splitting, the spontaneous initiation of a wave from the trailing edge of another wave, is found to occur for some relative diffusivities. Successive wave splittings give rise to stationary Turing patterns at long times.

Instabilities in propagating reaction‐diffusion fronts

Simple reaction‐diffusion fronts are examined in one and two dimensions. In one‐dimensional configurations, fronts arising from either quadratic or cubic autocatalysis typically choose the minimum allowable velocity from an infinite spectrum of possible wave speeds. These speeds depend on both the diffusion coefficient of the autocatalytic species and the pseudo‐first‐order rate constant for the autocatalytic reaction. In the mixed‐order case, where both quadratic and cubic channels contribute, the wave speed depends on the rate constants for both channels, provided the cubic channel dominates. Wave propagation is completely determined by the quadratic contribution when it is more heavily weighted. In two‐dimensional configurations, with unequal diffusion coefficients, the corresponding two‐variable planar fronts may become unstable to perturbations. The instability occurs when the ratio of the diffusion coefficient for the reactant to that for the autocatalyst exceeds some critical value. This critical v...