Target patterns in a realistic model of the Belousov–Zhabotinskii reaction

Abstract

Periodic expanding target patterns of chemical activity are observed in thin layers of solution containing bromate, malonic acid and ferroin in dilute sulfuric acid. Commonly these patterns appear as thin blue (oxidized) rings propagating out from a central point into red (reduced) bulk medium. Recently, the opposite pattern has been observed: red waves of reduction propagating through an oxidized bulk medium. We discuss both of these patterns under the assumption that there is a heterogeneity at the center of the pattern—most likely a dust particle or a scratch on the glass—which changes the kinetics locally from a stable excitable steady state to a stable periodic oscillatory state. The temporal oscillation at the origin triggers waves of chemical activity which propagate radially into the excitable medium. Our approach is to combine recent advances in the mathematical description of traveling wave front solutions of reaction–diffusion equations with a realistic model of the kinetics of the reaction medium. The model we use, the Oregonator, is based on known features of the reaction mechanism, gives an acceptable qualitative and semiquantitative account of the reaction dynamics, and yet is simple enough to yield to analytic techniques developed primarily for scalar reaction–diffusion equations. With this approach we can account in some detail for most of the important features of target patterns in the Belousov–Zhabotinskii reaction. In particular, the distinction between trigger waves and phase waves is clarified by our analysis, and the novel properties of reducing waves in an oxidized medium appear as natural consequences of our model.