Interaction of Turing and Hopf bifurcations in chemical systems

Abstract

When a Turing bifurcation occurs close to a Hopf bifurcation in the parameter space of a reaction-diffusion system, the Turing and Hopf modes may interact nonlinearly to form, a priori, a variety of complex spatiotemporal patterns. We have studied this type of interaction for three models of chemically active media: the Lengyel-Epstein model of the ${\mathrm{ClO}}_{2}^{\mathrm{\ensuremath{-}}}$\char21{}${\mathrm{I}}^{\mathrm{\ensuremath{-}}}$\char21{}malonic acid system, a model that describes the ferroin-catalyzed Belousov-Zhabotinsky reaction, and the Brusselator. One and two spatial dimensions are considered. The Poincar\'e-Birkhoff method was implemented for the reduction of the models to the Turing-Hopf normal forms. The normal-form analyses show that the stability regions of stationary periodic patterns and of homogeneous oscillations usually overlap over a wide region in parameter space, forming a domain of bistability. Mixed-mode (spatiotemporal) patterns do not occur in the models considered except for a very small region in the parameter space for two-dimensional hexagonal patterns.