Almost synchronization phenomena in the two and three coupled Brusselator systems

Abstract

We present a study of some temporal almost synchronization phenomena of systems of two and three coupled Brusselators: they are approximately synchronized during most of the dynamics, only losing synchronization for small times and quickly returning to an almost synchronized state. Here we show two situations where this phenomenon occurs, one related with codimension-two Hopf–pitchfork bifurcations, and the other one due to the existence of fast–slow dynamics. On the one hand, a detailed characterization of the codimension-two Hopf–pitchfork bifurcations in the model allows us to determine the regions of the parameter space in which this phenomenon occurs. On the other hand, a fast–slow analysis of the two coupled Brusselators, using singular perturbation theory, illustrates the second situation studied here. We next analyze this phenomenon numerically, by explicitly calculating the fraction of time during which different trajectories are almost synchronized. Our results are then extended to the case of three coupled Brusselators.