Turing instability-induced oscillations in coupled reaction–diffusion systems

A new type of localized oscillatory pattern is presented in a two-layer coupled reaction–diffusion system under conditions in which no Hopf instability can be discerned in either layer. The transitions from stationary patterns to asynchronous and synchronous oscillatory patterns are obtained. A novel method based on decomposing coupled systems into two associated subsystems has been proposed to elucidate the mechanism of formation of oscillating patterns. Linear stability analysis of the associated subsystems reveals that the Turing pattern in one layer induces the other layer locally, undergoes a supercritical Hopf bifurcation and gives rise to localized oscillations. It is found that the sizes and positions of oscillations are determined by the spatial distribution of the Turing patterns. When the size is large, localized traveling waves such as spirals and targets emerge. These results may be useful for deeper understanding of pattern formation in complex systems, particularly multilayered systems.