Echo waves and coexisting phenomena in coupled brusselators

In this paper we study the echo wave and coexisting phenomena in linearly coupled brusselators and put forward a new method to prove the existence of the echo wave. We find that the existence of the echo wave is determined by that of the periodic solution of the associate oscillator. In addition, we also find that the stable steady state may coexist with the echo wave and that the echo wave may coexist with the in-phase wave under suitable conditions, and specify the coexisting regimes on parameters.