Destabilization of synchronous periodic solutions for patch models

In this talk, I will introduce our recent results on the destabilization of synchronous periodic solutions for general patch-models with cross-diffusion-like couplings. In order to specify the problem and provide significant results, we focus on synchronous periodic solutions bifurcating from center-type equilibria, periodic solutions and double homoclinic loops. For the first two cases, the destabilization is determined by period functions associated with bifurcating periodic solutions. For the last case, the destabilization is determined by the characteristic function, which is derived by the Lyapunov-Schmidt reduction method.