Dynamics of two interfaces in a hybrid system with jump-type heterogeneity

We consider the dynamics of two interfaces that interact through a continuous medium with spatial heterogeneity. The dynamics of interface positions is governed by ordinary differential equations (ODEs), whereas that of the continuous field by a partial differential equation. The resulting mixed ODE-PDE system, which we call a hybrid system (HS), is derived as a singular limit of a certain bistable reaction-diffusion system (PDE), describing the dynamics of traveling pulses of front-back type. First, the traveling pulse dynamics in the heterogeneous medium is numerically studied both for the bistable reaction-diffusion system and for the hybrid system. Then, the hybrid system is analyzed to clarify the underlying mechanisms for the pulse behavior observed. In particular, we carry out a center manifold reduction for the hybrid system, which reveals not only the supercriticality of Hopf bifurcations but also the mechanism for sliding motion of an oscillating pulse observed in the heterogeneous medium.