Synchronization stability of a distributed hysteresis controlled system

The self-oscillatory system with hysteresis has been widely studied. However, most research deal with one hysteresis element in the system, seldom discuss the multi-hysteresis controllers, not to mention interactions between them. In the paper, we take a simple model with two coupled hysteresis controlled integrators as an example to study a synchronous phenomenon, that is similar to that happened in the large-scale distributed hysteresis controlled system, e.g. the supermarket refrigeration system. In the paper, synchronization is interpreted as a limit cycle in a state space created by switching among piecewise-affine dynamical system. Stability of the resultant limit cycle is examined by a Poincar'e map. We show that the synchronization takes place if the corresponding Poincar'e map is stable.