Consider the dynamics of complex systems with cyclic links. The systems have the ring structures composed of hysteresis-feedback oscillators. In one of these systems, each oscillator has an additional feedback with the next oscillator. Sufficient conditions for the existence of periodic and recurrent motions are established. The periodic motion corresponds to a synchronous process occurring in such cyclic structures. In particular cases, we obtain the conditions for synchronization, as well as the stability conditions for synchronous oscillatory processes.