Stabilization of a cycle in a coupled mechanical system

We study mechanical systems each of which admits a family of periodic motions when the systems are not coupled. It is proved that a necessary condition for the existence of a cycle in a coupled system is the nondegeneracy of periodic motions in all possibly but one subsystem. The structure and specific type of the coupling control are found. The problems of existence, stability, and natural stabilization of oscillations are solved. It is shown that the cycle synchronizes the oscillations of mechanical systems in frequency and phase. The paper develops the idea of stabilizing the oscillations of a coupled system by selecting a suitable coupling control between subsystems.