In this talk, synchronization dynamics of two distinct dynamical systems is presented through the theory of discontinuous dynamical systems. The necessary and sufficient conditions for the synchronization, de-synchronization and instantaneous synchronization (penetration or grazing) are presented. Using such a synchronization theory, the synchronization of a controlled pendulum with the Düffing oscillator is systematically discussed as a sampled problem, and the corresponding analytical conditions for the synchronization are presented. Finally, the partial and full synchronizations of the controlled pendulum with periodic and chaotic motions are presented to illustrate the analytical conditions. Through this example, the theory for dynamical systems synchronization developed recently is effective. The synchronization invariant domain can be obtained, which is an important hint for forming synchronization.