The Mechanism of a Controlled Pendulum Synchronizing with periodic Motions in a periodically forced, damped Duffing oscillator

In this paper, the analytical conditions for the controlled pendulum synchronizing with periodic motions in the Duffing oscillator are developed using the theory of discontinuous dynamical systems. From the analytical conditions, the synchronization invariant domain is obtained. The partial and full synchronizations of the controlled pendulum with periodic motions in the Duffing oscillator are discussed. The control parameter map for the synchronization is achieved from the analytical conditions, and numerical illustrations of the partial and full synchronizations are carried out to illustrate the analytical conditions. This synchronization is different from the controlled Duffing oscillator synchronizing with chaotic motion in the periodically excited pendulum. Because the periodically forced, damped Duffing oscillator possesses periodic and chaotic motions, further investigation on the controlled pendulum synchronizing with complicated periodic and chaotic motions in the Duffing oscillator will be accomplished in sequel.

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