Synchronization of asymmetrically coupled systems

In this paper, the dynamics of two mechanical oscillators interacting through an asymmetric coupling is considered. In the uncontrolled system, the asymmetry in the coupling prevents the oscillators from reaching synchronization. Therefore, a state feedback controller is designed and applied to the coupling system in order to remove the asymmetry. The resulting closed-loop dynamics is analyzed by using the Poincaré method, and analytic conditions for the existence of stable synchronous solutions are derived. It is demonstrated that two synchronous solutions coexist in the closed-loop system, namely in-phase and anti-phase synchronization. Additionally, analytic expressions for the amplitude, frequency, and phase of these solutions—which are not determined by a reference signal, but rather by the intrinsic properties of the coupled systems—are provided and a characteristic equation for determining the stability of the synchronous solution is given. Finally, the obtained theoretical results are illustrated by numerical simulations, including a numerical study on the robustness of the synchronous solution.