The Poincaré method: A powerful tool for analyzing synchronization of coupled oscillators

This paper focuses on the application of the Poincaré method of 'small parameter' for the study of coupled dynamical systems. Specifically, our attempt here is to show that, by using the Poincaré method, it is possible to derive conditions for the onset of synchronization in coupled (oscillatory) systems. A case of study is presented, in which conditions for the existence and stability of synchronous solutions, occurring in two nonlinear oscillators interacting via delayed dynamic coupling, are derived. Ultimately, it is demonstrated that the Poincaré method is indeed an effective tool for analyzing the synchronous behavior observed in coupled dynamical systems.