On the problem of synchronization of identical dynamical systems: the Huygens’s clocks

Abstract In 1665, Christiaan Huygens reported the observation of the synchro-nization of two pendulum clocks hanged on the wall of his workshop. After syn-chronization, the clocks swung exactly in the same frequency and 180 ◦ out of phase—anti-phase synchronization. Here, we propose and analyze a new interac-tion mechanism between oscillators leading to exact anti-phase and in-phase syn-chronization of pendulum clocks, and we determine a sufficient condition for the existence of an exact anti-phase synchronous state. We show that exact anti-phase and in-phase synchronous states can coexist in phase space, and the periods of the synchronous states are different from the eigenperiods of the individual oscillators. We analyze the robustness of the system when the parameters of the individual pendulum clocks are varied, and we show numerically that exact anti-phase and in-phase synchronous states exist in systems of coupled oscillators with different parameters. In 26 February 1665, Christiaan Huygens, in a letter to his father [1], reported th