Antiphase and in-phase synchronization of nonlinear oscillators: the Huygens's clocks system.

We introduce an interaction mechanism between oscillators leading to exact antiphase and in-phase synchronization. This mechanism is applied to the coupling between two nonlinear oscillators with a limit cycle in phase space, leading to a simple justification of the antiphase synchronization as observed in Huygens's pendulum clocks experiment. If the two coupled nonlinear oscillators reach the antiphase or the in-phase synchronized oscillatory state, the period of oscillation is different from the eigenperiods of the uncoupled oscillators.