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

TL;DR

This paper investigates the synchronization phenomena of identical pendulum clocks, proposing a new interaction mechanism that explains both in-phase and anti-phase synchronization, and analyzes their robustness under parameter variations.

Contribution

It introduces a novel interaction mechanism for oscillator synchronization and provides conditions for exact anti-phase and in-phase states, including coexistence and robustness analysis.

Findings

Exact anti-phase and in-phase synchronization states can coexist.

Synchronization periods differ from individual eigen-periods.

Synchronization persists under parameter variations.

Abstract

In 1665, Christiaan Huygens reported the observation of the synchronization of two pendulum clocks hanged on the wall of his workshop. After synchronization, the clocks swung exactly in the same frequency and $180^{o}$ out of phase -- anti-phase synchronization. Here, we propose and analyze a new interaction mechanism between oscillators leading to exact anti-phase and in-phase synchronization of pendulum clocks, and we determine a sufficient condition for the existence of an exact anti-phase synchronization state. We show that exact anti-phase and in-phase synchronization states can coexist in phase space, and the periods of the synchronized states are different from the eigen-periods 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 synchronization states exist in systems of coupled oscillators with different parameters.