On the Problem of Synchronization of Identical Dynamical Systems: The Huygens’s Clocks
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° 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 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.
KeywordsPhase Space Nonlinear Oscillator Couple Oscillator Attachment Point Positive Real Part
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I would like to thank the support of the Ettore Majorana Center for Scientific Culture and the hospitality of the organizers of the conference “Variational Analysis and Aerospace Engineering,” dedicated to Prof. Angelo Miele on his 85th birthday. This work has been partially supported by a Fundação para a Ciência e a Tecnologia (FCT) pluriannual funding grant to the NonLinear Dynamics Group (GDNL).
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