Synchronization of clocks and metronomes: A perturbation analysis based on multiple timescales
In 1665, Huygens observed that two pendulum clocks hanging from the same board became synchronized in antiphase after hundreds of swings. On the other hand, modern experiments with metronomes placed on a movable platform show that they often tend to synchronize in phase, not antiphase. Here we study both in-phase and antiphase synchronization in a model of pendulum clocks and metronomes and analyze their long-term dynamics with the tools of perturbation theory. Specifically, we exploit the separation of timescales between the fast oscillations of the individual pendulums and the much slower adjustments of their amplitudes and phases. By scaling the equations appropriately and applying the method of multiple timescales, we derive explicit formulas for the regimes in parameter space where either antiphase or in-phase synchronization are stable, or where both are stable. Although this sort of perturbative analysis is standard in other parts of nonlinear science, it has been applied surprisingly rarely in the context of Huygens's clocks. Unusual features of our approach include its treatment of the escapement mechanism, a small-angle approximation up to cubic order, and both a two- and three-timescale asymptotic analysis.
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