Synchronization of clocks and metronomes: A perturbation analysis based on multiple timescales
Abstract
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 inphase and antiphase synchronization in a model of pendulum clocks and metronomes and analyze their longterm 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 the parameter space where either antiphase or inphase synchronization is stable or where both are stable. Although this sort of perturbative analysis is standard in other parts of nonlinear science, surprisingly it has rarely been applied in the context of Huygens's clocks. Unusual features of our approach include its treatment of the escapement mechanism, a smallangle approximation up to cubic order, and both a two and threetimescale asymptotic analysis.
 Publication:

Chaos
 Pub Date:
 February 2021
 DOI:
 10.1063/5.0026335
 arXiv:
 arXiv:2008.02947
 Bibcode:
 2021Chaos..31b3109G
 Keywords:

 Nonlinear Sciences  Adaptation and SelfOrganizing Systems;
 Mathematics  Dynamical Systems;
 34C15
 EPrint:
 15 pages, 8 figures