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Synchronization of clocks and metronomes: a perturbation analysis based on multiple timescales. (English) Zbl 1458.34097
Summary: 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 the parameter space where either antiphase or in-phase 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 small-angle approximation up to cubic order, and both a two- and three-timescale asymptotic analysis.
©2021 American Institute of Physics
MSC:
34D06 Synchronization of solutions to ordinary differential equations
34D10 Perturbations of ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34N05 Dynamic equations on time scales or measure chains
Software:
MATCONT
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