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Using a combination of theory, experiment, and simulation, we revisit the dynamics of two coupled metronomes on a moving platform. Our experiments show that the platform's motion is damped by a dry friction force of Coulomb type, not the viscous linear friction force that has often been assumed in the past. Prompted by this result, we develop a new mathematical model that builds on previously introduced models but departs from them in its treatment of friction on the platform. We analyze the model by a two-timescale analysis and derive the slow-flow equations that determine its long-term dynamics. The derivation of the slow flow is challenging due to the stick-slip motion of the platform in some parameter regimes. Simulations of the slow flow reveal various kinds of long-term behavior including in-phase and antiphase synchronization of identical metronomes, phase locking and phase drift of non-identical metronomes, and metronome suppression and death. In these latter two states, one or both of the metronomes come to swing at such low amplitude that they no longer engage their escapement mechanisms. We find good agreement between our theory, simulations, and experiments, but stress that our exploration is far from exhaustive. Indeed, much still remains to be learned about the dynamics of coupled metronomes, despite their simplicity and familiarity.

The current definition of rate-induced tipping is tied to the idea of a pullback attractor limiting in forward and backward time to a stable quasi-static equilibrium. Here, we propose a new definition that encompasses the standard definition in the literature for certain scalar systems and includes previously excluded N-dimensional systems that exhibit rate-dependent critical transitions.

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.