Two Metronomes

Suppose the timekeeping abilities of two identical metronomes are compared over several hours. They will drift faster or slower at different rates. When both metronomes are placed on a skateboard that moves freely horizontally, their drifts change gradually as they tend to synchronize. Each metronome has been subjected to the driving force of the other, the result being the phenomenon called “phase-locking” or “modelocking.” Suppose now that each metronome on the skateboard begins with different initial conditions, but one of the two metronomes is driven by perturbations that fluctuate randomly in time. Can the metronomes become synchronized?

Answer

For the case of the periodic perturbation of one metronome by the other, the mode-locking occurs when the perturbing frequency is sufficiently close to the unperturbed frequency of the metronome. When a metronome is placed on the skateboard, the movement of the pendulum causes the skateboard itself to move slightly, usually in the opposite direction to the pendulum swing, since the metronome base is kept in place on the skateboard by static friction or could be bolted down. Some of the energy of the metronome base motion is transferred to the skateboard, and this very small amount of energy is further transferred along the skateboard in several directions, with some amount reaching the other identical metronome.

If at first this energy arrives at some random phase point in the oscillation of the second metronome, eventually its regular energy delivery becomes more and more effective in synchronizing the pendulum oscillations. Of course, the second metronome is acting on the first metronome in the same way simultaneously. The synchronization is normally in-phase, but anti phase synchronization can occur in special conditions. (See the second reference below for details.)

The behavior can be represented by two equations for two harmonically driven oscillators with a significant amount of dampening. If the dampening were not significant, then we would see two coupled pendulums alternating their swing behavior out of phase from maximum amplitude to nearly zero amplitude. In the actual case, the pendulums simply synchronize and keep nearly identical time.

For the case of one of the pendulums being driven by a force random in time, their fluctuating behavior can converge to an identical response. Both pendulums would exhibit the same random fluctuations eventually. For both periodic and aperiodic driving forces, asymptotic stability results for linear oscillators properly damped. That is, small changes in the parameters of the linear oscillator or the driving force result in only small changes in the asymptotic behavior. The equation of motion for each oscillator is mathematically equivalent to describing a linear spring in a viscous medium with a fluctuating driving force.

According to the first reference below, the mode-locking can occur also for a wide range of aperiodiccally driven nonlinear oscillators in the physical and biological sciences, from nonlinear electrical circuits to neural systems. As in the periodically driven systems, the synchronization of randomly driven nonlinear oscillators was found to be structurally stable, which means that even in the presence of small amounts of noise an approximate synchronization is achieved.