The Devil's Staircase

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Abstract
In the 17th century the Dutch physicist Christian Huyghens observed that two clocks hanging back to back on the wall tend to synchronize their motion. This phenomenon is known as phase locking, frequency locking or resonance, and is generally present in dynamical systems with two competing frequencies. The two frequencies may arise dynamically within the system, as with Huyghens's coupled clocks, or through the coupling of an oscillator to an external periodic force, as with the swing and attendant shown in figure 1. If some parameter is varied—the length of a pendulum or the frequency of the force that drives it, for instance—the system will pass through regimes that are phase locked and regimes that are not. When systems are phase locked the ratio between their frequencies is a rational number. For weak coupling the phase‐locked intervals are narrow, so that even if there is an infinity of intervals, the motion is quasiperiodic for most driving frequencies; that is, the ratio between the two frequencies is more likely to be irrational. When the coupling increases, the phase‐locked portions increase, and it becomes less likely that the motion is quasiperiodic. This is a unique situation, where it makes sense, despite experimental uncertainty, to ask whether a physical quantity is rational or irrational. When the interaction between an oscillator and its driver is strong enough, the oscillator will resonate at, or “lock” onto, an infinity of driving frequencies, giving rise to steps with a fractal dimension between 0 and 1.