known to be conveyed from the eyes to the pacemaker through a specialized neural pathway). If the light-dark cycle is 24 hours long, we expect the electrical rhythms of many individual pacemaker cells to successfully entrain to it. But what if we alter the period or strength of the external forcing, as has been done in countless experiments on mice, hamsters, primates, and human volunteers \[Moore-Ede et al. 1982\]? Or what happens if the experiment is conducted on mutant organisms \[Dunlap 1993, Takahashi 1995\] whose intrinsic periods are a few hours longer or shorter than normal, or which may be intrinsically arrhythmic, having almost no free-running circadian rhythm at all?

Questions like this can be addressed, in mathematically idealized form, within the framework of the periodically forced Kuramoto model \[Sakaguchi 1988, Antonsen et al. 2008, Ott and Antonsen 2008\]. Its governing equations are given by

$$\label{forcedKuramoto_labframe}
\frac{d \vartheta_i}{dt} = \omega_i+\frac{K}{N}\sum_{j=1}^N \sin(\vartheta_j-\vartheta_i)+F \sin(\sigma  t-\vartheta_i),$$

for $i = 1, \ldots, N$. Here $\vartheta_i$ is the phase of oscillator $i$, $\omega_i$ is its natural frequency, $K$ is the coupling strength, $F$ is the forcing strength, $\sigma$ is the forcing frequency, and $N \gg 1$ is the number of oscillators. The natural frequencies are 