collective rhythm. (For circadian rhythms, this would mean that the animal’s internal clock was drifting relative to the outside world.) Sakaguchi’s numerics further indicated how forced entrainment could be lost and give way to mutual entrainment. Such transitions were found to occur via two different mechanisms, corresponding to a pair of distinct bifurcation curves in parameter space. These curves appeared to join at a point, but Sakaguchi was unable to resolve the details of the cross-over region numerically.

More recently, Antonsen et al. \[2008\] gave an improved analytical treatment of the model. Their linear stability analysis and numerical simulations also revealed an intriguing set of bifurcation curves, but the way the various curves join together still remained unclear. The overall layout of the stability diagram suggested that an underlying two-dimensional system was controlling the dynamics—a remarkable finding, given that the model  is essentially infinite-dimensional (recall $N \gg 1$).

This tantalizing clue led Ott and Antonsen to an important discovery \[Ott and Antonsen 2008\]. They found that the Kuramoto model possesses an invariant manifold, a special family of states for which the macroscopic dynamics becomes *low*-dimensional. In particular they showed that on this invariant manifold, the order parameter for the forced 