the rotating frame. But unlike the limit cycle of Fig. 2(b) this cycle winds around the origin of the $z$-plane, marked by an asterisk in Fig. 2(d). This is an important distinction, because it implies that the phase of $z$ now increases monotonically relative to that of the drive. Consequently the order parameter $z(t)$ oscillates at a different average frequency from the drive signal, implying that a macroscopic fraction of the oscillator population has broken loose from the drive. In other words, the system has spontaneously mutually entrained itself, at least in part. This is therefore one mechanism by which forced entrainment can give way to mutual entrainment.

But there are other possible mechanisms as well. For example, consider Fig. 1(b) again, and now direct your attention to the upper right corner. By moving down along the right side of the picture, we can cross directly from C to E, without ever going through D. This happens when we cross through the portion of the lower saddle-node curve lying above and to the right of the saddle-node-loop point. In this case the bifurcation is not a SNIPER; it’s just an ordinary saddle-node bifurcation. To visualize this scenario, imagine sliding the 