saddle in the middle of Fig. 2(c) to the right along its unstable manifold until it collides with the node and annihilates it. During this process the limit cycle in Fig. 2(c) grows. And so the phase portrait now resembles the one shown in Fig. 2(e).

A third scenario is much simpler. Suppose $\Omega > \Omega_{\rm \, cusp}$, so that we’re well to the right of the cusp in Figs. 1(c) and 1(d). Then as we decrease $F$, we move directly from A to E. Forced entrainment gives way to mutual entrainment through a supercritical Hopf bifurcation.

Discussion
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Stability diagram
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The main result of the paper is the stability diagram shown in Fig. 1. We have focused on the analytical derivation of several of the curves in this picture and tried to clarify how they fit together and what they imply about the system’s overall dynamics. Having immersed ourselves in the details, it is worthwhile to step back and try to understand the broader lessons that this picture holds.

Figure 1 essentially divides into two big regions. One represents forced entrainment, wherein a macroscopic fraction of the population is phase-locked to the drive. The rest of the population 