emerge from the codimension-2 point, tangential to the saddle-node and Hopf curves. For the case $K=5$ shown in Fig. 1, $\Omega_{\rm \,TB} = \frac{75}{28}$ and $F_{\rm \,TB} = \frac{3}{4}\sqrt{\frac{87}{7}}$. The curve shown in the diagram was computed numerically. It is almost indistinguishable from the Hopf curve and thus produces a very small region between them, as shown in Fig. 1(b).

A striking feature of the homoclinic curve is that after moving parallel to the Hopf curve for a while, it makes a sharp backward turn and then joins onto the lower branch of the saddle-node/SNIPER curve, meeting that curve tangentially at a codimension-2 “saddle-node-loop” point \[Guckenheimer 1986, Izhikevich 2000\] marked by a filled diamond in Figs. 1(b) and 1(d).

Phase portraits and bifurcation scenarios
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As we have seen, the bifurcation curves in Fig. 1 partition the stability diagram into five regions, labeled A-E. We now give a fuller treatment of the dynamics associated with each region and the transitions from one to another.

### Region A: Forced entrainment

Here the order parameter $z$ approaches a stable fixed point for all initial conditions, as shown in Fig. 2(a). To interpret what this means, recall that all our analysis has assumed 