a homoclinic orbit. Beyond the bifurcation the phase portrait looks like that shown in Fig. 2(d). An invariant loop has been created, in which the saddle and the original stable node are now connected by both branches of the saddle’s unstable manifold. The stable node is the unique attractor. Hence the system again falls into a state of forced entrainment.

### Region E: Mutual entrainment

Forced entrainment is finally lost when we pass from Region D to E. When crossing the lower branch of the saddle-node curve, we need to be careful to specify exactly where the crossing occurs. Specifically, do we cross to the left or right of the codimension-2 saddle-node-loop point (filled diamond in Fig. 1(b)) at which the homoclinic curve meets the saddle-node curve?

Suppose first that we cross below and to the left of the saddle-node-loop point. Then in Fig. 2(d) the saddle and node would slide toward each other along the invariant loop, coalesce, and disappear, leaving a stable limit cycle in their wake. Thus, this saddle-node bifurcation is actually a SNIPER (saddle-node infinite-period) bifurcation.

The limit cycle created by the bifurcation is globally attracting. Hence the order parameter always settles into periodic motion in 