et al. \[2008\]; see Fig. 3 in their paper. Those authors generated their results based on direct simulations of Eq. for $N=1000$ oscillators. They also compared their numerics to analytical results they derived for the existence and stability of equilibrium points for , which correspond to entrained states in the original frame. Our approach, in contrast, is to analyze the reduced system Eqs. (\[eq:phi-rhoa\])-(\[eq:phi-rhob\]). We do not present numerical results for the full system because in every case we have checked, our results match those reported already by Antonsen et al. \[2008\], except in cases where the previous methods were inconclusive.

Saddle-node and SNIPER bifurcations
-----------------------------------

It is algebraically awkward to solve for the fixed points of Eqs. (\[eq:phi-rhoa\])-(\[eq:phi-rhob\]) in terms of the parameters. Fortunately, we do not need to solve for them. Since we are mainly interested in the bifurcation curves, we can make headway more easily by imposing an appropriate bifurcation condition and then solving for the parameters in terms of the fixed point, rather than the other way around. This is a standard trick in bifurcation theory, and it allows us to derive the bifurcation curves in closed form, either explicitly or as parametric equations.

For example, 