use $\sin^2 \phi+ \cos^2 \phi = 1$ to solve for $F$, now regarding $K$ and $\rho$ as parameters. The result is the following parametrization of the saddle-node curve: $$\begin{aligned}
\label{SNcurvesK}
\Omega_{\rm \,SN} &=& \frac{\left(\rho ^2+1\right)^{3/2}
   \sqrt{K \left(\rho ^2-1\right)
   \left(K \left(\rho
   ^2-1\right)^2-4\right)-4
   \left(\rho ^2+1\right)}}{2
   \left(\rho ^2-1\right)^2}\\
F_{\rm \,SN} &=& \frac{\sqrt{2} \rho ^2 \sqrt{K^2
   \left(\rho ^2-1\right)^3+2 K
   \left(\rho ^4-4 \rho
   ^2+3\right)-8}}{\left(\rho
   ^2-1\right)^2}.\end{aligned}$$ Figure 1 plots this saddle-node curve for the case $K=5$, as previously studied by Antonsen et al. \[2008\]. We compute the curve for all values $0<\rho  <1$, disregarding any values that yield non-real results for $\Omega$ or $F$.

The two branches of the saddle-node curve intersect tangentially at a codimension-2 cusp point, as highlighted in Fig. 1(c) and marked schematically in Fig. 1(d) by the solid square . For $K=5$, the parameter values at the cusp are $\Omega \approx 3.5445$ and $F \approx 3.4164$.

Along with local saddle-node bifurcations, the lower branch of the saddle-node curve(where $F \approx \Omega$) also includes a large section consisting of saddle-node infinite-period (SNIPER) bifurcations. These have important global implications, because they create or destroy limit cycles in the phase portrait of Eqs. (\[eq:phi-rhoa\])-(\[eq:phi-rhob\]).

Hopf bifurcation
----------------

Next we calculate the locus of parameter values 