at a saddle-node bifurcation, one of the eigenvalues equals 0 and hence the determinant of the Jacobian vanishes there. (The same would be true at transcritical or pitchfork bifurcations, but given the absence of the constraints or symmetries associated with these types of bifurcations, there’s no reason to expect either of them to occur here.)

Hence to find the locus of saddle-node bifurcations, we solve $\rho' = 0$, $\phi' = 0$ and $\delta=0$ simultaneously, where $\delta$ denotes the determinant of the Jacobian. The trick is to regard the unknown values of the variables $\rho$ and $\phi$ on equal footing with the parameters $K, \Omega$ and $F$. Then the resulting system of 3 equations in 5 unknowns can be solved explicitly to yield a parametrization of the saddle-node bifurcation surface. Various parametrizations are possible. One convenient choice is to express the parameters in terms of the fixed-point values of $\rho$ and $\phi$. We find that the saddle-node surface is then given by $$\begin{aligned}
\label{SNsurface}
K&=&\frac{2 \left(\rho ^4+2 \rho ^2 \cos 2 \phi  +1\right)}{\left(1-\rho
   ^2 \right)^2 \left(1+ \rho ^2 \cos 2 \phi \right)}\\
\Omega &=&\frac{\left(\rho ^3+\rho \right)^2 \sin 2 \phi }{\left(1-\rho
   ^2 \right)^2 \left(1+ \rho ^2 \cos 2 \phi \right)}\\
F&=&-\frac{4 \rho 