at the corresponding parameter values $(\Omega, F)$ . This will be the case if $\Omega$ and $F$ are sufficiently large. Specifically, their values must exceed those at the Takens-Bogdanov point $$\begin{aligned}
\Omega_{\rm \,TB} &=& \frac{(K-2) K^2}{4 (K+2)} \label{TB_Ka}\\
F_{\rm \,TB} &=& \frac{1}{4} (K-2)
   \sqrt{\frac{K^3-2 K^2+4
   K-8}{K+2}} \label{TB_Kb}\end{aligned}$$ obtained by solving four simultaneous equations: $\phi' = 0$, $\rho' = 0$, trace = 0, and determinant = 0.

The Takens-Boganov point is marked with a filled circle on Figs. 1(a) and 1(d). In addition to serving as the endpoint of the Hopf curve, it splits the upper branch of the saddle-node curve into two sections of different dynamical character. On the lower section (below the Takens-Boganov point), an *unstable* node collides with a saddle along the saddle-node bifurcation curve; this can be seen by comparing regions D and A, as shown in Figs. 2(d) and 2(a). The opposite situation occurs on the upper section of the saddle-node curve, above the Takens-Boganov point. Here a *stable* node collides with a saddle, corresponding to the transition between regions B and A; see Figs. 2(b) and 2(a).

Homoclinic bifurcation
----------------------

The theory of the Takens-Bogdanov bifurcation implies that a curve of homoclinic bifurcations must also 