I'm dealing with the following ODE,
$$ \left(\frac{dr}{d\lambda}\right)^2 = \left[1 + \frac{(C - 1)M}{(2r - 3M)}\right]\left[1 - \left(1-\frac{2M}{r}\right)\left(\frac{D}{r}\right)^2\right] \equiv f(r;C,D)\,\, , $$
where
$$ r \in \mathbb{R^{+}} \quad \textrm{and} \quad C,D,M \in \mathbb{R} \,\, . $$
Taking as bifurcation parameters $C$ and $D$, I wish to describe the bifurcations ocurring in this system, for this I used the following system to get the bifurcation curve,
$$ f(r;C,D) = 0 \,\, , \\ f_{r}(r;C,D) = 0 \,\, , $$
Which gives me the following curve,
Which suggests to me that there is a kind of cusp and saddle-node bifurcation here, but the conditions for saddle-node fail. On the other hand, the conditions for a cusp bifurcation holds, even though the non degenerancy conditions fail too. My question is, this bifurcation curve tells me that there is a more complex bifurcation happening here, not just a cusp one, which kind of bifurcation could be happening here? Especifically, I would like to know what kind of bifurcations goes in the black points. The $(r,C,D)$-coordinates of these points considering $M = 1$ are
$$ p_{1} = (3, -2, 3\sqrt{3}) \,\, , \\ p_{2} = (3, -2, -3\sqrt{3}) \,\, , \\ p_{3} = (0, 4, 0) \,\, , \\ p_{4} = (-6, 16, -3\sqrt{3}) \,\, , \\ p_{5} = (-6, 16, 3\sqrt{3}) \,\, . $$
Those with $r < 0$ are not in the domain of the system, but I would like to know the bifurcation anyway.