I was tasked with finding and classifying the two local bifurcations that occur in the following system: $$\frac{dx}{dt}=x^2-rx+9$$ I have found that there exists a bifurcation at $r=\pm6$. The system has equilibrium points at $x=\frac{r\pm\sqrt{r^2-36}}{2}$.
When I perform bifurcation analysis on these equilibrium points, I discover that the equilibrium point $x=\frac{r+\sqrt{r^2-36}}{2}$ goes from being unstable for $r<-6$ to not existing when $-6<r<6$ to being unstable again when $r>6$. The other equilibrium point follows the same pattern, but it goes from being stable to not existing to being stable again.
How can I classify these bifurcations when it would seem as though they only affect whether or not the equilibrium point exists or not?
Smaller, side question: Is there a resource available that lists all the types of bifurcations and their characteristics? My textbook (Sayama's Introduction to the Modeling and Analysis of Complex Systems) lists a few, but it isn't stated that those are the only types.