# Find homoclinic bifurcation points

given the following simple system

$$\dot{x}=x\left(1-ax-\frac{y}{1+x^2}\right)\\ \dot{y}=y\left(b\frac{x}{1+x^2}-c\right)$$

I am trying to produce a three-dimensional bifurcation diagram. Now, I managed to find bifurcations such as Hopf-bifurcations. However, I know from XPPAUT (numerical bifurcation software), that the limit cycle vanishes via a homoclinic bifurcation. How can I find this point? Is there any analytical method suitable for my problem?

Furthermore, see attached a phase diagram showing the homoclinic orbit (black lines: trajectories, green and red lines: nullclines, blue line: stable manifold). I know what a homoclinic bifurcation is. However, I am struggling to understand types of homoclinic bifurcations. Super/subcritical and type I/type II. I guess that it is supercritical as the vanishing limit cycle is stable?! Type II bifurcation means that the homoclinic orbit "traps" the other ends of the unstable and stable manifolds of the saddle. So is this a type II homoclinic bifurcation then? I don't really understand this.

Thank you for helping.