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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. enter image description here

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Numerically locating a homoclinic orbit: Generically, it is not guaranteed that a periodic orbit that bifurcates from a Hopf bifurcation collides with a saddle to form a homoclinic orbit. In your system that may be the case.

If you continue the periodic orbit that bifurcates from the Hopf point (using XPPAUT/Matcont) and measure the period, at some point the period can go to infinity. That means that the periodic orbit spends a lot of time close to an invariant set before returning. If there's a saddle close by, you can be sure that there is a homoclinic orbit nearby.

After you're there, a more rigorous numerical method of locating a homoclinic orbit to a hyperbolic saddle is via the method of shooting/homotopy. See here for a detailed tutorial on Matcont: http://www.staff.science.uu.nl/%7Ekouzn101/NBA/LAB5.pdf

Types of homoclinic bifurcations: I think you're confusing type 1 and type 2 homoclinics with supercritical/subcritical Hopf bifurcations. As I mentioned before, stable cycles form supercritical Hopf bifurcations do not generically become type 1 or type 2 homoclinic orbits. This may happen in a system, such as in your case.

I suppose by type 1 and type 2 you mean the return of the stable manifold, that can occur in one of two directions in a planar system such as yours. In your case it returns in such a way that the other unstable direction is free, that is to say that if you start on the unstable direction that does not belong to the homoclinic, you escape (right side of the saddle in your case).

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