There is this model created by Shilnikov and described in the paper: https://www.researchgate.net/publication/220580167_Shilnikov_bifurcation
The problem is that I do not understand how the system of equations (1) gives the homoclinic loop, which conditions for the system to give such loop. Are they only that $λ_{12}=ρ±iω$, $ρ>0$ and $ω≠0$, and $λ_3=γ>0$. Also $F_i$ are smooth functions with $F_i(0)=0$. With only these conditions we will get $Γ$?
If so, how to solve this system qualitatively and plot it? Do I neglect $F_i$ when I find the Jacobian to solve for the eigenvalues? I found the Jacobian of the system including $F_i$ and solve it in the Matlab and got very complicated eigenvalues.
Another issue, what is the deal of the bifurcation? Where does the paper take care of the bifurcation?