  In this paper we describe the bifurcation diagram of the$2$-parameter family
of vector fields $\dot z = z(z^k+\epsilon_1z+\epsilon_0)$ over $\mathbb
C\mathbb P^1$ for $(\epsilon_1,\epsilon_0)\in \mathbb C^2$. There are two kinds
of bifurcations: bifurcations of parabolic points and bifurcations of
homoclinic loops through infinity. The latter are studied using the tool of the
periodgon introduced in a particular case in \cite{CR}, and then generalized in
\cite{KR}. We apply the results to the bifurcation diagram of a generic germ of
2-parameter analytic unfolding preserving the origin of the vector field $\dot
z = z^{k+1} +o(z^{k+1})$ with a parabolic point at the origin.
