Generic $2$-parameter perturbations of parabolic singular points of vector fields in $\mathbb {C}$

Abstract:

We describe the equivalence classes of germs of generic $2$-parameter families of complex vector fields $\dot z = \omega _\epsilon (z)$ on $\mathbb {C}$ unfolding a singular parabolic point of multiplicity $k+1$: $\omega _0= z^{k+1} +o(z^{k+1})$. The equivalence is under conjugacy by holomorphic change of coordinate and parameter. As a preparatory step, we present the bifurcation diagram of the family of vector fields $\dot z = z^{k+1}+\epsilon _1z+\epsilon _0$ over $\mathbb {C}\mathbb {P}^1$. This presentation is done using the new tools of periodgon and star domain. We then provide a description of the modulus space and (almost) unique normal forms for the equivalence classes of germs.