Generic 2-parameter perturbations of parabolic singular points of vector fields in 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 $\dot{z}=z^{k+1}+{\epsilon }_{1}z+{\epsilon }_0$ over ${\mathbb{CP}}^1$. 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.

44 pages, 34 figures