Generic 2-parameter perturbations of parabolic singular points of vector fields in C

TL;DR

This paper classifies generic 2-parameter families of complex vector fields near parabolic singular points, providing a detailed description of their moduli space and normal forms using new geometric tools.

Contribution

It introduces a complete classification of 2-parameter unfoldings of parabolic points in complex vector fields, including a bifurcation diagram and novel geometric tools.

Findings

Bifurcation diagram of the family $ ext{z}^{k+1} + ext{epsilon}_1 ext{z} + ext{epsilon}_0$

Introduction of periodgon and star domain tools

Description of the moduli space and normal forms

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.