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

Abstract

We describe the equivalence classes of germs of generic 2 2 -parameter families of complex vector fields z ˙ = ω ϵ ( z ) \dot z = \omega _\epsilon (z) on C \mathbb {C} unfolding a singular parabolic point of multiplicity k + 1 k+1 : ω 0 = z k + 1 + o ( z 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 z ˙ = z k + 1 + ϵ 1 z + ϵ 0 \dot z = z^{k+1}+\epsilon _1z+\epsilon _0 over C P 1 \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.