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 fieldsż = ω (z) on C unfolding a singular parabolic point of multiplicity 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 fieldṡ z = z k+1 + 1 z + 0 over 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.

Analytic Classification of Families of Linear Differential Systems Unfolding a Resonant Irregular Singularity

“…is a rational vector field on C, but becomes a polynomial vector field in the coordinate s −1 on CP 1 = C ∪ {∞} with a regular point at s = ∞. The real dynamics of complex polynomial vector fields on CP 1 has been extensively studied in [6,9,11] (see also [24,29,37,38]). Some of the basic properties when applied to e iω χ can be summarized as: In particular, there are no limit cycles.…”

Two-parameter unfolding of a parabolic point of a vector field in ℂ fixing the origin

“…Hence, it is natural to integrate the two cases in a 2-parameter family and study the family of vector fieldsż = z K+1 + 1 z + 0 . This is what has been done in [4]. A different problem is to study what occurs in generic 2-parameter unfoldings g of g defined in (1.2), through the 2-parameter family f = g •q .…”

“…multiplicity k + 1). The paper [1] studied the case m = 1, and the paper [4], the case m = 2. When m = 1, the singular points ofż = z k+1 + are located at the vertices of a regular polygon.…”

“…The periodgon is a polygon with k + 1 sides, one for each fixed point, which completely characterizes the polynomial vector field up to a rotation of order k, provided it is monic and centered. The periodgon was later generalized in [4] to all generic polynomial vector fieldsż = dz/dt = P (z), where generic is understood in a different sense defined below. The periodgon is a polygon whose edges are given by oriented vectors corresponding to the periods of the different singular points of the vector field, in a proper order.…”

“…For instance, for the familyż = z k+1 + , 2(k + 1) open sets are needed for the Douady-Estrada-Sentenac description, while one open set is enough with the periodgon description (see [1]). One open set with slits is conjectured to be sufficient with the periodgon description for the familyż = z k+1 + 1 z + 0 (see [4]), while two are necessary forż = z 3 + 1 z + 0 with the Douady-Estrada-Sentenac point of view (see [6]). For the family (1.5) we conjecture that k − 1 open sets are sufficient: these domains are exactly the ones to which we can reduce our study using the symmetries of the system.…”