Characterization of the generic unfolding of a weak focus

TL;DR

This paper provides a geometric framework for understanding the unfolding of weak focus singularities in real analytic vector fields, establishing conditions for orbital equivalence via complexified Poincaré maps.

Contribution

It introduces a novel geometric description of the foliation in generic unfoldings of weak focus singularities and characterizes orbital equivalence through conjugacy of unfolding Poincaré maps.

Findings

Characterization of orbitally equivalent unfoldings via Poincaré map conjugacy

Construction of a complex manifold with elliptic foliation reflecting the unfolding dynamics

Use of quasiconformal surgery to analyze the foliation structure

Abstract

In this paper we give a geometric description of the foliation of a generic real analytic family unfolding a real analytic vector field with a weak focus at the origin, and show that two such families are orbitally analytically equivalent if and only if the families of diffeomorphisms unfolding the complexified Poincar\'e map of the singularities are conjugate. Moreover, by shifting the leaves of the formal normal form in the blow-up (quasiconformal surgery) by means of a fibered transformation along a convenient complex cross-section, one constructs an abstract manifold of complex dimension 2 equipped with an elliptic holomorphic foliation whose monodromy map coincides with a given family of admissible diffeomorphisms.