Planar vector fields in the kernel of a 1--form

TL;DR

This paper classifies planar vector fields that are in the kernel of a 1-form, providing local models and unfoldings to study their bifurcations, including integrable and non-integrable cases.

Contribution

It offers a comprehensive classification of planar vector fields in the kernel of a 1-form and constructs their unfoldings for bifurcation analysis.

Findings

Complete list of local models for such vector fields

Construction of transversal unfoldings

Analysis of bifurcations including integrable fields

Abstract

We classify, up to a natural equivalence relation, vector fields of the plane which belong to the kernel of a 1--form. This form can be closed, in which case the vector fields are integrable, or not, in which case the differential of the form defines a, possibly singular, symplectic form. In every case, we provide a fairly complete list of local models for such fields and construct their transversal unfoldings. Thus, the local bifurcations of vector fields of interest can be studied, among them being the integrable fields of the plane.