Darboux polynomials and global phase portraits for the D_2 vector field

TL;DR

This paper analyzes a specific symmetric vector field in three dimensions, identifying all Darboux polynomials and exploring its global behavior to facilitate comparisons between similar vector fields.

Contribution

It provides the complete list of Darboux polynomials for the D_2 symmetric vector field and discusses its global qualitative dynamics.

Findings

Complete set of Darboux polynomials identified

Insights into the global phase portrait of the D_2 field

Foundation for comparing vector fields via Darboux polynomial modules

Abstract

We study a vector field of R^3 equivariant under the D_2 symmetry group, called "the D_2 field" in the literature. We construct the complete list of Darboux polynomials for it, solving the partial differential equation defining them. We also use these polynomials to comment on its global qualitative behaviour. This is meant to be a first step towards the comparison of vector fields based on the module generated by their Darboux polynomials.