Global phase portraits of quadratic polynomial differential systems with a semi-elemental triple node

TL;DR

This paper provides a comprehensive global analysis of quadratic polynomial differential systems with a semi-elemental triple node, classifying their phase portraits and bifurcations using algebraic invariants and geometric representations.

Contribution

It introduces a detailed bifurcation diagram for the family QTN, revealing 28 distinct phase portraits and employing algebraic invariants to analyze connections of separatrices.

Findings

28 phase portraits identified, with and without limit cycles

Bifurcation set includes algebraic and numerical components

Connections of separatrices are characterized on the bifurcation surface

Abstract

Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers have been written on these systems, a complete understanding of this family is still missing. Classical problems, and in particular, Hilbert's 16th problem, are still open for this family. In this article we make a global study of the family QTN of all real quadratic polynomial differential systems which have a semi-elemental triple node (triple node with exactly one zero eigenvalue). This family modulo the action of the affine group and time homotheties is three-dimensional and we give its bifurcation diagram with respect to a normal form, in the three-dimensional real space of the parameters of this form. This bifurcation diagram yields 28 phase portraits for systems in QTN counting phase portraits with and without limit cycles. Algebraic invariants are used to construct the bifurcation set. The phase portraits are represented on the Poincar\'e disk. The bifurcation set is not only algebraic due to the presence of a surface found numerically. All points in this surface correspond to connections of separatrices.