The geometry of quadratic polynomial differential systems with a finite and an infinite saddle-node (A,B)

TL;DR

This paper provides a comprehensive geometric and bifurcation analysis of quadratic polynomial differential systems with specific saddle-node configurations, classifying phase portraits and bifurcation diagrams for two subfamilies.

Contribution

It offers a complete geometric and bifurcation classification of two subfamilies of quadratic systems with finite and infinite saddle-nodes, including detailed bifurcation diagrams and phase portraits.

Findings

29 phase portraits for subfamily (A)

16 phase portraits for subfamily (B)

Identification of algebraic bifurcation surfaces

Abstract

The goal is to make a global study of the family QsnSN of all real quadratic polynomial differential systems which have a finite semi-elemental saddle-node and an infinite saddle-node formed by the collision of two infinite singular points. This family can be divided into three different subfamilies, all of them with the finite saddle-node in the origin of the plane with the eigenvectors on the axes and (A) with the infinite saddle-node in the horizontal axis, (B) with the infinite saddle-node in the vertical axis and (C) with the infinite saddle-node in the bisector of the first and third quadrants. These three subfamilies modulo the action of the affine group and time homotheties are three-dimensional and we give their bifurcation diagram with respect to a normal form, in the three-dimensional real space of the parameters of these forms. In this paper we provide the complete study of the geometry of the first two families, (A) and (B). The bifurcation diagram for the subfamily (A) yields 29 phase portraits for systems in QsnSN(A) counting phase portraits with and without limit cycles, while the bifurcation diagram for the subfamily (B) yields 16 phase portraits for systems in QsnSN(B) under the same conditions. Case (C) will yield quite more cases and will have an independent paper in short. Algebraic invariants are used to construct the bifurcation set. The phase portraits are represented on the Poincar\'e disk. The bifurcation set of QsnSN(A) is not only algebraic due to the presence of a surface found numerically. All points in this surface correspond to connections of separatrices.