Quadratic Differential Systems with a Finite Saddle-Node and an Infinite Saddle-Node (1, 1) SN-(B)

This paper presents a global study of the class QsnSN11 of all real quadratic polynomial differential systems which have a finite semi-elemental saddle-node and an infinite saddle-node formed by the coalescence of a finite singularity and an infinite singularity. This class can be divided into two different families, namely, QsnSN11(A) phase portraits possessing a finite saddle-node as the only finite singularity and QsnSN11(B) phase portraits possessing a finite saddle-node and also a simple finite elemental singularity. Each one of these two families is given by a specific normal form. The study of family QsnSN11(A) was reported in [Artés et al., 2020b] where the authors obtained 36 topologically distinct phase portraits for systems in the closure QsnSN11(A)¯. In this paper, we provide the complete study of the geometry of family QsnSN11(B). This family which modulo the action of the affine group and time homotheties is three-dimensional and we give the bifurcation diagram of its closure with respect to a specific normal form, in the three-dimensional real projective space. The respective bifurcation diagram yields 631 subsets with 226 topologically distinct phase portraits for systems in the closure QsnSN11(B)¯ within the representatives of QsnSN11(B) given by a specific normal form. Some of these phase portraits are proven to have at least three limit cycles.