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

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 [Hilbert, 1900, 1902], are still open for this family. In this paper, we study the bifurcation diagram of the family QsnSN which is the set of all quadratic systems which have at least one finite semi-elemental saddle-node and one infinite semi-elemental saddle-node formed by the collision of two infinite singular points. We study this family with respect to a specific normal form which puts the finite saddle-node at the origin and fixes its eigenvectors on the axes. Our aim is to make a global study of the family $\overline{QsnSN}$ which is the closure of the set of representatives of QsnSN in the parameter space of that specific normal form. This family can be divided into three different subfamilies according to the position of the infinite saddle-node, namely: (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 times homotheties are four-dimensional. Here, we provide the complete study of the geometry with respect to a normal form of the first two families, (A) and (B). The bifurcation diagram for the subfamily (A) yields 38 phase portraits for systems in $\overline{QsnSN(A)}$ (29 in QsnSN(A)) out of which only three have limit cycles and 13 possess graphics. The bifurcation diagram for the subfamily (B) yields 25 phase portraits for systems in $\overline{QsnSN(B)}$ (16 in QsnSN(B)) out of which 11 possess graphics. None of the 25 portraits has limit cycles. Case (C) will yield many more phase portraits and will be written separately in a forthcoming new paper. Algebraic invariants are used to construct the bifurcation set. The phase portraits are represented on the Poincaré disk. The bifurcation set of $\overline{QsnSN(A)}$ is the union of algebraic surfaces and one surface whose presence was detected numerically. All points in this surface correspond to connections of separatrices. The bifurcation set of $\overline{QsnSN(B)}$ is formed only by algebraic surfaces. © 2014 World Scientific Publishing Company.