The Geometry of Quadratic Polynomial Differential Systems with a Finite and an Infinite Saddle-Node (A, B)

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 [Hilbert, 1900, Hilbert, 1902, are still open for this family. Our 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 Poincaré 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.

Structurally unstable quadratic vector fields of codimension two: families possessing one finite saddle-node and a separatrix connection

“…In summary, the bifurcations from phase portraits in [10] with a separatrix connection into class (AD) do not bring any new phase portrait from those obtained from [11]. [15] and [16] with an invariant straight line…”

“…Then we obtain phase portrait U 2 AD,72 . By the way, there is a small mistake in the image of phase portrait V 1 in Figure 3 of [15] since it does not deploys the saddle-node at [0 : 1 : 0]. We add here the correct image as well as its bifurcation into U 2 AD,72 , see Figure 66.…”

“…So we can obtain a system of class (AD) in four different forms. We may keep the horizontal straight line, break the loop in two different ways, and split the infinite saddle-node in two real singularities, but all this is equivalent to make the bifurcations from cases V 3 and V 9 from [15] (A) that we have already described. So, what we need to do is break the straight line and keep the loop.…”

“…Concretely we will take some families having a finite saddle-node and an infinite saddle-node 0 2 SN and families which apart from the finite saddle-node have an infinite saddle-node 1 1 SN . These families are studied in two papers each, [15] and [16] for the first case and [10] and [11] for the second. All four papers are done using similar techniques, and the notation used to describe the phase portraits is similar.…”

“…In a very similar way, we are going to use papers [15] and [16] to obtain most of the rest of phase portraits of class (AD).…”