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

Abstract

This paper presents a global study of the class [Formula: see text] 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, [Formula: see text] phase portraits possessing a finite saddle-node as the only finite singularity and [Formula: see text] 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 [Formula: see text] was reported in [Artés et al., 2020b] where the authors obtained [Formula: see text] topologically distinct phase portraits for systems in the closure [Formula: see text]. In this paper, we provide the complete study of the geometry of family [Formula: see text]. 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 [Formula: see text] within the representatives of [Formula: see text] given by a specific normal form. Some of these phase portraits are proven to have at least three limit cycles.

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

“…Please note that several mistypes were detected in [16] and many of them were corrected in the Appendix A of [11]. In fact here we have found another mistype that Lin theorem may correspond to a region where the anti-saddle is a node instead of a focus.…”

“…Note that we have used a still unnumbered phase portrait of codimension three as an intermediate step to describe the bifurcation. 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…”

“…Take system (5) with (c, e, h, m, ε) = (0, 10, 1, 4, 0). This system has phase portrait 4S 1 from [11] (see Figure 61). If we perturb ε ̸ = 0 the vertical invariant straight line (which is connection of separatrices) persists.…”

“…As well as with the papers [10] and [11] now we must focus our attention in the cases from [15] and [16] where we have a separatrix connection which is not an invariant straight line. To know, we have just one case in [15] (A) which is 7S 1 , none in [15] (B) and 48 in [16] (all those named 7S x ).…”

“…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.…”