2026-04-13T08:23:28Zhttps://recercat.cat/oai/request

oai:recercat.cat:2072/4134142024-11-24T23:43:18Zcom_2072_98col_2072_378192

00925njm 22002777a 4500 dc Artés Ferragud, Joan Carles author Rezende, Alex C. author 2015 Agraïments: The second author is supported by CAPES/DGU grant number BEX 9439/12-9 and CAPES/CSF-PVE's 88887.068602/2014-00 and CAPES/CSF-PVE's 88881.030454/2013-01 and CNPq grant "Projeto Universal 472796/2013-5". 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 with the eigenvector associated with the zero eigenvalue on the horizontal axis 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 the bifurcation diagram of their closure with respect to specific normal forms, in the three-dimensional real projective space. The subfamilies (A) and (B) have already been studied [Artés et al., 2013b] and in this paper we provide the complete study of the geometry of the last family (C). The bifurcation diagram for the subfamily (C) yields 371 topologically distinct phase portraits with and without limit cycles for systems in the closure QsnSN(C) within the representatives of QsnSN(C) given by a chosen normal form. Algebraic invariants are used to construct the bifurcation set. The phase portraits are represented on the Poincaré disk. The bifurcation set of QsnSN(C) is not only algebraic due to the presence of some surfaces found numerically. All points in these surfaces correspond to either connections of separatrices, or the presence of a double limit cycle. Algebraic invariants Bifurcation diagram Finite saddle-node Infinite saddle-node Phase portrait Quadratic differential systems The geometry of quadratic polynomial differential systems with a finite and an infinite saddle-node (C)