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, Hi...
| Authors: | , |
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| Format: | article |
| Publication Date: | 2014 |
| Country: | España |
| Institution: | Universitat Autònoma de Barcelona |
| Repository: | Dipòsit Digital de Documents de la UAB |
| Language: | English |
| OAI Identifier: | oai:ddd.uab.cat:150702 |
| Online Access: | https://ddd.uab.cat/record/150702 https://dx.doi.org/urn:doi:10.1142/S0218127414500448 |
| Access Level: | Open access |
| Keyword: | Algebraic invariants Bifurcation diagram Phase portrait Quadratic vector fields |
| Summary: | 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 aim is to make a global study of the family QsnSN which is the closure within real quadratic differential systems of the family QsnSN of all such systems which have two semi-elemental saddle-nodes, one finite and one infinite 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 at 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 (the closure is four-dimensional) and we give their bifurcation diagram with respect to a normal form. 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 38 phase portraits for systems in QsnSN(A) (29 in QsnSN(A)) out of which only 3 have limit cycles and 13 possess graphics. The bifurcation diagram for the subfamily (B) yields 25 phase portraits for systems in 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 QsnSN(A) is formed by algebraic surfaces and one surface whose presence was detected numerically. All points in this surface correspond to connections of separatrices. The bifurcation set of QsnSN(B) is formed only by algebraic surfaces. |
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