Structurally stable quadratic vector fields

Artés, Joan C.; Kooij, Robert E.; Llibre, Jaume

doi:10.1090/memo/0639

Cited by 27 publications

(58 citation statements)

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“…We observe also that the knowledge of quadratic vector fields with saddle connections is important because they are in the boundary of the structurally stables quadratic vector fields, classified by Artés, Kooij and Libbre [3].…”

Section: Introductionmentioning

confidence: 82%

Planar Quadratic Vector Fields with Finite Saddle Connection on a Straight Line (Non-convex Case)

Carrião

Gomes

Ruas

2008

Qual. Theory Dyn. Syst.

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Section: Introductionmentioning

confidence: 82%

Planar Quadratic Vector Fields with Finite Saddle Connection on a Straight Line (Non-convex Case)

Carrião

Gomes

Ruas

2008

Qual. Theory Dyn. Syst.

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“…Picture C 2 .2(a) (1) We now consider perturbations (C 2ε,δ .2) of the systems in the above normal form found in [59, p. 771]:…”

Section: Normal Forms and Bifurcation Diagramsmentioning

confidence: 99%

Topological and polynomial invariants, moduli spaces, in classification problems of polynomial vector fields

Schlomiuk¹

2014

Publicacions Matemàtiques

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Abstract:We describe the origin and evolution of ideas on topological and polynomial invariants and their interaction, in problems of classification of polynomial vector fields. The concept of moduli space is discussed in the last section and we indicate its value in understanding the dynamics of families of such systems. Our interest here is in the concepts and the way they interact in the process of topologically classifying polynomial vector fields. We survey the literature giving an ample list of references and we illustrate the ideas on the testing ground of families of quadratic vector fields. In particular, the role of polynomial invariants is illustrated in the proof of our theorem in the section next to last. These concepts have proven their worth in a number of classification results, among them the most recent work on the geometric classification of the whole class of quadratic vector fields, according to their configurations of infinite singularities. An analog work including both finite and infinite singularities of the whole quadratic class, joint work with J. C. Artés, J. Llibre, and N. Vulpe, is in progress.

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“…In [Artés et al, 1998] the authors classified all the structurally stable quadratic planar systems modulo limit cycles, also known as the codimension-zero quadratic systems (roughly speaking, those systems whose all singularities, finite and infinite, are simple, with no separatrix connection, and where any nest of limit cycles is considered a single point with the stability of the outer limit cycle) by proving the existence of 44 topologically different phase portraits for these systems. The natural continuation in this idea is the classification of the structurally unstable quadratic systems of codimension-one, i.e.…”

Section: Introduction Brief Review Of the Literature And Statement Omentioning

confidence: 99%

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

Artés

Rezende

Oliveira

2014

Int. J. Bifurcation Chaos

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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.

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Structurally stable quadratic vector fields

Cited by 27 publications

References 0 publications

Planar Quadratic Vector Fields with Finite Saddle Connection on a Straight Line (Non-convex Case)

Planar Quadratic Vector Fields with Finite Saddle Connection on a Straight Line (Non-convex Case)

Topological and polynomial invariants, moduli spaces, in classification problems of polynomial vector fields

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

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