PHASE PORTRAITS OF THE QUADRATIC SYSTEMS WITH A POLYNOMIAL INVERSE INTEGRATING FACTOR

2009 ◽  
Vol 19 (03) ◽  
pp. 765-783 ◽  
Author(s):  
BARTOMEU COLL ◽  
ANTONI FERRAGUT ◽  
JAUME LLIBRE
Keyword(s):  
Quadratic Polynomial ◽  
Phase Portraits ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Polynomial Differential Systems ◽  
Integrating Factor ◽  
Inverse Integrating Factor

We classify the phase portraits of all planar quadratic polynomial differential systems having a polynomial inverse integrating factor.

scholarly journals Limit Cycles Bifurcating from a Family of Reversible Quadratic Centers via Averaging Theory

2020 ◽  
Vol 30 (04) ◽  
pp. 2050051
Author(s):  
Jaume Llibre ◽  
Arefeh Nabavi ◽  
Marzieh Mousavi
Keyword(s):  
Periodic Orbits ◽  
Limit Cycles ◽  
Quadratic Polynomial ◽  
Averaging Theory ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Polynomial Differential Systems ◽  
Seventh Order ◽  
Whole Class

Consider the class of reversible quadratic systems [Formula: see text] with [Formula: see text]. These quadratic polynomial differential systems have a center at the point [Formula: see text] and the circle [Formula: see text] is one of the periodic orbits surrounding this center. These systems can be written into the form [Formula: see text] with [Formula: see text]. For all [Formula: see text] we prove that the averaging theory up to seventh order applied to this last system perturbed inside the whole class of quadratic polynomial differential systems can produce at most two limit cycles bifurcating from the periodic orbits surrounding the center (0,0) of that system. Up to now this result was only known for [Formula: see text] (see Li, 2002; Liu, 2012).

scholarly journals Quadratic systems with an invariant conic having Darboux invariants

2018 ◽  
Vol 20 (04) ◽  
pp. 1750033
Author(s):  
Jaume Llibre ◽  
Regilene Oliveira
Keyword(s):  
Vector Fields ◽  
Normal Forms ◽  
Complete Characterization ◽  
Phase Portraits ◽  
Affine Transformations ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Quadratic Vector Fields ◽  
Polynomial Differential Systems

The complete characterization of the phase portraits of real planar quadratic vector fields is very far from being accomplished. As it is almost impossible to work directly with the whole class of quadratic vector fields because it depends on twelve parameters, we reduce the number of parameters to five by using the action of the group of real affine transformations and time rescaling on the class of real quadratic differential systems. Using this group action, we obtain normal forms for the class of quadratic systems that we want to study with at most five parameters. Then working with these normal forms, we complete the characterization of the phase portraits in the Poincaré disc of all planar quadratic polynomial differential systems having an invariant conic [Formula: see text]: [Formula: see text], and a Darboux invariant of the form [Formula: see text] with [Formula: see text].

scholarly journals Topological Classification of Quadratic Polynomial Differential Systems with a Finite Semi-Elemental Triple Saddle

2016 ◽  
Vol 26 (11) ◽  
pp. 1650188 ◽  
Author(s):  
Joan C. Artés ◽  
Regilene D. S. Oliveira ◽  
Alex C. Rezende
Keyword(s):  
Normal Form ◽  
Bifurcation Diagram ◽  
Limit Cycles ◽  
Three Dimensional ◽  
Quadratic Polynomial ◽  
Phase Portraits ◽  
Structure Stability ◽  
Differential Systems ◽  
Bifurcation Set ◽  
Polynomial Differential Systems

The study of planar quadratic differential systems is very important not only because they appear in many areas of applied mathematics but due to their richness in structure, stability and questions concerning limit cycles, for example. Even though many papers have been written on this class of systems, a complete understanding of this family is still missing. Classical problems, and in particular Hilbert’s 16th problem [Hilbert, 1900, 1902], are still open for this family. In this article, we make a global study of the family [Formula: see text] of all real quadratic polynomial differential systems which have a finite semi-elemental triple saddle (triple saddle with exactly one zero eigenvalue). This family modulo the action of the affine group and time homotheties is three-dimensional and we give its bifurcation diagram with respect to a normal form, in the three-dimensional real space of the parameters of this normal form. This bifurcation diagram yields 27 phase portraits for systems in [Formula: see text] counting phase portraits with and without limit cycles. Algebraic invariants are used to construct the bifurcation set and we present the phase portraits on the Poincaré disk. The bifurcation set is not just algebraic due to the presence of a surface found numerically, whose points correspond to connections of separatrices.

scholarly journals The Geometry of Quadratic Polynomial Differential Systems with a Finite and an Infinite Saddle-Node (C)

2015 ◽  
Vol 25 (03) ◽  
pp. 1530009 ◽  
Author(s):  
Joan C. Artés ◽  
Alex C. Rezende ◽  
Regilene D. S. Oliveira
Keyword(s):  
Bifurcation Diagram ◽  
Applied Mathematics ◽  
Three Dimensional ◽  
Horizontal Axis ◽  
Quadratic Polynomial ◽  
Phase Portraits ◽  
Differential Systems ◽  
Bifurcation Set ◽  
Polynomial Differential Systems ◽  
Saddle Node

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, 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 [Formula: see text] 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 [Formula: see text] 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.

scholarly journals Bifurcations of the Riccati Quadratic Polynomial Differential Systems

2021 ◽  
Vol 31 (06) ◽  
pp. 2150094
Author(s):  
Jaume Llibre ◽  
Bruno D. Lopes ◽  
Paulo R. da Silva
Keyword(s):  
Phase Portrait ◽  
Differential System ◽  
Quadratic Polynomial ◽  
Phase Portraits ◽  
Topological Equivalence ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Polynomial Differential System ◽  
Global Phase Portrait ◽  
Poincaré Disk

In this paper, we characterize the global phase portrait of the Riccati quadratic polynomial differential system [Formula: see text] with [Formula: see text], [Formula: see text] nonzero (otherwise the system is a Bernoulli differential system), [Formula: see text] (otherwise the system is a Liénard differential system), [Formula: see text] a polynomial of degree at most [Formula: see text], [Formula: see text] and [Formula: see text] polynomials of degree at most 2, and the maximum of the degrees of [Formula: see text] and [Formula: see text] is 2. We give the complete description of the phase portraits in the Poincaré disk (i.e. in the compactification of [Formula: see text] adding the circle [Formula: see text] of the infinity) modulo topological equivalence.

scholarly journals Phase portraits of quadratic Lotka–Volterra systems with a Darboux invariant in the Poincaré disc

2014 ◽  
Vol 16 (06) ◽  
pp. 1350041 ◽  
Author(s):  
Yudy Bolaños ◽  
Jaume Llibre ◽  
Claudia Valls
Keyword(s):  
Quadratic Polynomial ◽  
Phase Portraits ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Volterra Systems

We characterize the global phase portraits in the Poincaré disc of all the planar Lotka–Volterra quadratic polynomial differential systems having a Darboux invariant.

scholarly journals GLOBAL PHASE PORTRAITS OF QUADRATIC POLYNOMIAL DIFFERENTIAL SYSTEMS WITH A SEMI-ELEMENTAL TRIPLE NODE

2013 ◽  
Vol 23 (08) ◽  
pp. 1350140 ◽  
Author(s):  
JOAN C. ARTÉS ◽  
ALEX C. REZENDE ◽  
REGILENE D. S. OLIVEIRA
Keyword(s):  
Bifurcation Diagram ◽  
Applied Mathematics ◽  
Three Dimensional ◽  
Real Space ◽  
Quadratic Polynomial ◽  
Phase Portraits ◽  
Zero Eigenvalue ◽  
Differential Systems ◽  
Bifurcation Set ◽  
Polynomial Differential Systems

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, 1902], are still open for this family. In this article, we make a global study of the family [Formula: see text] of all real quadratic polynomial differential systems which have a semi-elemental triple node (triple node with exactly one zero eigenvalue). This family modulo the action of the affine group and time homotheties is three-dimensional and we give its bifurcation diagram with respect to a normal form, in the three-dimensional real space of the parameters of this form. This bifurcation diagram yields 28 phase portraits for systems in [Formula: see text] counting phase portraits with and without limit cycles. Algebraic invariants are used to construct the bifurcation set. The phase portraits are represented on the Poincaré disk. The bifurcation set is not only algebraic due to the presence of a surface found numerically. All points in this surface correspond to connections of separatrices.

scholarly journals Quadratic systems with invariant straight lines of total multiplicity two having Darboux invariants

2015 ◽  
Vol 17 (03) ◽  
pp. 1450018 ◽  
Author(s):  
Jaume Llibre ◽  
Regilene D. S. Oliveira
Keyword(s):  
Polynomial Systems ◽  
Quadratic Polynomial ◽  
Phase Portraits ◽  
Quadratic Systems ◽  
Total Multiplicity ◽  
Straight Lines

In this paper, we present the global phase portraits in the Poincaré disc of the planar quadratic polynomial systems which admit invariant straight lines with total multiplicity two and Darboux invariants.

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