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

2021 ◽  
Vol 31 (09) ◽  
pp. 2130026
Author(s):  
Joan C. Artés ◽  
Marcos C. Mota ◽  
Alex C. Rezende
Keyword(s):  
Normal Form ◽  
Bifurcation Diagram ◽  
Three Dimensional ◽  
Phase Portraits ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Saddle Node ◽  
Complete Study ◽  
Real Quadratic ◽  
Finite Singularity

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.

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

2021 ◽  
Vol 31 (02) ◽  
pp. 2150026 ◽  
Author(s):  
Joan C. Artés ◽  
Marcos C. Mota ◽  
Alex C. Rezende
Keyword(s):  
Normal Form ◽  
Projective Space ◽  
Bifurcation Diagram ◽  
Phase Portraits ◽  
Real Projective Space ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Saddle Node ◽  
Complete Study ◽  
First Time

Our goal is to make 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 and infinite singularities. This class can be divided into two different families, being (A) possessing the finite saddle-node as the only finite singularity and (B) possessing the finite saddle-node and also a finite simple elemental singularity. In this paper we provide the complete study of the geometry of family (A). The family (A) modulo the action of the affine group and time homotheties are four-dimensional and we give the bifurcation diagram of its closure with respect to a specific normal form, in the four-dimensional real projective space [Formula: see text]. As far as we know, this is the first time that a complete family is studied in the four-dimensional real projective space. The respective bifurcation diagram yields 36 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.

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

2014 ◽  
Vol 24 (04) ◽  
pp. 1450044 ◽  
Author(s):  
Joan C. Artés ◽  
Alex C. Rezende ◽  
Regilene D. S. Oliveira
Keyword(s):  
Normal Form ◽  
Bifurcation Diagram ◽  
Limit Cycles ◽  
Applied Mathematics ◽  
Algebraic Surfaces ◽  
Phase Portraits ◽  
Differential Systems ◽  
Bifurcation Set ◽  
Saddle Node ◽  
The Family

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 paper, we study the bifurcation diagram of the family QsnSN which is the set of all quadratic systems which have at least one finite semi-elemental saddle-node and one infinite semi-elemental saddle-node formed by the collision of two infinite singular points. We study this family with respect to a specific normal form which puts the finite saddle-node at the origin and fixes its eigenvectors on the axes. Our aim is to make a global study of the family [Formula: see text] which is the closure of the set of representatives of QsnSN in the parameter space of that specific normal form. This family can be divided into three different subfamilies according to the position of the infinite saddle-node, namely: (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 times homotheties are four-dimensional. Here, we provide the complete study of the geometry with respect to a normal form of the first two families, (A) and (B). The bifurcation diagram for the subfamily (A) yields 38 phase portraits for systems in [Formula: see text] (29 in QsnSN(A)) out of which only three have limit cycles and 13 possess graphics. The bifurcation diagram for the subfamily (B) yields 25 phase portraits for systems in [Formula: see text] (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 [Formula: see text] is the union of algebraic surfaces and one surface whose presence was detected numerically. All points in this surface correspond to connections of separatrices. The bifurcation set of [Formula: see text] is formed only by algebraic surfaces.

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.

THE GEOMETRY OF QUADRATIC POLYNOMIAL DIFFERENTIAL SYSTEMS WITH A WEAK FOCUS AND AN INVARIANT STRAIGHT LINE

2010 ◽  
Vol 20 (11) ◽  
pp. 3627-3662 ◽  
Author(s):  
JOAN C. ARTÉS ◽  
JAUME LLIBRE ◽  
DANA SCHLOMIUK
Keyword(s):  
Phase Portrait ◽  
Bifurcation Diagram ◽  
Quadratic Differential ◽  
Three Dimensional ◽  
Phase Portraits ◽  
Differential Systems ◽  
Weak Focus ◽  
Bifurcation Set ◽  
Total Multiplicity ◽  
Straight Lines

Planar quadratic differential systems occur in many areas of applied mathematics. Although more than a thousand papers were 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 conduct a global study of the class QWI of all real quadratic differential systems which have a weak focus and invariant straight lines of total multiplicity of at least two. 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 projective space of the parameters of this form. The bifurcation diagram yields 73 phase portraits for systems in QWI plus 26 additional phase portraits with the center at its border points. Algebraic invariants are used to construct the bifurcation set. We show that all systems in QWI necessarily have their weak focus of order one and invariant straight lines of total multiplicity exactly two. The phase portraits are represented on the Poincaré disk. The bifurcation set is algebraic and all points in this set are points of bifurcation of singularities. We prove that there is no phase portrait with limit cycles in this class but that there is a total of five phase portraits with graphics, four having the invariant line as a regular orbit and one phase portrait with an infinity of graphics which are all homoclinic loops inside a heteroclinic graphic with two singularities, both at infinity.

Three-Dimensional Hopf Bifurcation for a Class of Cubic Kolmogorov Model

2014 ◽  
Vol 24 (03) ◽  
pp. 1450036 ◽  
Author(s):  
Chaoxiong Du ◽  
Qinlong Wang ◽  
Wentao Huang
Keyword(s):  
Hopf Bifurcation ◽  
Singular Point ◽  
Three Dimensional ◽  
Singular Values ◽  
Bifurcation Problem ◽  
Disturbed Condition ◽  
Differential Systems ◽  
Kolmogorov Model ◽  
Polynomial Differential Systems ◽  
Fine Focus

We study the Hopf bifurcation for a class of three-dimensional cubic Kolmogorov model by making use of our method (i.e. singular values method). We show that the positive singular point (1, 1, 1) of an investigated model can become a fine focus of 5 order, and moreover, it can bifurcate five small limit cycles under certain coefficients with disturbed condition. In terms of three-dimensional cubic Kolmogorov model, published references can hardly be seen, and our results are new. At the same time, it is worth pointing out that our method is valid to study the Hopf bifurcation problem for other three-dimensional polynomial differential systems.

scholarly journals Centers and limit cycles for a generalized kukles differential systems of degree 8

2021 ◽  
Author(s):  
Loubna Damene ◽  
Rebiha Benterki
Keyword(s):  
Limit Cycles ◽  
Main Tool ◽  
Upper Bounds ◽  
First Integrals ◽  
Averaging Theory ◽  
Phase Portraits ◽  
Differential Systems ◽  
Polynomial Differential Systems

Abstract In this paper we provide all the global phase portraits of the generalized kukles differential systems x= y; y = x + ax8 + bx6y2 + cx4y4 + dx2y6 + ey8; symmetric with respect to the x{axis, with a2 + b2 + c2 + d2 + e2 6= 0, and by using the averaging theory up to seven order, we give the upper bounds of limit cycles which can bifurcate from its center when we perturb it inside the class of all polynomial differential systems of degree 8. The main tool used for proving these results is based in the first integrals of the systems which form the discontinuous piecewise differential systems.

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