scholarly journals Global Phase Portraits of Kukles Differential Systems with Homogeneous Polynomial Nonlinearities of Degree 6 Having a Center and Their Small Limit Cycles

2016 ◽  
Vol 26 (03) ◽  
pp. 1650044 ◽  
Author(s):  
Jaume Llibre ◽  
Maurício Fronza da Silva
Keyword(s):  
Limit Cycles ◽  
Homogeneous Polynomial ◽  
Averaging Theory ◽  
Sixth Order ◽  
Phase Portraits ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
The Family ◽  
Poincaré Disk

We provide the nine topological global phase portraits in the Poincaré disk of the family of the centers of Kukles polynomial differential systems of the form [Formula: see text] [Formula: see text] where [Formula: see text] and [Formula: see text] are real parameters satisfying [Formula: see text] Using averaging theory up to sixth order we determine the number of limit cycles which bifurcate from the origin when we perturb this system first inside the class of all homogeneous polynomial differential systems of degree [Formula: see text] and second inside the class of all polynomial differential systems of degree [Formula: see text]

scholarly journals Centers and Limit Cycles of Generalized Kukles Polynomial Differential Systems: Phase Portraits and Limit Cycles

2020 ◽  
pp. 378-398
Author(s):  
Ahlam Belfar ◽  
Rebiha Benterki
Keyword(s):  
Differential System ◽  
Limit Cycles ◽  
Averaging Theory ◽  
Sixth Order ◽  
Phase Portraits ◽  
Differential Systems ◽  
Polynomial Differential Systems

In this work, we give the seven global phase portraits in the Poincar´e disc of the Kukles differential system given by x˙ = −y, y˙ = x + ax8 + bx4y4 + cy8, where x, y ∈ R and a, b, c ∈ R with a2 + b2 + c2 ̸= 0. Moreover, we perturb these system inside all classes of polynomials of eight degrees, then we use the averaging theory up sixth order to study the number of limit cycles which can bifurcate from the origin of coordinates of the Kukles differential system

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.

scholarly journals On the number of limit cycles of a class of polynomial differential systems

2012 ◽  
Vol 468 (2144) ◽  
pp. 2347-2360 ◽  
Author(s):  
Jaume Llibre ◽  
Clàudia Valls
Keyword(s):  
Periodic Orbits ◽  
Limit Cycles ◽  
Upper Bound ◽  
Second Order ◽  
Averaging Theory ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Generalized Polynomial

We study the number of limit cycles of polynomial differential systems of the form where g 1 , f 1 , g 2 and f 2 are polynomials of a given degree. Note that when g 1 ( x )= f 1 ( x )=0, we obtain the generalized polynomial Liénard differential systems. We provide an accurate upper bound of the maximum number of limit cycles that the above system can have bifurcating from the periodic orbits of the linear centre , using the averaging theory of first and second order.

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 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).

THE GEOMETRY OF QUADRATIC DIFFERENTIAL SYSTEMS WITH A WEAK FOCUS OF SECOND ORDER

2006 ◽  
Vol 16 (11) ◽  
pp. 3127-3194 ◽  
Author(s):  
JOAN C. ARTÉS ◽  
JAUME LLIBRE ◽  
DANA SCHLOMIUK
Keyword(s):  
Limit Cycles ◽  
Quadratic Differential ◽  
Second Order ◽  
Phase Portraits ◽  
Nonlinear Phenomena ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Weak Focus ◽  
Bifurcation Set ◽  
The Family

Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers were written on these systems, a complete understanding of this class is still missing. Classical problems, and in particular, Hilbert's 16th problem [Hilbert, 1900, 1902], are still open for this class. In this article we make an interdisciplinary global study of the subclass [Formula: see text] which is the closure within real quadratic differential systems, of the family QW2 of all such systems which have a weak focus of second order. This class [Formula: see text] also includes the family of all quadratic differential systems possessing a weak focus of third order and topological equivalents of all quadratic systems with a center. The bifurcation diagram for this class, done in the adequate parameter space which is the three-dimensional real projective space, is quite rich in its complexity and yields 373 subsets with 126 phase portraits for [Formula: see text], 95 for QW2, 20 having limit cycles but only three with the maximum number of limit cycles (two) within this class. The phase portraits are always represented in the Poincaré disc. The bifurcation set is formed by an algebraic set of bifurcations of singularities, finite or infinite and by a set of points which we suspect to be analytic corresponding to global separatrices which have connections. Algebraic invariants were needed to construct the algebraic part of the bifurcation set, symbolic computations to deal with some quite complex invariants and numerical calculations to determine the position of the analytic bifurcation set of connections. The global geometry of this class [Formula: see text] reveals interesting bifurcations phenomena; for example, all phase portraits with limit cycles in this class can be produced by perturbations of symmetric (reversible) quadratic systems with a center. Many other nonlinear phenomena displayed here form material for further studies.

scholarly journals A number of limit cycle of sextic polynomial differential systems via the averaging theory

2021 ◽  
Vol 39 (4) ◽  
pp. 181-197
Author(s):  
Amour Menaceur ◽  
Salah Boulaaras
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Averaging Theory ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Period Annulus ◽  
Cubic System ◽  
Cubic Polynomial

The main purpose of this paper is to study the number of limit cycles of sextic polynomial differential systems (SPDS) via the averaging theory which is an extension to the study of cubic polynomial vector fields in (Nonlinear Analysis 66 (2007), 1707--1721), where we provide an accurate upper bound of the maximum number of limit cycles that SPDS can have bifurcating from the period annulus surrounding the origin of a class of cubic system.

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