scholarly journals The cyclicity of the period annulus of a reversible quadratic system

2021 ◽  
pp. 1-10
Author(s):  
Changjian Liu ◽  
Chengzhi Li ◽  
Jaume Llibre
Keyword(s):  
Limit Cycles ◽  
Mathematical Proof ◽  
First Integral ◽  
Quadratic System ◽  
Quadratic Polynomial ◽  
Unperturbed System ◽  
Computer Assisted ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Period Annulus

We prove that perturbing the periodic annulus of the reversible quadratic polynomial differential system $\dot x=y+ax^2$ , $\dot y=-x$ with a ≠ 0 inside the class of all quadratic polynomial differential systems we can obtain at most two limit cycles, including their multiplicities. Since the first integral of the unperturbed system contains an exponential function, the traditional methods cannot be applied, except in Figuerasa, Tucker and Villadelprat (2013, J. Diff. Equ., 254, 3647–3663) a computer-assisted method was used. In this paper, we provide a method for studying the problem. This is also the first purely mathematical proof of the conjecture formulated by Dumortier and Roussarie (2009, Discrete Contin. Dyn. Syst., 2, 723–781) for q ⩽ 2. The method may be used in other problems.

scholarly journals The non-existence, existence and uniqueness of limit cycles for quadratic polynomial differential systems

2017 ◽  
Vol 149 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Jaume Llibre ◽  
Xiang Zhang
Keyword(s):  
Differential System ◽  
Limit Cycles ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Quadratic Polynomial ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Polynomial Differential System ◽  
Uniqueness Of Limit Cycles

AbstractWe provide sufficient conditions for the non-existence, existence and uniqueness of limit cycles surrounding a focus of a quadratic polynomial differential system in the plane.

scholarly journals Integrability and Limit Cycles via First Integrals

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1736
Author(s):  
Jaume Llibre
Keyword(s):  
Phase Portrait ◽  
Differential System ◽  
Limit Cycles ◽  
Applied Mathematics ◽  
First Integral ◽  
First Integrals ◽  
Differential Systems ◽  
Polynomial Differential Systems

In many problems appearing in applied mathematics in the nonlinear ordinary differential systems, as in physics, chemist, economics, etc., if we have a differential system on a manifold of dimension, two of them having a first integral, then its phase portrait is completely determined. While the existence of first integrals for differential systems on manifolds of a dimension higher than two allows to reduce the dimension of the space in as many dimensions as independent first integrals we have. Hence, to know first integrals is important, but the following question appears: Given a differential system, how to know if it has a first integral? The symmetries of many differential systems force the existence of first integrals. This paper has two main objectives. First, we study how to compute first integrals for polynomial differential systems using the so-called Darboux theory of integrability. Furthermore, second, we show how to use the existence of first integrals for finding limit cycles in piecewise differential systems.

scholarly journals On the birth and death of algebraic limit cycles in quadratic differential systems

2020 ◽  
pp. 1-20
Author(s):  
JAUME LLIBRE ◽  
REGILENE OLIVEIRA ◽  
YULIN ZHAO
Keyword(s):  
Limit Cycles ◽  
Open Problem ◽  
Quadratic Differential ◽  
Quadratic Polynomial ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Algebraic Limit Cycles ◽  
Algebraic Limit

In 1958 started the study of the families of algebraic limit cycles in the class of planar quadratic polynomial differential systems. In the present we known one family of algebraic limit cycles of degree 2 and four families of algebraic limit cycles of degree 4, and that there are no limit cycles of degree 3. All the families of algebraic limit cycles of degree 2 and 4 are known, this is not the case for the families of degree higher than 4. We also know that there exist two families of algebraic limit cycles of degree 5 and one family of degree 6, but we do not know if these families are all the families of degree 5 and 6. Until today it is an open problem to know if there are algebraic limit cycles of degree higher than 6 inside the class of quadratic polynomial differential systems. Here we investigate the birth and death of all the known families of algebraic limit cycles of quadratic polynomial differential systems.

Limit Cycles of Perturbed Cubic Isochronous Center via the Second Order Averaging Method

2014 ◽  
Vol 24 (03) ◽  
pp. 1450035 ◽  
Author(s):  
Shimin Li ◽  
Yulin Zhao
Keyword(s):  
Limit Cycles ◽  
Averaging Method ◽  
Second Order ◽  
Isochronous Center ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Period Annulus ◽  
Isochronous System ◽  
Cubic Polynomial

In this paper, we bound the number of limit cycles for a class of cubic reversible isochronous system inside the class of all cubic polynomial differential systems. By applying the averaging method of second order to this system, it is proved that at most eight limit cycles can bifurcate from the period annulus. Moreover, this bound is sharp.

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 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.

scholarly journals On the Limit Cycles for Continuous and Discontinuous Cubic Differential Systems

2016 ◽  
Vol 2016 ◽  
pp. 1-11
Author(s):  
Ziguo Jiang
Keyword(s):  
Periodic Orbits ◽  
Limit Cycles ◽  
Quadratic Polynomial ◽  
Averaging Theory ◽  
Isochronous Center ◽  
Discontinuous Systems ◽  
Differential Systems ◽  
First Order ◽  
Polynomial Differential Systems ◽  
Cubic Polynomial

We study the number of limit cycles for the quadratic polynomial differential systemsx˙=-y+x2,y˙=x+xyhaving an isochronous center with continuous and discontinuous cubic polynomial perturbations. Using the averaging theory of first order, we obtain that 3 limit cycles bifurcate from the periodic orbits of the isochronous center with continuous perturbations and at least 7 limit cycles bifurcate from the periodic orbits of the isochronous center with discontinuous perturbations. Moreover, this work shows that the discontinuous systems have at least 4 more limit cycles surrounding the origin than the continuous ones.

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 Algebraic Limit Cycles Bifurcating from Algebraic Ovals of Quadratic Centers

2018 ◽  
Vol 28 (12) ◽  
pp. 1850145 ◽  
Author(s):  
Jaume Llibre ◽  
Yun Tian
Keyword(s):  
Limit Cycles ◽  
Algebraic Curves ◽  
Polynomial Systems ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Period Annulus ◽  
Invariant Algebraic Curves ◽  
Relevant Role ◽  
Algebraic Limit Cycles ◽  
Algebraic Limit

In the integrability of polynomial differential systems it is well known that the invariant algebraic curves play a relevant role. Here we will see that they can also play an important role with respect to limit cycles.In this paper, we study quadratic polynomial systems with an algebraic periodic orbit of degree [Formula: see text] surrounding a center. We show that there exists only one family of such systems satisfying that an algebraic limit cycle of degree [Formula: see text] can bifurcate from the period annulus of the mentioned center under quadratic perturbations.

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