scholarly journals Fifteen Limit Cycles Bifurcating from a Perturbed Cubic Center

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Amor Menaceur ◽  
Mufda Alrawashdeh ◽  
Sahar Ahmed Idris ◽  
Hala Abd-Elmageed
Keyword(s):  
Limit Cycles ◽  
Averaging Theory ◽  
Bifurcation Of Limit Cycles ◽  
Differential Systems ◽  
First Order ◽  
Polynomial Differential Systems ◽  
Period Annulus ◽  
Cubic Polynomial

In this work, we study the bifurcation of limit cycles from the period annulus surrounding the origin of a class of cubic polynomial differential systems; when they are perturbed inside the class of all polynomial differential systems of degree six, we obtain at most fifteenth limit cycles by using the averaging theory of first order.

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.

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.

Bifurcation of Limit Cycles from the Center of a Family of Cubic Polynomial Vector Fields

2018 ◽  
Vol 28 (05) ◽  
pp. 1850063 ◽  
Author(s):  
Shiyou Sui ◽  
Liqin Zhao
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Abelian Integral ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Bifurcation Of Limit Cycles ◽  
First Order ◽  
Number Of Zeros ◽  
Period Annulus ◽  
Cubic Polynomial

In this paper, we consider the number of zeros of Abelian integral for the system [Formula: see text], [Formula: see text], where [Formula: see text], [Formula: see text], and [Formula: see text] are arbitrary polynomials of degree [Formula: see text]. We obtain that [Formula: see text] if [Formula: see text] and [Formula: see text] if [Formula: see text], where [Formula: see text] is the maximum number of limit cycles bifurcating from the period annulus up to the first order in [Formula: see text]. So, the bounds for [Formula: see text] or [Formula: see text], [Formula: see text], [Formula: see text] are exact.

scholarly journals On the Number of Limit Cycles for Discontinuous Generalized Liénard Polynomial Differential Systems

2015 ◽  
Vol 25 (10) ◽  
pp. 1550131 ◽  
Author(s):  
Fangfang Jiang ◽  
Junping Shi ◽  
Jitao Sun
Keyword(s):  
Differential System ◽  
Limit Cycles ◽  
Upper Bound ◽  
Averaging Theory ◽  
Differential Systems ◽  
First Order ◽  
Polynomial Differential Systems ◽  
Polynomial Differential System

In this paper, we investigate the number of limit cycles for a class of discontinuous planar differential systems with multiple sectors separated by many rays originating from the origin. In each sector, it is a smooth generalized Liénard polynomial differential system x′ = -y + g1(x) + f1(x)y and y′ = x + g2(x) + f2(x)y, where fi(x) and gi(x) for i = 1, 2 are polynomials of variable x with any given degree. By the averaging theory of first-order for discontinuous differential systems, we provide the criteria on the maximum number of medium amplitude limit cycles for the discontinuous generalized Liénard polynomial differential systems. The upper bound for the number of medium amplitude limit cycles can be attained by specific examples.

scholarly journals Limit Cycles for the Class ofD-Dimensional Polynomial Differential Systems

2016 ◽  
Vol 2016 ◽  
pp. 1-5
Author(s):  
Zouhair Diab ◽  
Amar Makhlouf
Keyword(s):  
Differential System ◽  
Limit Cycles ◽  
Averaging Theory ◽  
Differential Systems ◽  
First Order ◽  
Polynomial Differential Systems ◽  
Perturbed System

We perturb the differential systemx˙1=-x2(1+x1),x˙2=x1(1+x1), andx˙k=0fork=3,…,dinside the class of all polynomial differential systems of degreeninRd, and we prove that at mostnd-1limit cycles can be obtained for the perturbed system using the first-order averaging theory.

Bifurcation of Limit Cycles from a Class of Piecewise Smooth Systems with Two Vertical Straight Lines of Singularity

2017 ◽  
Vol 27 (10) ◽  
pp. 1750157 ◽  
Author(s):  
Yunfei Gao ◽  
Linping Peng ◽  
Changjian Liu
Keyword(s):  
Limit Cycles ◽  
Averaging Theory ◽  
Piecewise Smooth Systems ◽  
Bifurcation Of Limit Cycles ◽  
Argument Principle ◽  
First Order ◽  
Piecewise Polynomials ◽  
Period Annulus ◽  
Number Formula ◽  
Straight Lines

Perturbing the following systems [Formula: see text] having a linear center with two vertical straight lines ([Formula: see text]) of singularity inside the class of all piecewise polynomials of degree [Formula: see text], we give the estimate of the maximum number [Formula: see text] of limit cycles bifurcating from the period annulus around the center based on the argument principle and the first order averaging theory. It shows that [Formula: see text] is at least [Formula: see text] bigger than the maximum number of limit cycles bifurcating from the period annulus around the linear center with two parallel straight lines of singularity in [Li & Liu, 2015].

scholarly journals The Cubic Polynomial Differential Systems with two Circles as Algebraic Limit Cycles

2018 ◽  
Vol 18 (1) ◽  
pp. 183-193 ◽  
Author(s):  
Jaume Giné ◽  
Jaume Llibre ◽  
Claudia Valls
Keyword(s):  
Limit Cycles ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Cubic Polynomial ◽  
Algebraic Limit Cycles ◽  
Algebraic Limit

AbstractIn this paper we characterize all cubic polynomial differential systems in the plane having two circles as invariant algebraic limit cycles.

scholarly journals Limit Cycles of a Class of Perturbed Differential Systems via the First-Order Averaging Method

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Amor Menaceur ◽  
Salah Mahmoud Boulaaras ◽  
Amar Makhlouf ◽  
Karthikeyan Rajagobal ◽  
Mohamed Abdalla
Keyword(s):  
Hamiltonian System ◽  
Periodic Orbits ◽  
Limit Cycles ◽  
Averaging Method ◽  
Differential Systems ◽  
First Order ◽  
Polynomial Differential Systems

By means of the averaging method of the first order, we introduce the maximum number of limit cycles which can be bifurcated from the periodic orbits of a Hamiltonian system. Besides, the perturbation has been used for a particular class of the polynomial differential systems.

Sign in / Sign up

Export Citation Format

Share Document