The number of limit cycles bifurcating from the periodic orbits of an isochronous center

2018 ◽  
Vol 42 (3) ◽  
pp. 821-829 ◽  
Author(s):  
Meryem Bey ◽  
Sabrina Badi ◽  
Khairedine Fernane ◽  
Amar Makhlouf
Keyword(s):  
Periodic Orbits ◽  
Limit Cycles ◽  
Isochronous Center

ON THE NUMBER OF ZEROS OF ABELIAN INTEGRAL FOR A CUBIC ISOCHRONOUS CENTER

2012 ◽  
Vol 22 (01) ◽  
pp. 1250016 ◽  
Author(s):  
KUILIN WU ◽  
YUNLIN ZHAO
Keyword(s):  
Periodic Orbits ◽  
Limit Cycles ◽  
Upper Bound ◽  
Abelian Integral ◽  
Isochronous Center ◽  
Number Of Zeros ◽  
Period Annulus

In this paper, we study the number of limit cycles that bifurcate from the periodic orbits of a cubic reversible isochronous center under cubic perturbations. It is proved that in this situation the least upper bound for the number of zeros (taking into account the multiplicity) of the Abelian integral associated with the system is equal to four. Moreover, for each k = 0, 1, …, 4, there are perturbations that give rise to exactly k limit cycles bifurcating from the period annulus.

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.

BIFURCATION OF LIMIT CYCLES AND ISOCHRONOUS CENTERS FOR A QUARTIC SYSTEM

2013 ◽  
Vol 23 (10) ◽  
pp. 1350172 ◽  
Author(s):  
WENTAO HUANG ◽  
AIYONG CHEN ◽  
QIUJIN XU
Keyword(s):  
Limit Cycles ◽  
Necessary Conditions ◽  
Polynomial System ◽  
Isochronous Center ◽  
Eighth Order ◽  
Bifurcation Of Limit Cycles ◽  
Quartic Polynomial ◽  
Fine Focus ◽  
Sufficient And Necessary Conditions ◽  
Isochronous Centers

For a quartic polynomial system we investigate bifurcations of limit cycles and obtain conditions for the origin to be a center. Computing the singular point values we find also the conditions for the origin to be the eighth order fine focus. It is proven that the system can have eight small amplitude limit cycles in a neighborhood of the origin. To the best of our knowledge, this is the first example of a quartic system with eight limit cycles bifurcated from a fine focus. We also give the sufficient and necessary conditions for the origin to be an isochronous center.

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.

Bifurcation of Periodic Orbits Crossing Switching Manifolds Multiple Times in Planar Piecewise Smooth Systems

2019 ◽  
Vol 29 (12) ◽  
pp. 1950160
Author(s):  
Zhihui Fan ◽  
Zhengdong Du
Keyword(s):  
Limit Cycle ◽  
Periodic Orbits ◽  
Limit Cycles ◽  
Sufficient Conditions ◽  
Piecewise Smooth ◽  
Unperturbed System ◽  
Piecewise Smooth Systems ◽  
Smooth Curves ◽  
Switching Curve ◽  
Smooth System

In this paper, we discuss the bifurcation of periodic orbits in planar piecewise smooth systems with discontinuities on finitely many smooth curves intersecting at the origin. We assume that the unperturbed system has either a limit cycle or a periodic annulus such that the limit cycle or each periodic orbit in the periodic annulus crosses every switching curve transversally multiple times. When the unperturbed system has a limit cycle, we give the conditions for its stability and persistence. When the unperturbed system has a periodic annulus, we obtain the expression of the first order Melnikov function and establish sufficient conditions under which limit cycles can bifurcate from the annulus. As an example, we construct a concrete nonlinear planar piecewise smooth system with three zones with 11 limit cycles bifurcated from the periodic annulus.

Center, limit cycles and isochronous center of a Z 4-equivariant quintic system

2010 ◽  
Vol 26 (6) ◽  
pp. 1183-1196 ◽  
Author(s):  
Chao Xiong Du ◽  
Hei Long Mi ◽  
Yi Rong Liu
Keyword(s):  
Limit Cycles ◽  
Isochronous Center

SEVEN LARGE-AMPLITUDE LIMIT CYCLES IN A CUBIC POLYNOMIAL SYSTEM

2006 ◽  
Vol 16 (02) ◽  
pp. 473-485 ◽  
Author(s):  
YIRONG LIU ◽  
WENTAO HUANG
Keyword(s):  
Singular Point ◽  
Limit Cycles ◽  
Large Amplitude ◽  
Polynomial System ◽  
Isochronous Center ◽  
Rational System ◽  
Cubic Polynomial

In this paper, the problem of limit cycles bifurcated from the equator for a cubic polynomial system is investigated. The best result so far in the literature for this problem is six limit cycles. By using the method of singular point value, we prove that a cubic polynomial system can bifurcate seven limit cycles from the equator. We also find that a rational system has an isochronous center at the equator.

scholarly journals Limit Cycles Bifurcating from the Periodic Orbits of a Discontinuous Piecewise Linear Differentiable Center with Two Zones

2015 ◽  
Vol 25 (11) ◽  
pp. 1550144 ◽  
Author(s):  
Jaume Llibre ◽  
Douglas D. Novaes ◽  
Marco A. Teixeira
Keyword(s):  
Equilibrium Point ◽  
Periodic Solutions ◽  
Periodic Orbits ◽  
Limit Cycles ◽  
Homoclinic Orbit ◽  
Piecewise Linear ◽  
Differential Systems ◽  
Linear Differential Systems ◽  
Straight Line ◽  
Linear Differential

We study a class of discontinuous piecewise linear differential systems with two zones separated by the straight line x = 0. In x > 0, we have a linear saddle with its equilibrium point living in x > 0, and in x < 0 we have a linear differential center. Let p be the equilibrium point of this linear center, when p lives in x < 0, we say that it is real, and when p lives in x > 0 we say that it is virtual. We assume that this discontinuous piecewise linear differential system formed by the center and the saddle has a center q surrounded by periodic orbits ending in a homoclinic orbit of the saddle, independent if p is real, virtual or p is in x = 0. Note that q = p if p is real or p is in x = 0. We perturb these three classes of systems, according to the position of p, inside the class of all discontinuous piecewise linear differential systems with two zones separated by x = 0. Let N be the maximum number of limit cycles which can bifurcate from the periodic solutions of the center q with these perturbations. Our main results show that N = 2 when p is on x = 0, and N ≥ 2 when p is a real or virtual center. Furthermore, when p is a real center we found an example satisfying N ≥ 3.

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 Bifurcation of Limit Cycles of a Class of Piecewise Linear Differential Systems in with Three Zones

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Yanyan Cheng
Keyword(s):  
Periodic Orbits ◽  
Limit Cycles ◽  
Averaging Method ◽  
Piecewise Linear ◽  
Dimensional System ◽  
Bifurcation Of Limit Cycles ◽  
Linear Differential Systems ◽  
Main Technique ◽  
Perturbed Systems ◽  
Linear Differential

We study the bifurcation of limit cycles from periodic orbits of a four-dimensional system when the perturbation is piecewise linear with two switching boundaries. Our main result shows that when the parameter is sufficiently small at most, six limit cycles can bifurcate from periodic orbits in a class of asymmetric piecewise linear perturbed systems, and, at most, three limit cycles can bifurcate from periodic orbits in another class of asymmetric piecewise linear perturbed systems. Moreover, there are perturbed systems having six limit cycles. The main technique is the averaging method.

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