scholarly journals Sharp Bound of the Number of Zeros for a Liénard System with a Heteroclinic Loop

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Junning Cai ◽  
Minzhi Wei ◽  
Guoping Pang
Keyword(s):  
Limit Cycles ◽  
Upper Bound ◽  
Sharp Bound ◽  
Abelian Integral ◽  
Heteroclinic Loop ◽  
Liénard System ◽  
Algebraic Criterion ◽  
Number Of Zeros ◽  
Lienard System ◽  
Nilpotent Saddle

In the presented paper, the Abelian integral I h of a Liénard system is investigated, with a heteroclinic loop passing through a nilpotent saddle. By using a new algebraic criterion, we try to find the least upper bound of the number of limit cycles bifurcating from periodic annulus.

On the Distribution of Limit Cycles in a Liénard System with a Nilpotent Center and a Nilpotent Saddle

2016 ◽  
Vol 26 (02) ◽  
pp. 1650025 ◽  
Author(s):  
R. Asheghi ◽  
A. Bakhshalizadeh
Keyword(s):  
Limit Cycles ◽  
Upper Bound ◽  
Integral Formula ◽  
Abelian Integral ◽  
Liénard System ◽  
Algebraic Criterion ◽  
Period Annulus ◽  
Lienard System ◽  
Nilpotent Center ◽  
Nilpotent Saddle

In this work, we study the Abelian integral [Formula: see text] corresponding to the following Liénard system, [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] are real bounded parameters. By using the expansion of [Formula: see text] and a new algebraic criterion developed in [Grau et al., 2011], it will be shown that the sharp upper bound of the maximal number of isolated zeros of [Formula: see text] is 4. Hence, the above system can have at most four limit cycles bifurcating from the corresponding period annulus. Moreover, the configuration (distribution) of the limit cycles is also determined. The results obtained are new for this kind of Liénard system.

scholarly journals Poincaré Bifurcation of Limit Cycles from a Liénard System with a Homoclinic Loop Passing through a Nilpotent Saddle

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Minzhi Wei ◽  
Junning Cai ◽  
Hongying Zhu
Keyword(s):  
Hamiltonian System ◽  
Limit Cycles ◽  
Abelian Integral ◽  
Chebyshev System ◽  
Homoclinic Loop ◽  
Bifurcation Of Limit Cycles ◽  
Liénard System ◽  
Number Of Zeros ◽  
Lienard System ◽  
Nilpotent Saddle

In present paper, the number of zeros of the Abelian integral is studied, which is for some perturbed Hamiltonian system of degree 6. We prove the generating elements of the Abelian integral from a Chebyshev system of accuracy of 3; therefore there are at most 6 zeros of the Abelian integral.

Bifurcation of Limit Cycles in Small Perturbation of a Class of Liénard Systems

2014 ◽  
Vol 24 (01) ◽  
pp. 1450004 ◽  
Author(s):  
Xianbo Sun ◽  
Hongjian Xi ◽  
Hamid R. Z. Zangeneh ◽  
Rasool Kazemi
Keyword(s):  
Limit Cycle ◽  
Small Perturbation ◽  
Limit Cycles ◽  
Upper Bound ◽  
Bifurcation Of Limit Cycles ◽  
Heteroclinic Loop ◽  
Algebraic Criterion ◽  
Liénard Systems ◽  
Limit Cycle Bifurcation ◽  
Nilpotent Saddle

In this article, we study the limit cycle bifurcation of a Liénard system of type (5,4) with a heteroclinic loop passing through a hyperbolic saddle and a nilpotent saddle. We study the least upper bound of the number of limit cycles bifurcated from the periodic annulus inside the heteroclinic loop by a new algebraic criterion. We also prove at least three limit cycles will bifurcate and six kinds of different distributions of these limit cycles are given. The methods we use and the results we obtain are new.

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 Nine Limit Cycles in a 5-Degree Polynomials Liénard System

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Junning Cai ◽  
Minzhi Wei ◽  
Hongying Zhu
Keyword(s):  
Limit Cycles ◽  
Melnikov Function ◽  
Homoclinic Bifurcation ◽  
Heteroclinic Bifurcation ◽  
Homoclinic Loop ◽  
Heteroclinic Loop ◽  
Liénard System ◽  
Period Annulus ◽  
Liénard Systems ◽  
Lienard System

In this article, we study the limit cycles in a generalized 5-degree Liénard system. The undamped system has a polycycle composed of a homoclinic loop and a heteroclinic loop. It is proved that the system can have 9 limit cycles near the boundaries of the period annulus of the undamped system. The main methods are based on homoclinic bifurcation and heteroclinic bifurcation by asymptotic expansions of Melnikov function near the singular loops. The result gives a relative larger lower bound on the number of limit cycles by Poincaré bifurcation for the generalized Liénard systems of degree five.

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