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.

HOPF BIFURCATION OF LIÉNARD SYSTEMS BY PERTURBING A NILPOTENT CENTER

2012 ◽  
Vol 22 (08) ◽  
pp. 1250203 ◽  
Author(s):  
JING SU ◽  
JUNMIN YANG ◽  
MAOAN HAN
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Small Amplitude ◽  
Nonlinear Oscillators ◽  
Melnikov Function ◽  
Liénard System ◽  
First Order ◽  
Liénard Systems ◽  
Lienard System ◽  
Nilpotent Center

As we know, Liénard system is an important model of nonlinear oscillators, which has been widely studied. In this paper, we study the Hopf bifurcation of an analytic Liénard system by perturbing a nilpotent center. We develop an efficient method to compute the coefficients bl appearing in the expansion of the first order Melnikov function by finding a set of equivalent quantities B2l+1 which are able to calculate directly and can be used to study the number of small-amplitude limit cycles of the system. As an application, we investigate some polynomial Liénard systems, obtaining a lower bound of the maximal number of limit cycles near a nilpotent center.

Poincaré Bifurcation of Some Nonlinear Oscillator of Generalized Liénard Type Using Symbolic Computation Method

2018 ◽  
Vol 28 (08) ◽  
pp. 1850096 ◽  
Author(s):  
Hongying Zhu ◽  
Bin Qin ◽  
Sumin Yang ◽  
Minzhi Wei
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Nonlinear Oscillator ◽  
Melnikov Function ◽  
Computation Method ◽  
Rigorous Proof ◽  
Heteroclinic Loop ◽  
Period Annulus ◽  
Domain Iii ◽  
Weak Damping

In this paper, we study the Poincaré bifurcation of a nonlinear oscillator of generalized Liénard type by using the Melnikov function. The oscillator has weak damping terms. When the damping terms vanish, the oscillator has a heteroclinic loop connecting a nilpotent cusp to a hyperbolic saddle. Our results reveal that: (i) the oscillator can have at most four limit cycles bifurcating from the corresponding period annulus. (ii) There are some parameters such that three limit cycles emerge in the original periodic orbit domain. (iii) Especially, we give a rigorous proof that [Formula: see text] limit cycle(s) can emerge near the original singular loop and [Formula: see text] limit cycle(s) can emerge near the original elementary center with [Formula: see text].

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.

Bifurcation of Limit Cycles in a Near-Hamiltonian System with a Cusp of Order Two and a Saddle

2016 ◽  
Vol 26 (11) ◽  
pp. 1650180 ◽  
Author(s):  
Ali Bakhshalizadeh ◽  
Hamid R. Z. Zangeneh ◽  
Rasool Kazemi
Keyword(s):  
Asymptotic Expansion ◽  
Hamiltonian System ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Bifurcation Of Limit Cycles ◽  
Heteroclinic Loop ◽  
First Order ◽  
Period Annulus

In this paper, the asymptotic expansion of first-order Melnikov function of a heteroclinic loop connecting a cusp of order two and a hyperbolic saddle for a planar near-Hamiltonian system is given. Next, we consider the limit cycle bifurcations of a hyper-elliptic Liénard system with this kind of heteroclinic loop and study the least upper bound of limit cycles bifurcated from the period annulus inside the heteroclinic loop, from the heteroclinic loop itself and the center. We find that at most three limit cycles can be bifurcated from the period annulus, also we present different distributions of bifurcated limit cycles.

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.

LIMIT CYCLE BIFURCATIONS OF SOME LIÉNARD SYSTEMS WITH A NILPOTENT CUSP

2010 ◽  
Vol 20 (11) ◽  
pp. 3829-3839 ◽  
Author(s):  
JUNMIN YANG ◽  
MAOAN HAN
Keyword(s):  
Lower Bound ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Liénard System ◽  
Liénard Systems ◽  
Lienard System

In this paper, we study the number of limit cycles of a kind of polynomial Liénard system with a nilpotent cusp and obtain some new results on the lower bound of the maximal number of limit cycles for this kind of systems.

THE NUMBER OF LIMIT CYCLES IN A Z2-EQUIVARIANT LIÉNARD SYSTEM

2013 ◽  
Vol 23 (05) ◽  
pp. 1350085 ◽  
Author(s):  
YANQIN XIONG ◽  
HUI ZHONG
Keyword(s):  
Limit Cycle ◽  
Lower Bounds ◽  
Limit Cycles ◽  
Bifurcation Theory ◽  
Heteroclinic Bifurcation ◽  
Liénard System ◽  
Lienard System ◽  
Limit Cycle Bifurcation

In this paper, we consider the problem of limit cycle bifurcation near center points and a Z2-equivariant compound cycle in a polynomial Liénard system. Using the methods of Hopf, homoclinic and heteroclinic bifurcation theory, we found some new and better lower bounds of the maximal number of limit cycles for this system.

Limit Cycle Bifurcations Near a Piecewise Smooth Generalized Homoclinic Loop with a Saddle-Fold Point

2017 ◽  
Vol 27 (05) ◽  
pp. 1750071 ◽  
Author(s):  
Feng Liang ◽  
Dechang Wang
Keyword(s):  
Asymptotic Expansion ◽  
Hamiltonian System ◽  
Limit Cycles ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
Homoclinic Loop ◽  
First Order ◽  
Period Annulus ◽  
Straight Line ◽  
Smooth Perturbation

In this paper, we suppose that a planar piecewise Hamiltonian system, with a straight line of separation, has a piecewise generalized homoclinic loop passing through a Saddle-Fold point, and assume that there exists a family of piecewise smooth periodic orbits near the loop. By studying the asymptotic expansion of the first order Melnikov function corresponding to the period annulus, we obtain the formulas of the first six coefficients in the expansion, based on which, we provide a lower bound for the maximal number of limit cycles bifurcated from the period annulus. As applications, two concrete systems are considered. Especially, the first one reveals that a quadratic piecewise Hamiltonian system can have five limit cycles near a generalized homoclinic loop under a quadratic piecewise smooth perturbation. Compared with the smooth case [Horozov & Iliev, 1994; Han et al., 1999], three more limit cycles are found.

Limit Cycle Bifurcations by Perturbing a Hamiltonian System with a Cuspidal Loop of Order m

2015 ◽  
Vol 25 (06) ◽  
pp. 1550083 ◽  
Author(s):  
Yanqing Xiong
Keyword(s):  
Hamiltonian System ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Melnikov Function ◽  
Liénard System ◽  
First Order ◽  
Lienard System

This paper is concerned with the expansion of the first-order Melnikov function for general Hamiltonian systems with a cuspidal loop having order m. Some criteria and formulas are derived, which can be used to obtain first-order coefficients in the expansion. In particular, we deduce the first-order coefficients for the case m = 3 and give the corresponding conditions of existing several limit cycles. As an application, we study a Liénard system of type (n, 9) and prove that it can have 14 limit cycles near a cuspidal loop of order 3 for n = 8.

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