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.

scholarly journals Existence of Limit Cycles for Liénard System with Application

2021 ◽  
pp. 33-37
Author(s):  
Ali Bakur Barsham ALmurad ◽  
Elamin Mohammed Saeed Ali
Keyword(s):  
Mathematical Method ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Planar System ◽  
Liénard System ◽  
System Form ◽  
Lienard System

This paper is part of a wider study limit cycle problems and planar system; The aims of this is to study the existence of limit cycle for Liénard system. We followed the historical analytical mathematical method to present a proof of a result on the existence of limit cycle for Liénard system form x ̇=y-F(x) ,y ̇=-g(x)

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.

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.

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.

scholarly journals ON THE NUMBER OF LIMIT CYCLES IN PIECEWISE-LINEAR LIÉNARD SYSTEMS

2005 ◽  
Vol 15 (04) ◽  
pp. 1417-1422 ◽  
Author(s):  
A. TONNELIER
Keyword(s):  
Limit Cycles ◽  
General Class ◽  
Piecewise Linear ◽  
Liénard System ◽  
Liénard Systems ◽  
Lienard System ◽  
General Class Of Functions ◽  
Class Of Functions

In a previous paper [Tonnelier, 2002] we conjectured that a Liénard system of the form ẋ = p(x) - y, ẏ = x where p is piecewise linear on n + 1 intervals has up to 2n limit cycles. We construct here a general class of functions p satisfying this conjecture. Limit cycles are obtained from the bifurcation of the linear center.

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.

Limit Cycle Bifurcations for Piecewise Smooth Hamiltonian Systems with a Generalized Eye-Figure Loop

2016 ◽  
Vol 26 (12) ◽  
pp. 1650204 ◽  
Author(s):  
Jihua Yang ◽  
Liqin Zhao
Keyword(s):  
Lower Bound ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
First Order ◽  
Period Annulus

This paper deals with the limit cycle bifurcations for piecewise smooth Hamiltonian systems. By using the first order Melnikov function of piecewise near-Hamiltonian systems given in [Liu & Han, 2010], we give a lower bound and an upper bound of the number of limit cycles that bifurcate from the period annulus between the center and the generalized eye-figure loop up to the first order of Melnikov function.

Periodic Orbits Analysis in a Class of Planar Liénard Systems with State-Triggered Jumps

2016 ◽  
Vol 26 (09) ◽  
pp. 1650153
Author(s):  
Fangfang Jiang ◽  
Wei D. Lu ◽  
Jitao Sun
Keyword(s):  
Periodic Orbits ◽  
Impulsive Differential Equations ◽  
Successor Function ◽  
Liénard System ◽  
Geometrical Properties ◽  
Liénard Systems ◽  
Closed Orbits ◽  
Lienard System ◽  
Abrupt Changes ◽  
New Criteria

In this paper, we investigate the existence problem of periodic orbits for a planar Liénard system, whose solution mappings are interrupted by abrupt changes of state. We first present the geometrical properties of solutions for the planar Liénard system with state impulses, then by using Bendixson theorem of impulsive differential equations and successor function method, several new criteria on the closed orbits and discontinuous periodic orbits are established in the impulsive Liénard system.

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