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)

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 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.

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 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.

scholarly journals Small amplitude limit cycles for the polynomial Liénard system

2008 ◽  
Vol 245 (9) ◽  
pp. 2522-2533 ◽  
Author(s):  
Maciej Borodzik ◽  
Henryk Żołądek
Keyword(s):  
Limit Cycles ◽  
Small Amplitude ◽  
Liénard System ◽  
Lienard System
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