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.

scholarly journals Global Analysis of a Liénard System with Quadratic Damping

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Feng Guo
Keyword(s):  
Necessary Conditions ◽  
Global Analysis ◽  
Global Bifurcation ◽  
Phase Portraits ◽  
Liénard System ◽  
Closed Orbits ◽  
Lienard System ◽  
Sufficient And Necessary Conditions ◽  
Global Phase Portrait ◽  
Quadratic Damping

In this paper, the global analysis of a Liénard equation with quadratic damping is studied. There are 22 different global phase portraits in the Poincaré disc. Every global phase portrait is given as well as the complete global bifurcation diagram. Firstly, the equilibria at finite and infinite of the Liénard system are discussed. The properties of the equilibria are studied. Then, the sufficient and necessary conditions of the system with closed orbits are obtained. The degenerate Bogdanov-Takens bifurcation is studied and the bifurcation diagrams of the system are given.

scholarly journals Existence of periodic solutions for nonlinear Lienard systems

1995 ◽  
Vol 18 (2) ◽  
pp. 265-272 ◽  
Author(s):  
Wan Se Kim
Keyword(s):  
Periodic Solutions ◽  
Liénard System ◽  
Existence And Multiplicity ◽  
Liénard Systems ◽  
Lienard System ◽  
Existence Of Periodic Solutions

We prove the existence and multiplicity of periodic solutions for nonlinear Lienard System of the typex″(t)+ddt[∇F(x(t))]+g(x(t))+h(t,x(t))=e(t)under various conditions upon the functionsg,hande.

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.

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.

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.

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