scholarly journals On the existence of a periodic solution of the Lienard system

2021 ◽  
Vol 66 (1) ◽  
pp. 47-53
Author(s):  
Alexander Ignatyev
Keyword(s):  
Periodic Solution ◽  
Liénard System ◽  
Lienard System

The Li\'{e}nard system $\frac{dx}{dt}=y,\ \frac{dy}{dt}=-f(x)y-g(x)$ is considered. Under some assumptions on functions $f(x)$ and $g(x)$, we prove the existence of a periodic solution of this 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.

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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