Existence of Limit Cycles for Liénard System with Application

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)

On the existence of a periodic solution of the 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.

Sharp Bound of the Number of Zeros for a Liénard System with a Heteroclinic Loop

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.

Nine Limit Cycles in a 5-Degree Polynomials Liénard 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.