A Simple Criterion for the Non-Existence of Limit Cycles of a Lienard System

Abstract

In this paper, as an application in our results, the non-existence of limit cycles for the Liénard system x ̇ = y –F (x), y ̇=-g(x) with F (x)=(x^2-x) e^(-x) (x≥-1) and 5(x^2+x) e^(x+2)+2e (x≤-1),g(x)=x is discussed by the simple criterion. Graef [1] in 1971 has studied the uniformly boundedness of the solution orbits under the condition (C1) and further proved the existence of limit cycles under the conditions (C1) and (C2) . Recently, Cioni and Villari [2] in 2015 gave the same result as in Graef [1] under the conditions (C1) and (C3) includes (C2). Our aim is to discuss on the case of which (C1) is satisfied, but (C3) is not satisfied. As the result, we shall give the simple criterion for the non-existence of limit cycles for a Liénard system with these conditions.