On the uniqueness of limit cycle for certain Liénard systems without symmetry

Abstract

The problem of the uniqueness of limit cycles for Liénard systems is investigated in connection with the properties of the function $F(x)$. When $\alpha$ and $\beta$~$(\alpha<0<\beta)$ are the unique nontrivial solutions of the equation $F(x)=0$, necessary and sufficient conditions in order that all the possible limit cycles of the system intersect the lines $x=\alpha$ and $x=\beta$ are given. Therefore, in view of classical results, the limit cycle is unique. Some examples are presented to show the applicability of our results in situations with lack of symmetry.

Keywords

• liénard system
• uniqueness
• limit cycles
• invariant curves.