Gabriele Villari and Fabio Zanolin
Existence and Nonexistence of Limit Cycles for a Certain Class of Planar Systems of Lienard Type
Dynamic Systems and Applications 31 (2022) No.4, 257-264

https://doi.org/10.46719/dsa202231.04.03

ABSTRACT.
The phase-portrait of a {L}i\'{e}nard system $\dot{x} = y – F(x),$$\dot{y} = – g(x)$ is investigated under a classical assumption on $F(x),$ namely that there are $\alpha < 0 < \beta$ such that $F(\alpha)=F(0)=F(\beta)=0$ with $F(x)x<0$ for $x\in (\alpha,\beta)\setminus{0}$ and $F$ is monotone increasing outside $(\alpha,\beta).$ It is well known that such a system has at least a limit cycle provided that $G(x)\to +\infty$ or $F(x){\rm sign}(x)\to +\infty$ for $x\to\pm\infty$. Clearly, such assumptions imply that $G(x)\pm F(x)\to +\infty$ as $x\to \pm\infty.$ In \cite{Vi-1987} it has been proved that this is actually a necessary and sufficient condition for the intersection of the trajectories with the vertical isocline. In this paper we treat the case in which this assumption is not fulfilled and prove that there are cases where both existence and nonexistence may occur.

AMS (MOS) Subject Classification. 34C05, 34C25, 34C15.

Key Words and Phrases. Lienard equation, limit cycles, existence/non-existence.