The domain of location of a limit cycle of the Li\'{e}nard system

Abstract

The Li\'{e}nard system $\frac{dx}{dt}=y,\quad \frac{dy}{dt}=-f(x)y-g(x)$ is considered.

Under some assumptions on functions $f(x)$ and $g(x)$, we estimate the domain of location of the unique stable limit cycle of the Li\'{e}nard system.

This estimation has the form $\alpha_2<x<\alpha_1$, where $\alpha_1$ and $\alpha_2$ are respectively the positive the negative roots of the equation $\int_0^{\alpha}\left[\int_0^xf(s)ds\right] g(x)dx=0$.