Number of limit cycles of the Liénard equation

In this paper, we study a Liénard system of the form $\dot{x}=y-F\left(x\right),\hspace{0.5em}\dot{y}=-x$, where $F\left(x\right)\mathrm{}$ is an odd polynomial. We introduce a method that gives a sequence of algebraic approximations to the equation of each limit cycle of the system. This sequence seems to converge to the exact equation of each limit cycle. We obtain also a sequence of polynomials $R_n\mathrm{}\left(x\right)\mathrm{}$ whose roots of odd multiplicity are related to the number and location of the limit cycles of the system.