Improving a method for the study of limit cycles of the Liénard equation

Abstract

In recent papers we have introduced a method for the study of limit cycles of the Li\'enard system, $\mathrm{x\ifmmode \dot{}\else \.{}\fi{}}=y\ensuremath{-}F(x)$, $\mathrm{y\ifmmode \dot{}\else \.{}\fi{}}=\ensuremath{-}x$, where $F(x)$ is an odd polynomial. The method gives a sequence of polynomials ${R}_{n}(x)$, whose roots are related to the number and location of the limit cycles, and a sequence of algebraic approximations to the bifurcation set of the system. In this paper, we present a variant of the method that gives very important qualitative and quantitative improvements.