Approximating the Amplitude and Form of Limit Cycles in the Weakly Nonlinear Regime of Lienard Systems

Li'{e}nard equations, $\ddot{x}+ϵf(x)\dot{x}+x=0$, with $f(x)$ an even continuous function are considered. In the weakly nonlinear regime ($ϵ\to 0$), the number and an order zero in $ϵ$ approximation of the amplitude of limit cycles present in this type of systems can be obtained by applying a methodology recently proposed by the authors [L'opez-Ruiz R, L'opez JL. Bifurcation curves of limit cycles in some Li'enard systems. Int J Bifurcat Chaos 2000; 10:971-980]. In the present work, that method is carried forward to higher orders in $ϵ$ and is embedded in a general recursive algorithm capable to approximate the form of the limit cycles and to correct their amplitudes as an expansion in powers of $ϵ$. Several examples showing the application of this scheme are given.