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

Li\'{e}nard equations, $\ddot{x}+\epsilon f(x)\dot{x}+x=0$, with $f(x)$ an even continuous function are considered. In the weakly nonlinear regime ($\epsilon\to 0$), the number and an order zero in $\epsilon$ 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 $\epsilon$ 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 $\epsilon$. Several examples showing the application of this scheme are given.