Approximating the amplitude and form of limit cycles in the weakly nonlinear regime of Liénard systems

Liénard equations, x¨+ϵf(x)x˙+x=0, with f(x) an even continuous function are considered. In the weak nonlinear regime (ϵ → 0), the number and O(ϵ0) 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 [López-Ruiz R, López JL. Bifurcation curves of limit cycles in some Liénard systems. Int J Bifurcat Chaos 2000;10:971-80]. 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.