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

Abstract

Liénard equations, x ¨ + ϵ f ( x ) x ˙ + x = 0 , with f ( x ) an even continuous function are considered. In the weak nonlinear regime ( ϵ → 0), the number and O ( ϵ 0 ) 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.