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

Abstract

Liénard 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 [López-Ruiz R, López JL. Bifurcation curves of limit cycles in some Liénard 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.

24 pages, 6 figures, 1 table

References in corpus (2)

• Transition between Two Oscillation Modes
• Number and Amplitude of Limit Cycles emerging from {\it Topologically Equivalent} Perturbed Centers

Cited by in corpus (3)

• Formulas for the amplitude of the van der Pol limit cycle
• On the determination of exact number of limit cycles in Lienard Systems
• The homotopy analysis method and the Lienard equation