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

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

2006-03-31

arxiv.org/abs/nlin/0603076v1

Nonlinear Sciences

Adaptation and Self-Organizing Systems

24 pages, 6 figures, 1 table

Scientific paper

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.

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