Analysis of the limit cycle of a generalised van der Pol equation by a time transformation method. (English) Zbl 0807.34037

Moremedi, G. M.; Mason, D. P.

Quaest. Math. 17, No. 3, 349-379 (1994).

Summary: The properties of the limit cycle of a generalized van der Pol equation of the form \(\ddot u+ u= \varepsilon(1- u^{2n})\dot u\), where \(\varepsilon\) is small and \(n\) is any positive integer, are investigated by applying a time transformation perturbation method due to Burton. It is found that as \(n\) increases the amplitude of the limit cycle oscillation decreases and its period increases. The time transformation solution is compared with the solution derived using the method of multiple scales and with a numerical solution. It is found that, to first order in \(\varepsilon\), the time transformation solution for the limit cycle agrees better with the numerical solution than the multiple scales solution. Both perturbation solutions give the same result for the period of the limit cycle to second order in \(\varepsilon\). The accuracy of the time transformation solution decreases as \(n\) increases.

MSC:

34C05	Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34E15	Singular perturbations for ordinary differential equations
65J99	Numerical analysis in abstract spaces

Keywords:

limit cycle; generalized van der Pol equation; time transformation perturbation method; method of multiple scales; numerical solution

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