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

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.


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
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[1] DOI: 10.1016/0020-7462(74)90008-0 · Zbl 0298.34053
[2] Bender C. M., Methods for Scientists and Engineers (1978) · Zbl 0417.34001
[3] DOI: 10.1016/0020-7462(82)90033-6 · Zbl 0499.70031
[4] DOI: 10.1016/0022-460X(83)90504-7 · Zbl 0565.70028
[5] DOI: 10.1016/0022-460X(83)90505-9 · Zbl 0562.70024
[6] DOI: 10.1016/0020-7462(91)90086-9 · Zbl 0734.70016
[7] DOI: 10.1137/0144063 · Zbl 0568.65048
[8] DOI: 10.1016/0020-7462(67)90011-X · Zbl 0145.12001
[9] Dorodnicyn A. A., Prikl. Mat. Mekh. 11 pp 313– (1947)
[10] Gradshteyn, I. S. and Ryzhik, I. M. 1980.Table of Integrals Series and Products25New York: Academic. · Zbl 0521.33001
[11] DOI: 10.1016/S0022-460X(86)80206-1 · Zbl 1235.70088
[12] DOI: 10.1016/0020-7462(80)90031-1 · Zbl 0453.70015
[13] IMSL MATH/LIBRARY. 1987.FORTRAN Subroutines for Mathematical Applications633–651.
[14] Minorsky, N. 1962.Nonlinear Oscillations557–559. Princeton: D van Nostrand.
[15] DOI: 10.1016/0020-7462(93)90060-X · Zbl 0776.70014
[16] Nayfeh A. H., Methods pp 8– (1973)
[17] Nayfeh A. H., Introduction to Perturbation Techniques (1981) · Zbl 0449.34001
[18] Nguyen V. D., J. Tech. Phys. (Poland) 17 pp 435– (1976)
[19] DOI: 10.1137/0131028 · Zbl 0346.34027
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