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On the periodic solution of the Van der Pol oscillator with large damping

Published online by Cambridge University Press:  14 November 2011

E. M. El-Abbasy
E. M. El-Abbasy
Affiliation:
Department of Pure Mathematics, The University College of Wales, Aberystwyth, Dyfed S723 3BZ
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Synopsis

Littlewood showed that the forced Van der Pol oscillator with 0<b<⅔ and k large normally has subharmonic solutions of order 2n + l where n ≅ O([⅔−b]k). Numerical experiments suggest that n ≅ (⅔ –b)k/3 as k →∞. A refinement of Littlewood's calculation is given which leads to this result.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 100 , Issue 1-2 , 1985 , pp. 103 - 106
Copyright
Copyright © Royal Society of Edinburgh 1985

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References

1Cartwright, M. L. and Littlewood, J. E.. On nonlinear differential equations of the second order. I. The equation ×+k(×2– l)– + – = kb cos (μt + a), k large. J. London Math. Soc. 20 (1945), 180–189.Google Scholar
2Flaherty, J. E. and Hopensteadt, F. C.. Frequency entrainment of a forced Van der Pol oscillator. Stud. Appl. Math. 18 (1978) 5–15.Google Scholar
3Littlewood, J. E.. On nonlinear differential equations of the second order: III. The equation × + k(×2 – l)× + × = kbμ cos (μt + a) for k large and its applications. Ada Math. 97 (1957), 267–308.Google Scholar
4Littlewood, J. E.. Some Problems in Real and Complex Analysis (Lexington, Mass.: Heath, 1968).Google Scholar