Hostname: page-component-7bb8b95d7b-w7rtg Total loading time: 0 Render date: 2024-09-16T12:17:01.627Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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