SINGULAR PERTURBATIONS FOR THE FORCED VAN DER POL OSCILLATOR

Authors

  • E. M. ELABBASY Dept. Maths., Faculty of Science, Mansoura Univ., Mansoura, Egypt.

DOI:

https://doi.org/10.5556/j.tkjm.24.1993.4514

Keywords:

subharmonic solutions, large sinusoidal forcing term, asymptotic solutions

Abstract

This paper deals with the Van der Pol oscillator with large si­ nusoidal forcing term. By using singular perturbation techniques a.symp­ totic solutions of such a system are constructed. We considered the case $b$ small and then we find the range of values of $b$ for which the Van der Pol oscillatore may have 3 stable subharmonic solutions for the same values of the parameters. We have integrated the equation numerically; then a comparison of the numerical results obtained with analytical results of this paper is given.

References

M. L. Cartwright, "Forced Oscillations in Nonlinear Systems", Contributions to the The­ory of Nonlinear Oscillations, Vol. 1. Annals of Math. Studies No. 20.

A. A. Dorodnicyn, "Asymptotic Solution of the Van der Pol Equation", Prikl. Mat. Mekh., 11 (1947), p. 313-328; Am. Math. Soc. Transl., series 1, 4 (1962), p. 1-23.

E. M. Elabbasy, "Periodic Solutions of Nonlinear Differential Equations'', a Numerical Investigation. Ph. D. thesis, Univ. of Wales UK, 1980.

E. M. Elabbasy, "On the periodic solution of the Van der Pol oscillator with large damp­ing", Proceedings of the Royal Society of Edinburgh, 100A, 103-106, 1985.

J. Kevorkian and J. D. Cole, "Perturbation methods in Applied Mathematics", Springer Verlag., New York, 1981.

J. E. Littlewood, "On nonlinear differential equations of the second order: III. The equa­tion x +k(x^2 -1)+x = kbμcos(μt) for k large and its generalization", Acta. Math. 97, 267-308.

J J. E. Littlewood, "Some problems in real and complex analysis", Mass. Healh, 1968.

M. Urabe, "Periodic Solutions of Van der Pol's Equation with Damping Coefficient k = 0~10",IRE Transactions on Circuit Theory., 1960, p.382-386.

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Published

1993-12-01

How to Cite

ELABBASY, E. M. (1993). SINGULAR PERTURBATIONS FOR THE FORCED VAN DER POL OSCILLATOR. Tamkang Journal of Mathematics, 24(4), 417–430. https://doi.org/10.5556/j.tkjm.24.1993.4514

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Section

Papers