SINGULAR PERTURBATIONS FOR THE FORCED VAN DER POL OSCILLATOR

Main Article Content

E. M. ELABBASY

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.




Article Details

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
Section
Papers

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.