Relaxation oscillations of a van der Pol equation with large critical forcing term
Author:
J. Grasman
Journal:
Quart. Appl. Math. 38 (1980), 9-16
MSC:
Primary 70K99; Secondary 34D15, 58F22
DOI:
https://doi.org/10.1090/qam/575829
MathSciNet review:
575829
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Abstract: A van der Pol equation with sinusoidal forcing term is analyzed with singular perturbation methods for large values of the parameter. Asymptotic approximations of (sub)harmonic solutions with period $T = 2\pi \left ( {2n - 1} \right ),n = 1, 2, ...$ are constructed under certain restricting conditions for the amplitude of the forcing term. These conditions are such that always two solutions with period $T = 2\pi \left ( {2n \pm 1} \right )$ coexist.
H. Bavinck and J. Grasman, The method of matched asymptotic expansions for the periodic solution of the Van der Pol equation, Int. J. Nonlin. Mech. 9, 421β434 (1974)
J. E. Flaherty and F. C. Hoppensteadt, Frequency entrainment of a forced Van der Pol oscillator, Studies Appl. Math. 18, 5β15 (1978)
- J. Grasman, E. J. M. Veling, and G. M. Willems, Relaxation oscillations governed by a Van der Pol equation with periodic forcing term, SIAM J. Appl. Math. 31 (1976), no. 4, 667β676. MR 432975, DOI https://doi.org/10.1137/0131059
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- John Guckenheimer, Symbolic dynamics and relaxation oscillations, Phys. D 1 (1980), no. 2, 227β235. MR 581353, DOI https://doi.org/10.1016/0167-2789%2880%2990014-7
- Mark Levi, QUALITATIVE ANALYSIS OF THE PERIODICALLY FORCED RELAXATION OSCILLATIONS, ProQuest LLC, Ann Arbor, MI, 1978. Thesis (Ph.D.)βNew York University. MR 2628470
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H. Bavinck and J. Grasman, The method of matched asymptotic expansions for the periodic solution of the Van der Pol equation, Int. J. Nonlin. Mech. 9, 421β434 (1974)
J. E. Flaherty and F. C. Hoppensteadt, Frequency entrainment of a forced Van der Pol oscillator, Studies Appl. Math. 18, 5β15 (1978)
J. Grasman, E. J. M. Veling and G. M. Willems, Relaxation oscillations governed by a Van der Pol equation with periodic forcing term, SIAM J. Appl. Math. 31, 667β676 (1976)
J. Grasman, M. J. W. Jansen and E. J. M. Veling, Asymptotic methods for relaxation oscillations, in Proceedings of the third Scheveningen conference on differential equations, W. Eckhaus and E. M. Jager (eds.), North Holland Math. Studies 31, 93β111 (1978)
J. Guckenheimer, Symbolic dynamics and relaxation oscillations, preprint
M. Levi, Qualitative analysis of the periodically forced relaxation oscillations, Ph.D. Thesis, New York University, New York (1978)
N. Levinson, A second-order differential equation with singular solutions, Ann. Math. 50, 127β152 (1949)
J. E. Littlewood, On Van der Polβs equation with large k, in Nonlinear problems, R. E. Langer (ed.), Univ. of Wisconsin Press, Madison, 161β165 (1963)
N. G. Lloyd, On the non-autonomous Van der Pol equation with large parameter, Proc. Camb. Phil. Soc. 72, 213β227 (1972)
E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge, Cambridge University Press (1935)
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Article copyright:
© Copyright 1980
American Mathematical Society