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J. Phys. A: Math. Gen. 35 (2 August 2002) 6209-6231

Subharmonic resonances of the parametrically driven pendulum

Eugene I Butikov

St Petersburg State University, St Petersburg, Russia

E-mail: butikov@spb.runnet.ru

Received 31 January 2002, in final form 28 May 2002
Published 19 July 2002

Abstract. A simple qualitative physical explanation is suggested for the phenomenon of subharmonic resonances of a rigid planar pendulum whose axis is forced to oscillate with a high frequency in the vertical direction. An approximate quantitative theory based on the suggested approach is developed. The spectral composition of the subharmonic resonances is investigated quantitatively, and the boundaries of these modes in the parameter space are determined. New related modes of regular behaviour are described and explained. The conditions of the inverted pendulum stability are determined with greater precision than previous results. A computer program simulating the physical system supports the analytical investigation.

PACS numbers: 05.45.-a, 05.10.-a, 07.05.Tp, 45.10.-b

URL: stacks.iop.org/0305-4470/35/6209
DOI: 10.1088/0305-4470/35/30/301
PII: S0305-4470(02)33399-7

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