Duffing--van der Pol--type oscillator system and its first integrals

Zhaosheng Feng; Guangyue Gao; Jing Cui

doi:10.3934/cpaa.2011.10.1377

Article Contents

2011, Volume 10, Issue 5: 1377-1391. Doi: 10.3934/cpaa.2011.10.1377

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Duffing--van der Pol--type oscillator system and its first integrals

1.
Department of Mathematics, University of Texas-Pan American, Edinburg, TX 78539
2.
Department of Mathematics, University of Texas{Pan American, Edinburg, Texas 78539

Received: February 28, 2009

Revised: January 31, 2010

Published: August 2011

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Abstract

In this paper, under certain parametric conditions we are concerned with the first integrals of the Duffing--van der Pol--type oscillator system, which include the van der Pol oscillator and the damped Duffing oscillator etc as particular cases. We apply the Lie symmetry method to find two nontrivial infinitesimal generators and use them to construct canonical variables. Through the inverse transformations we obtain the first integrals of the original oscillator system under the given parametric conditions, and some particular cases such as the damped Duffing equation and the van der Pol oscillator system are discussed accordingly.

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Mathematics Subject Classification: Primary: 76M60; Secondary: 37K10, 34L30.

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