Electron. J. Diff. Equ.,
Vol. 2013 (2013), No. 96, pp. 1-12.
First integral method for an oscillator system
Xiaoqian Gong, Jing Tian, Jiaoyan Wang
Abstract:
In this article, we consider the nonlinear Duffing-van der Pol-type
oscillator system by means of the first integral method.
This system has physical relevance as a model in certain flow-induced
structural vibration problems, which includes the van der Pol oscillator
and the damped Duffing oscillator etc as particular cases.
Firstly, we apply the Division Theorem for two variables in the complex
domain, which is based on the ring theory of commutative algebra, to explore
a quasi-polynomial first integral to an equivalent autonomous system.
Then, through solving an algebraic system we derive the first integral of
the Duffing-van der Pol-type oscillator system under certain parametric
condition.
Submitted December 10, 2012. Published April 16, 2013.
Math Subject Classifications: 34A25, 34L30.
Key Words: First integral; Duffing oscillator; van der Pol oscillator;
autonomous system; division theorem.
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Xiaoqian Gong
College of Science, Tianjin University of Technology and Education
Tianjin 300222, China.
Department of Mathematics, University of Texas-Pan American
Edinburg, TX 78539, USA
email: xgong@broncs.utpa.edu, fax: +1 (956) 665-5091
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Jing Tian
Department of Mathematics, Texas A & M University
College Station, TX 77843, USA
email: jtian@math.tamu.edu
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Jiaoyan Wang
College of Science, Tianjin University of Technology and Education
Tianjin 300222, China.
Department of Mathematics, University of Texas-Pan American
Edinburg, TX 78539, USA
email: jiaoyanwang@163.com, fax: +1 (956) 665-5091
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