Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 96, pp. 1-12.

Title: First integral method for an oscillator system
        
Authors: Xiaoqian Gong (Univ. of Texas-Pan American, Edinburg, TX, USA)
         Jing Tian (Texas A & M Univ., College Station, TX, USA
         Jiaoyan Wang (Univ. of Texas-Pan American, Edinburg, TX, USA)

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.