Technical Brief

First Integrals and Solutions of Duffing–Van der Pol Type Equations

[+] Author and Article Information
Firdaus E. Udwadia

Departments of Aerospace and Mechanical Engineering, Civil Engineering, Mathematics,
Systems Architecture Engineering,
and Information and Operations Management,
University of Southern California,
430K Olin Hall,
Los Angeles, CA 90089
e-mail: fudwadia@usc.edu

Hancheol Cho

Department of Aerospace and Mechanical Engineering,
University of Southern California,
Los Angeles, CA 90089
e-mail: hancheoc@usc.edu

Manuscript received December 14, 2012; final manuscript received May 18, 2013; accepted manuscript posted May 29, 2013; published online September 18, 2013. Assoc. Editor: Wei-Chau Xie.

J. Appl. Mech 81(3), 034501 (Sep 18, 2013) (4 pages) Paper No: JAM-12-1559; doi: 10.1115/1.4024673 History: Received December 14, 2012; Revised May 18, 2013; Accepted May 29, 2013

A simple transformation is used to obtain the first integrals and the solutions of the Duffing–van der Pol type equation under certain conditions. It is shown that the system can be totally integrable and this total integrability admits new solutions. The new solutions require weaker conditions on the system's parameters than hereto known.

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Grahic Jump Location
Fig. 2

Time history of the displacement of the system (left) and the difference between the numerical and the new analytical solutions Eq. (3.4) (right)

Grahic Jump Location
Fig. 1

Phase diagram of the Duffing–van der Pol system Eq. (2.9) with α = 1, β = 3, γ = −4, and λ = 1



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