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Technical Brief

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

[+] Author and Article Information
Firdaus E. Udwadia

Professor
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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References

Santilli, R. M., 1978, Foundations of Theoretical Mechanics I: The Inverse Problem in Newtonian Mechanics, Springer-Verlag, New York.
Leitmann, G., 1963, “Some Remarks on Hamilton's Principle,” ASME J. Appl. Mech., 30, pp. 623–625. [CrossRef]
Udwadia, F. E., Leitmann, G., and Cho, H., 2011, “Some Further Remarks on Hamilton's Principle,” ASME J. Appl. Mech., 78, p. 011014. [CrossRef]
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Pars, L. A., 1972, A Treatise on Analytical Dynamics, Ox Bow Press, Woodbridge, CT.
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Duarte, L. G. S., Duarte, S. E. S., da Mota, A. C. P., and Skea, J. E. F., 2001, “Solving the Second-Order Ordinary Differential Equations by Extending the Prelle–Singer Method,” J. Phys. A, 34, pp. 3015–3024. [CrossRef]
Hydon, P. E., 2000, Symmetry Methods for Differential Equations, Cambridge University Press, New York.
Gao, G., and Feng, Z., 2010, “First Integrals for the Duffing–van der Pol Type Oscillator,” Electron. J. Differ. Equations, 19, pp. 1–12. [CrossRef]
Feng, Z., Gao, G., and Cui, J., 2011, “Duffing–van der Pol Type Oscillator System and Its First Integrals,” Commun. Pure Appl. Anal., 10, pp. 1377–1391. [CrossRef]

Figures

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

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)

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