A qualitative study of the damped duffing equation and applications

Zhaosheng Feng; Goong Chen; Sze-Bi Hsu

doi:10.3934/dcdsb.2006.6.1097

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2006, Volume 6, Issue 5: 1097-1112. Doi: 10.3934/dcdsb.2006.6.1097

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A qualitative study of the damped duffing equation and applications

1.
Department of Mathematics, University of Texas-Pan American, Edinburg, TX 78541
2.
Department of Mathematics, Texas A&M University, College Station, TX 77843
3.
Department of Mathematics, National Tsing-Hua University, Hsin-Chu 30043

Received: March 31, 2005

Revised: February 28, 2006

Published: August 2006

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Abstract

In this paper, we analyze the damped Duffing equation by means of qualitative theory of planar systems. Under certain parametric choices, the global structure in the Poincaré phase plane of an equivalent two-dimensional autonomous system is plotted. Exact solutions are obtained by using the Lie symmetry and the coordinate transformation method, respectively. Applications of the second approach to some nonlinear evolution equations such as the two-dimensional dissipative Klein-Gordon equation are illustrated.

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Mathematics Subject Classification: Primary: 34C05, 34C14, 34C20; Secondary: 35B40.

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