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September  2006, 6(5): 1097-1112. doi: 10.3934/dcdsb.2006.6.1097

A qualitative study of the damped duffing equation and applications

Zhaosheng Feng 1, , Goong Chen 2, and  Sze-Bi Hsu 3,

1. 

Department of Mathematics, University of Texas-Pan American, Edinburg, TX 78541, United States
2. 

Department of Mathematics, Texas A&M University, College Station, TX 77843
3. 

Department of Mathematics, National Tsing-Hua University, Hsin-Chu 30043

Received  April 2005 Revised  March 2006 Published  June 2006

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.
Keywords: equilibrium point, Duffing's equation, global structure, autonomous system, phase plane, Lie symmetry..
Mathematics Subject Classification: Primary: 34C05, 34C14, 34C20; Secondary: 35B4.
Citation: Zhaosheng Feng, Goong Chen, Sze-Bi Hsu. A qualitative study of the damped duffing equation and applications. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1097-1112. doi: 10.3934/dcdsb.2006.6.1097
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