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Shock and Vibration
Volume 1, Issue 3, Pages 233-240

Normal Form Analysis of Hopf Bifurcation Exemplified by Duffing’s Equation

A.Y.T. Leung,1 Zhang Qichang,2 and Chen Yushu2

1Department of Civil and Structural Engineering, University of Hong Kong, Hong Kong
2Department of Mechanics, Tianjin University, China

Received 23 July 1993; Accepted 30 August 1993

Copyright © 1994 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A method is proposed for calculating the normal form coefficients of the degenerate Hopf bifurcation system and the steady periodic solutions of a nonlinear vibration system. The results obtained by this method are the same as those obtained by the classical one. The present method is much simpler and can easily be implemented: that is, given the coefficients of the governing equations, the response is obtained directly by substitution.