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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 813957, 6 pages
http://dx.doi.org/10.1155/2013/813957
Research Article

The Application of the Undetermined Fundamental Frequency Method on the Period-Doubling Bifurcation of the 3D Nonlinear System

1School of Mechanical Engineering, Tianjin Polytechnic University, Tianjin 300072, China
2Department of Mechanics, Tianjin University, Tianjin 300072, China

Received 28 April 2013; Revised 10 August 2013; Accepted 10 August 2013

Academic Editor: Ren Yong

Copyright © 2013 Gen Ge and Wei Wang. 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.

Abstract

The analytical method to predict the period-doubling bifurcation of the three-dimensional (3D) system is improved by using the undetermined fundamental frequency method. We compute the stable response of the system subject to the quadratic and cubic nonlinearity by introducing the undetermined fundamental frequency. For the occurrence of the first and second period-doubling bifurcation, the new bifurcation criterion is accomplished. It depends on the stability of the limit cycle on the central manifold. The explicit applications show that the new results coincide with the results of the numerical simulation as compared with the initial methods.