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Shock and Vibration
Volume 2018, Article ID 7213606, 12 pages
Research Article

Heteroclinic Bifurcation Behaviors of a Duffing Oscillator with Delayed Feedback

1Department of Traffic and Transportation, Shijiazhuang Tiedao University, Shijiazhuang 050043, China
2Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China
3Department of Engineering Mechanics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China

Correspondence should be addressed to Shao-Fang Wen; moc.361@11893fsw

Received 17 July 2017; Revised 22 November 2017; Accepted 21 December 2017; Published 21 January 2018

Academic Editor: Gianluca Gatti

Copyright © 2018 Shao-Fang Wen et al. 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.


The heteroclinic bifurcation and chaos of a Duffing oscillator with forcing excitation under both delayed displacement feedback and delayed velocity feedback are studied by Melnikov method. The Melnikov function is analytically established to detect the necessary conditions for generating chaos. Through the analysis of the analytical necessary conditions, we find that the influences of the delayed displacement feedback and delayed velocity feedback are separable. Then the influences of the displacement and velocity feedback parameters on heteroclinic bifurcation and threshold value of chaotic motion are investigated individually. In order to verify the correctness of the analytical conditions, the Duffing oscillator is also investigated by numerical iterative method. The bifurcation curves and the largest Lyapunov exponents are provided and compared. From the analysis of the numerical simulation results, it could be found that two types of period-doubling bifurcations occur in the Duffing oscillator, so that there are two paths leading to the chaos in this oscillator. The typical dynamical responses, including time histories, phase portraits, and Poincare maps, are all carried out to verify the conclusions. The results reveal some new phenomena, which is useful to design or control this kind of system.