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Advances in Acoustics and Vibration
Volume 2012, Article ID 872498, 5 pages
Research Article

Control of Bistability in a Delayed Duffing Oscillator

Laboratory of Mechanics, Hassan II University Casablanca, Morocco

Received 25 June 2011; Accepted 19 September 2011

Academic Editor: Marek Pawelczyk

Copyright © 2012 Mustapha Hamdi and Mohamed Belhaq. 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 effect of a high-frequency excitation on nontrivial solutions and bistability in a delayed Duffing oscillator with a delayed displacement feedback is investigated in this paper. We use the technique of direct partition of motion and the multiple scales method to obtain the slow dynamic of the system and its slow flow. The analysis of the slow flow provides approximations of the Hopf and secondary Hopf bifurcation curves. As a result, this study shows that increasing the delay gain, the system undergoes a secondary Hopf bifurcation. Further, it is indicated that as the frequency of the excitation is increased, the Hopf and secondary Hopf bifurcation curves overlap giving birth in the parameter space to small regions of bistability where a stable trivial steady state and a stable limit cycle coexist. Numerical simulations are carried out to validate the analytical finding.