Abstract
In this paper, a Duffing oscillator model with delayed velocity feedback is considered. Applying the time delayed feedback control method and delayed differential equation theory, we establish some criteria which ensure the stability and the existence of Hopf bifurcation of the model. By choosing the delay as bifurcation parameter and analyzing the associated characteristic equation, the existence of bifurcation parameter point is determined. We found that if the time delay is chosen as a bifurcation parameter, Hopf bifurcation occurs when the time delay is changed through a series of critical values. Some numerical simulations show that the designed feedback controllers not only delay the onset of Hopf bifurcation, but also enlarge the stability region for the model.
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This work was supported by National Natural Science Foundation of China (Nos. 11261010 and 11101126), Natural Science and Technology Foundation of Guizhou Province(No. J[2015]2025) and 125 Special Major Science and Technology of Department of Education of Guizhou Province (No. [2012]011), and Natural Science Innovation Team Project of Guizhou Province (No. [2013]14).
Recommended by Associate Editor Ivan Zelinka
Chang-Jin Xu graduated from Huaihua University, China, in 1994. He received the M. Sc. degree from Kunming University of Science and Technology in 2004 and the Ph.D. degree from Central South University, China in 2010. He is currently a professor at the Guizhou Key Laboratory of economic system simulation, Guizhou University of Finance and Economics. He has published about 100 refereed journal papers. He is a reviewer of the journal Mathematical Reviews and Zentralbatt-Math.
His research interests include stability and bifurcation theory of delayed differential equation.
ORCID iD: 0000-0001-5844-2985
Yu-Sen Wu graduated from Liaocheng University, China in 2004. He received the M. Sc. degree from Central South University, China in 2007 and the Ph.D. degree from Central South University, China in 2010. He is currently an associate professor at School of Mathematics and Statistics of Henan University of Science and Technology. He has published about 30 refereed journal papers. He is a reviewer of the journal Mathematical Reviews.
His research interests include the qualitative theory of ordinary differential equation and computer symbol calculation.
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Xu, CJ., Wu, YS. Bifurcation control for a Duffing oscillator with delayed velocity feedback. Int. J. Autom. Comput. 13, 596–606 (2016). https://doi.org/10.1007/s11633-015-0944-4
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DOI: https://doi.org/10.1007/s11633-015-0944-4