Applied and Computational Mathematics
Volume 5, Issue 5, October 2016, Pages: 186-192
Received: Sep. 1, 2016;
Accepted: Sep. 13, 2016;
Published: Oct. 8, 2016
Views 3281 Downloads 141
Mohammad Darvishi, Department of Mathematics, Faculty of Mathematics and Statistics, University of Birjand, Birjand, Iran
Haji Mohammad Mohammadinejad, Department of Mathematics, Faculty of Mathematics and Statistics, University of Birjand, Birjand, Iran
In this paper we study the Hopf bifurcation control problem of a Internet Model with Time-delay. The stable region and stability condition of the congestion control model are given by use of the linear stability analysis. When the system gain passes through a critical value, the system loses the stability and Hopf bifurcation occurs. Considering the negative influence caused by Hopf bifurcation, we apply Proportional - Derivative (PD) controller to postpone the onset of undesirable Hopf bifurcation. Numerical simulation results and figures confirm that the control strategy is efficient in controlling the Hopf bifurcation.
Haji Mohammad Mohammadinejad,
Control of Bifurcation in Internet Model with Time-Delay, Applied and Computational Mathematics.
Vol. 5, No. 5,
2016, pp. 186-192.
Copyright © 2016 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/
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B. D. Hassard, N. D. Kazarinoff, Y. H. Wan, “Theory and Applications of Hopf Bifurcation,” Cambridge University Press, Cambridge, 1981.
D. Ding, J. Zhu, X. Luo, and Y. Liu, “Delay induced Hopf bifurcation in a dual model of Internet congestion control algorithm,” Nonlinear Analysis: Real World Applications 10.2009, 2873-2883.
D. Ding, X. Zhang, J. Cao, N. Wang, D. Liang, “Bifurcation control of complex networks model via PD controller,” Neurocomputing 175. 2016, 1-9.
Al-Bahadili, H. “Simulation in computer network design and modeling: Use and analysis”, Hershey, PA: IGI Global. pp. 282, 2012.
C. Jin, D. X. Wei, and S. H. Low, “BFAST TCP: Motivation, architecture, algorithms, and performance,” in Proc. IEEE INFOCOM, Mar. 2004.
F. Zhao, J. Zhou, N. Lu, and M. Zhang, “Stability analysis of FAST TCP based on Lyapunov function, ” in Proceedings of the 7th World Congress on Intelligent Control and Automation (WCICA 08). 2008, pp: 2136-2140.
H. Zhang and B. Liu, “Stability of FAST TCP with time-delay in single-link multi-source networks, ” in Proceedings of the IEEE International Conference on Information Networking and Automation (ICINA 10). 2010 V2-259-V2-263.
J. Wang, D. X. Wei, J.-Y. Choi, and S. H. Low, “Modelling and stability of FAST TCP,” in Wireless Communications 143. 2007 331-356.
J.-Y. Choi, H. Kim, and S. Kwon, “Global asymptotic stability analysis of fast tcp using nonlinear small gain theorem,” IEEE Communications Letters 18. 2014, 523-526.
J.-Y. Choi, S.-Y. Kim, and J.-W. Kim, “Parameter condition for global asymptotic stability of FAST TCP in the presence of cross traffics,” IEEE Communications Letters 14. 2010, 584-586.
L. Feng, X. Gaoxiang, G. Zhi-Hong et al, “Stability analysis and control Hopf bifurcation in a FAST TCP model, ” in Proceedings of the 32nd IEEE Chinese Control Conference (CCC13). 2013, 1076-1080.
M. Xiao, W. X. Zheng, and J. Cao, “Bifurcation control of a congestion control model via state feedback,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 23. 31, 2013.
Z.-Q. Zhan, J. Zhu, and W. Li, “Stability and bifurcation analysis in a FAST TCP model with feedback delay,” Nonlinear Dynamics 70. 2012, 255-267.