Control of Bifurcation in Internet Model with Time-Delay

Mohammad Darvishi; Haji Mohammad Mohammadinejad

doi:doi:10.11648/j.acm.20160505.11

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Control of Bifurcation in Internet Model with Time-Delay

Mohammad Darvishi, Haji Mohammad Mohammadinejad

Published in Applied and Computational Mathematics (Volume 5, Issue 5)

Received: 1 September 2016 Accepted: 13 September 2016 Published: 8 October 2016

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Abstract

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.

Published in	Applied and Computational Mathematics (Volume 5, Issue 5)
DOI	10.11648/j.acm.20160505.11
Page(s)	186-192
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2016. Published by Science Publishing Group

Keywords

Hopf Bifurcation, Bifurcation Control, PD Controller

References

[1]	B. D. Hassard, N. D. Kazarinoff, Y. H. Wan, “Theory and Applications of Hopf Bifurcation,” Cambridge University Press, Cambridge, 1981.
[2]	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.
[3]	D. Ding, X. Zhang, J. Cao, N. Wang, D. Liang, “Bifurcation control of complex networks model via PD controller,” Neurocomputing 175. 2016, 1-9.
[4]	Al-Bahadili, H. “Simulation in computer network design and modeling: Use and analysis”, Hershey, PA: IGI Global. pp. 282, 2012.
[5]	C. Jin, D. X. Wei, and S. H. Low, “BFAST TCP: Motivation, architecture, algorithms, and performance,” in Proc. IEEE INFOCOM, Mar. 2004.
[6]	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.
[7]	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.
[8]	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.
[9]	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.
[10]	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.
[11]	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.
[12]	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.
[13]	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.

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Mohammad Darvishi, Haji Mohammad Mohammadinejad. (2016). Control of Bifurcation in Internet Model with Time-Delay. Applied and Computational Mathematics, 5(5), 186-192. https://doi.org/10.11648/j.acm.20160505.11

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ACS Style

Mohammad Darvishi; Haji Mohammad Mohammadinejad. Control of Bifurcation in Internet Model with Time-Delay. Appl. Comput. Math. 2016, 5(5), 186-192. doi: 10.11648/j.acm.20160505.11

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AMA Style

Mohammad Darvishi, Haji Mohammad Mohammadinejad. Control of Bifurcation in Internet Model with Time-Delay. Appl Comput Math. 2016;5(5):186-192. doi: 10.11648/j.acm.20160505.11

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@article{10.11648/j.acm.20160505.11,
  author = {Mohammad Darvishi and Haji Mohammad Mohammadinejad},
  title = {Control of Bifurcation in Internet Model with Time-Delay},
  journal = {Applied and Computational Mathematics},
  volume = {5},
  number = {5},
  pages = {186-192},
  doi = {10.11648/j.acm.20160505.11},
  url = {https://doi.org/10.11648/j.acm.20160505.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20160505.11},
  abstract = {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.},
 year = {2016}
}

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TY  - JOUR
T1  - Control of Bifurcation in Internet Model with Time-Delay
AU  - Mohammad Darvishi
AU  - Haji Mohammad Mohammadinejad
Y1  - 2016/10/08
PY  - 2016
N1  - https://doi.org/10.11648/j.acm.20160505.11
DO  - 10.11648/j.acm.20160505.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 186
EP  - 192
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20160505.11
AB  - 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.
VL  - 5
IS  - 5
ER  -

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Author Information

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

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APA Style

Mohammad Darvishi, Haji Mohammad Mohammadinejad. (2016). Control of Bifurcation in Internet Model with Time-Delay. Applied and Computational Mathematics, 5(5), 186-192. https://doi.org/10.11648/j.acm.20160505.11

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ACS Style

Mohammad Darvishi; Haji Mohammad Mohammadinejad. Control of Bifurcation in Internet Model with Time-Delay. Appl. Comput. Math. 2016, 5(5), 186-192. doi: 10.11648/j.acm.20160505.11

Copy | Download

AMA Style

Mohammad Darvishi, Haji Mohammad Mohammadinejad. Control of Bifurcation in Internet Model with Time-Delay. Appl Comput Math. 2016;5(5):186-192. doi: 10.11648/j.acm.20160505.11

Copy | Download

@article{10.11648/j.acm.20160505.11,
  author = {Mohammad Darvishi and Haji Mohammad Mohammadinejad},
  title = {Control of Bifurcation in Internet Model with Time-Delay},
  journal = {Applied and Computational Mathematics},
  volume = {5},
  number = {5},
  pages = {186-192},
  doi = {10.11648/j.acm.20160505.11},
  url = {https://doi.org/10.11648/j.acm.20160505.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20160505.11},
  abstract = {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.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - Control of Bifurcation in Internet Model with Time-Delay
AU  - Mohammad Darvishi
AU  - Haji Mohammad Mohammadinejad
Y1  - 2016/10/08
PY  - 2016
N1  - https://doi.org/10.11648/j.acm.20160505.11
DO  - 10.11648/j.acm.20160505.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 186
EP  - 192
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20160505.11
AB  - 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.
VL  - 5
IS  - 5
ER  -

Copy | Download