Local Bifurcation Analysis of a Delayed Fractional-order Dynamic Model of Dual Congestion Control Algorithms

Min Xiao; Guoping Jiang; Jinde Cao; Weixing Zheng

doi:10.1109/JAS.2016.7510151

Volume 4 Issue 2

Apr. 2017

IEEE/CAA Journal of Automatica Sinica

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Article Navigation > IEEE/CAA Journal of Automatica Sinica > 2017 > 4(2): 361-369

Min Xiao, Guoping Jiang, Jinde Cao and Weixing Zheng, "Local Bifurcation Analysis of a Delayed Fractional-order Dynamic Model of Dual Congestion Control Algorithms," IEEE/CAA J. Autom. Sinica, vol. 4, no. 2, pp. 361-369, Apr. 2017. doi: 10.1109/JAS.2016.7510151

Citation:

Min Xiao, Guoping Jiang, Jinde Cao and Weixing Zheng, "Local Bifurcation Analysis of a Delayed Fractional-order Dynamic Model of Dual Congestion Control Algorithms," IEEE/CAA J. Autom. Sinica, vol. 4, no. 2, pp. 361-369, Apr. 2017. doi: 10.1109/JAS.2016.7510151

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Local Bifurcation Analysis of a Delayed Fractional-order Dynamic Model of Dual Congestion Control Algorithms

doi: 10.1109/JAS.2016.7510151

Min Xiao^{1
,
,},
Guoping Jiang^1
,,
Jinde Cao^2
,,
Weixing Zheng^3
,

1.
College of Automation, Nanjing University of Posts and Telecommunications, Nanjing 210023, China
2.
Research Center for Complex Systems and Network Sciences, Southeast University, Nanjing 210096, China
3.
School of Computing, Engineering and Mathematics, Western Sydney University, Sydney, NSW 2751, Australia

Funds:

This work was supported by National Natural Science Foundation of China 61573194

This work was supported by National Natural Science Foundation of China 61374180

This work was supported by National Natural Science Foundation of China 61573096

China Postdoctoral Science Foundation Funded Project 2013M530229

China Postdoctoral Science Special Foundation Funded Project 2014T70463

Six Talent Peaks High Level Project of Jiangsu Province ZNDW-004

Science Foundation of Nanjing University of Posts and Telecommunications NY213095

and Australian Research Council DP120104986

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Author Bio:
Guoping Jiang (M'03-SM'13) received the B.E. degree in Electrical Engineering from Hohai University, Nanjing, China, in 1988, and the Ph.D. degree in control theory and engineering from Southeast University, Nanjing, China, in 1997. He is currently a professor and the vice president of Nanjing University of Posts and Telecommunications, Nanjing, China. He has authored or coauthored more than 200 published academic articles. Professor Jiang received the Academic Leader for the Youthful and MiddleAged Scholars of Jiangsu Province, China, in 2002, and the New Century Excellent Talents of Ministry of Education, China, in 2006. His research interests include chaos synchronization and control, and complex dynamical networks.(e-mail: jianggp@njupt.edu.cn)

Jinde Cao received the B.S. degree from Anhui Normal University, Wuhu, China, the M.S. degree from Yunnan University, Kunming, China, and the Ph.D. degree from Sichuan University, Chengdu, China, all in mathematics/applied mathematics, in 1986, 1989, and 1998, respectively. He is currently a distinguished professor, the dean of Department of Mathematics and the Director of the Research Center for Complex Systems and Network Sciences at Southeast University, Nanjing, China. Professor Cao was an associate editor of the IEEE Transactions on Neural Networks, Journal of the Franklin Institute and Neurocomputing. He is an associate editor of the IEEE Transactions on Cybernetics, Differential Equations and Dynamical Systems, Mathematics and Computers in Simulation, and Neural Networks. He is a fellow of IEEE (2016), and a member of the Academy of Europe (2016). He has been named as Highly-Cited Researcher in Mathematics, Computer Science and Engineering listed by Thomson Reuters. His research interests include nonlinear systems, neural networks, complex systems and complex networks, stability theory, and applied mathematics.(e-mail: jdcao@seu.edu.cn)

Weixing Zheng received the B.S. degree in 1982, the M.S. degree in 1984, and the Ph.D. degree in 1989, all from the Southeast University, Nanjing, China. He is currently a professor at the Western Sydney University, Sydney, Australia. Dr. Zheng serves as an associate editor of Automatica, IEEE Transactions on Automatic Control, and other IEEE journals. He is a fellow of IEEE and a Thomson Reuters Highly Cited Researcher. His research interests include system identification, networked control systems, multi-agent systems, complex networks and signal processing. (e-mail: w.zheng@westernsydney.edu.au)
Corresponding author: Min Xiao (M'11) received the B.S. and M.S. degrees in mathematics and fundamental mathematics from Nanjing Normal University, Nanjing, China, in 1998 and 2001, respectively, and the Ph.D. degree in applied mathematics from Southeast University, Nanjing, China, in 2007. He is currently a professor at the College of Automation, Nanjing University of Posts and Telecommunications, Nanjing, China. His current research interests include memristor-based neural networks, fractional order systems, networked control systems, bifurcation control, and smart networks of power.(e-mail: candymanxm2003@aliyun.com)
Received Date: 2015-09-07
Accepted Date: 2016-06-02

Abstract

Abstract

In this paper, we propose a delayed fractional-order congestion control model which is more accurate than the original integer-order model when depicting the dual congestion control algorithms. The presence of fractional orders requires the use of suitable criteria which usually make the analytical work so harder. Based on the stability theorems on delayed fractionalorder differential equations, we study the issue of the stability and bifurcations for such a model by choosing the communication delay as the bifurcation parameter. By analyzing the associated characteristic equation, some explicit conditions for the local stability of the equilibrium are given for the delayed fractionalorder model of congestion control algorithms. Moreover, the Hopf bifurcation conditions for general delayed fractional-order systems are proposed. The existence of Hopf bifurcations at the equilibrium is established. The critical values of the delay are identified, where the Hopf bifurcations occur and a family of oscillations bifurcate from the equilibrium. Same as the delay, the fractional order normally plays an important role in the dynamics of delayed fractional-order systems. It is found that the critical value of Hopf bifurcations is crucially dependent on the fractional order. Finally, numerical simulations are carried out to illustrate the main results.
- Congestion control algorithm,
- fractional-order congestion control algorithm model,
- Hopf bifurcation,
- stability

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Local Bifurcation Analysis of a Delayed Fractional-order Dynamic Model of Dual Congestion Control Algorithms

doi: 10.1109/JAS.2016.7510151

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