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

JCR Impact Factor: 11.8, Top 4% (SCI Q1)

CiteScore: 17.6, Top 3% (Q1)
Google Scholar h5-index: 77， TOP 5

Turn off MathJax

Article Contents

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

Citation:

PDF( 994 KB)

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

More Information

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

FullText(HTML)

References(42)

References

[1]	B. I. Henry and S. L. Wearne, "Existence of Turing instabilities in a two-species fractional reaction-diffusion system, " SIAM J. Appl. Math. , vol. 62, no. 3, pp. 870-887, Feb. 2002. doi: 10.1137/S0036139900375227
[2]	N. Engheia, "On the role of fractional calculus in electromagnetic theory, " IEEE Antennas Propag. Mag. , vol. 39, no. 4, pp. 35-46, Aug. 1997. http://ieeexplore.ieee.org/document/632994/
[3]	N. Heymans and J. C. Bauwens, "Fractal rheological models and fractional differential equations for viscoelastic behavior, " Rheol. Acta, vol. 33, no. 3, pp. 210-219, May1994. doi: 10.1007/BF00437306
[4]	H. Sun, A. Abdelwahab, and B. Onaral, "Linear approximation of transfer function with a pole of fractional power, " IEEE Trans. Automat. Control, vol. 29, no. 5, pp. 441-444, May1984. https://www.mathworks.com/help/slcontrol/ug/linearize.html
[5]	D. Baleanu, J. A. T. Machado, and A. C. J. Luo, " Fractional Dynamics and Control. Berlin: Springer, 2012.
[6]	V. D. Djordjević, J. Jarić, B. Fabry, J. J. Fredberg, and D. Stamenović, "Fractional derivatives embody essential features of cell rheological behavior, " Ann. Biomed. Eng. , vol. 31, no. 6, pp. 692-699, Jun. 2003. doi: 10.1114/1.1574026
[7]	Y. H. Lim, K. K. Oh, and H. S. Ahn, "Stability and stabilization of fractional-order linear systems subject to input saturation, " IEEE Trans. Automat. Control, vol. 58, no. 4, pp. 1062-1067, Apr. 2013. http://ieeexplore.ieee.org/document/6246678/
[8]	M. Xiao, W. X. Zheng, G. P. Jiang, and J. D. Cao, "Undamped oscillations generated by Hopf bifurcations in fractional-order recurrent neural networks with Caputo derivative, " IEEE Trans. Neural Networks Learn. Syst. , vol. 26, no. 12, pp. 3201-3214, Dec. 2015. https://www.semanticscholar.org/paper/Undamped-Oscillations-Generated-by-Hopf-Xiao-Zheng/3efc5b73a1229987be63a910c61682ba16cf7917
[9]	T. T. Hartley, C. F. Lorenzo, and H. K. Qammer, "Chaos in a fractional order Chua's system, " IEEE Trans. Circuits Syst. Ⅰ Fund. Theory Appl. , vol. 42, no. 8, pp. 485-490, Aug. 1995. http://ieeexplore.ieee.org/document/404062/
[10]	I. N'Doye, H. Voos, and M. Darouach, "Observer-based approach for fractional-order chaotic synchronization and secure communication, " IEEE J. Emerg. Sel. Topics Circuits Syst. , vol. 3, no. 3, pp. 442-450, Sep. 2013. http://ieeexplore.ieee.org/document/6669423/
[11]	A. Papachristodoulou and A. Jadbabaie, "Delay robustness of nonlinear Internet congestion control schemes, " IEEE Trans. Automat. Control, vol. 55, no. 6, pp. 1421-1427, Jun. 2010. http://ieeexplore.ieee.org/document/5422746/
[12]	J. Barrera and A. Garcia, "Dynamic incentives for congestion control, " IEEE Trans. Automat. Control, vol. 60, no. 2, pp. 299-310, Feb. 2015. http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6879269
[13]	X. Zhang and A. Papachristodoulou, "Improving the performance of network congestion control algorithms, " IEEE Trans. Automat. Control, vol. 60, no. 2, pp. 522-527, Feb. 2015. http://ieeexplore.ieee.org/iel7/9/4601496/06849427.pdf?arnumber=6849427
[14]	F. P. Kelly, A. K. Maulloo, and D. K. H. Tan, "Rate control for communication networks: shadow prices, proportional fairness and stability, " J. Oper. Res. Soc. , vol. 49, no. 3, pp. 237-252, Mar. 1998.
[15]	Y. P. Zhang and D. Loguinov, "Local and global stability of delayed congestion control systems, " IEEE Trans. Automat. Control, vol. 53, no. 10, pp. 2356-2360, Nov. 2008. http://ieeexplore.ieee.org/document/4668502/
[16]	P. Ranjan, R. J. La, and E. H. Abed, "Global stability conditions for rate control with arbitrary communication delays, " IEEE/ACM Trans. Network. , vol. 14, no. 1, pp. 94-107, Feb. 2006. http://www.academia.edu/412465/Global_Stability_Conditions_for_Rate_Control_With_Arbitrary_Communication_Delays
[17]	M. L. Sichitiu and P. H. Bauer, "Asymptotic stability of congestion control systems with multiple sources, " IEEE Trans. Automat. Control, vol. 51, no. 2, pp. 292-298, Feb. 2006. http://ieeexplore.ieee.org/document/1593903/
[18]	Y. P. Tian, "Stability analysis and design of the second-order congestion control for networks with heterogeneous delays, " IEEE/ACM Trans. Network. , vol. 13, no. 5, pp. 1082-1093, Oct. 2005. http://ieeexplore.ieee.org/iel5/90/32642/01528496.pdf?arnumber=1528496
[19]	G. Raina, "Local bifurcation analysis of some dual congestion control algorithms, " IEEE Trans. Automat. Control, vol. 50, no. 8, pp. 1135-1146, Aug. 2005. http://www.amm.shu.edu.cn/CN/abstract/abstract10711.shtml
[20]	C. G. Li, G. R. Chen, X. F. Liao, and J. B. Yu, "Hopf bifurcation in an Internet congestion control model, " Chaos Solitons Fractals, vol. 19, no. 4, pp. 853-862, Mar. 2004. http://www.sciencedirect.com/science/article/pii/S0960077903002698
[21]	M. Xiao, W. X. Zheng, and J. D. Cao, "Bifurcation control of a congestion control model via state feedback, " Int. J. Bifurc. Chaos, vol. 23, no. 6, pp. 1330018, Jun. 2013. doi: 10.1142/S0218127413300188
[22]	W. Y. Xu, J. D. Cao, and M. Xiao, "Bifurcation analysis of a class of (n+1)-dimension Internet congestion control systems, " Int. J. Bifurc. Chaos, vol. 25, no. 2, pp. 1550019, Feb. 2015. http://or.nsfc.gov.cn/handle/00001903-5/334670
[23]	D. W. Ding, J. Zhu, X. S. Luo, and Y. L. Liu, "Delay induced Hopf bifurcation in a dual model of Internet congestion control algorithm, " Nonlinear Anal. Real World Appl. , vol. 10, no. 5, pp. 2873-2883, Oct. 2009. http://www.sciencedirect.com/science/article/pii/S1468121808001983
[24]	M. Xiao, G. P. Jiang, and L. D. Zhao, "State feedback control at Hopf bifurcation in an exponential RED algorithm model, " Nonlinear Dyn. , vol. 76, no. 2, pp. 1469-1484, Apr. 2014. https://www.researchgate.net/publication/271740392_State_feedback_control_at_Hopf_bifurcation_in_an_exponential_RED_algorithm_model
[25]	F. Liu F, H. O. Wang, and Z. H. Guan, "Hopf bifurcation control in the XCP for the Internet congestion control system, " Nonlinear Anal. Real World Appl. , vol. 13, no. 3, pp. 1466-1479, Jun. 2012. http://www.sciencedirect.com/science/article/pii/S146812181100318X
[26]	M. Xiao and J. D. Cao, "Delayed feedback-based bifurcation control in an Internet congestion model, " J. Math. Anal. Appl. , vol. 332, no. 2, pp. 1010-1027, Aug. 2007. http://www.sciencedirect.com/science/article/pii/S0022247X06012157
[27]	I. Podlubny, Fractional Differential Equations. New York: Academic Press, 1999.
[28]	G. Q. Chen and E. G. Friedman, "An RLC interconnect model based on fourier analysis, " IEEE Trans. Comp. Aided Des. Integr. Circuits Syst. , vol. 24, no. 2, pp. 170-183, Feb. 2005. http://dl.acm.org/citation.cfm?id=2298529.2301127
[29]	V. G. Jenson and G. V. Jeffreys, Mathematical Methods in Chemical Engineering. 2nd ed. New York: Academic Press, 1977.
[30]	R. L. Magin, "Fractional calculus models of complex dynamics in biological tissues, " Comput. Math. Appl. , vol. 59, no. 5, pp. 1586-1593, Mar. 2010. http://www.sciencedirect.com/science/article/pii/S0898122109005343
[31]	N. Laskin, "Fractional market dynamics, " Phys. A, vol. 287, no. 3-4, pp. 482-492, Dec. 2000. http://www.sciencedirect.com/science/article/pii/S0378437100003873
[32]	W. H. Deng, C. P. Li, and J. H. Lü, "Stability analysis of linear fractional differential system with multiple time delays, " Nonlinear Dyn. , vol. 48, no. 4, pp. 409-416, Jun. 2007. doi: 10.1007%2Fs11071-006-9094-0
[33]	D. Matignon, "Stability results for fractional differential equations with applications to control processing, " in Multiconference: Computational Engineering in Systems and Application, Lille, France, 1996, pp. 963-968. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.40.4859
[34]	Y. A. Kuznetsov, Elements of Applied Bifurcation Theory. New York: Springer-Verlag, 2004.
[35]	H. A. El-Saka, E. Ahmed, M. I. Shehata, and A. M. A. El-Sayed, "On stability, persistence, and Hopf bifurcation in fractional order dynamical systems, " Nonlinear Dyn. , vol. 56, no. 1-2, pp. 121-126, Jul. 2008. doi: 10.1007/s11071-008-9383-x
[36]	M. S. Abdelouahab, N. E. Hamri, and J. W. Wang, "Hopf bifurcation and chaos in fractional-order modified hybrid optical system, " Nonlinear Dyn. , vol. 69, no. 1-2, pp. 275-284, Jul. 2012. doi: 10.1007%2Fs11071-011-0263-4
[37]	S. S. Kunniyur and R. Srikant, "Stable, scalable, fair congestion control and AQM schemes that achieve high utilization in the Internet, " IEEE Trans. Automat. Control, vol. 48, no. 11, pp. 2024-2028, Nov. 2003. http://ieeexplore.ieee.org/document/1245195/
[38]	S. Liu, T. Basar, and R. Srikant, "Exponential-RED: a stabilizing AQM scheme for low-and high-speed TCP protocols, " IEEE/ACM Trans. Network. , vol. 13, no. 5, pp. 1068-1081, Oct. 2005. http://ieeexplore.ieee.org/document/1528495/
[39]	P. Ranjan, E. H. Abed, and R. J. La, "Nonlinear instabilities in TCP-RED, " IEEE/ACM Trans. Network. , vol. 12, no. 6, pp. 1079-1092, Dec. 2004. http://dl.acm.org/citation.cfm?id=1046023
[40]	F. Kelly, "Fairness and stability of end-to-end congestion control, " Eur. J. Control, vol. 9, no. 2-3, pp. 159-176, 2003. http://www.statslab.cam.ac.uk/~frank/PAPERS/fse2ecc.html
[41]	D. W. Ding, X. M. Qin, N. Wang, T. T. Wu, and D. Liang, "Hybrid control of Hopf bifurcation in a dual model of Internet congestion control system, " Nonlinear Dyn. , vol. 76, no. 2, pp. 1041-1050, Apr. 2014. doi: 10.1007/s11071-013-1187-y
[42]	S. Bhalekar and V. Daftardar-Gejji, "A predictor-corrector scheme for solving nonlinear delay differential equations of fractional order, " J. Fract. Calculus Appl. , vol. 1, no. 5, pp. 1-9, Jul. 2011. https://www.researchgate.net/publication/282942655_A_Predictor-Corrector_Scheme_for_Solving_a_Nonlinear_Circuit

Supplements(0)

Cited By

Proportional views

Proportional views

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

Figures(7) / Tables(1)

Get Citation

PDF

XML

Article Metrics

Article views (1317) PDF downloads(139)

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

doi: 10.1109/JAS.2016.7510151

Abstract

References

Proportional views

Catalog

通讯作者: 陈斌, bchen63@163.com

Article Metrics

Export File

Citation

Format

Content