Stability Analysis and Hopf Bifurcation Control for a Wireless Network Model
Feng Liu*, Xiang Yin*, Xinmei Wang*, Wei Liu*, Longsheng Wei*, and Hua O. Wang**
*School of Automation, China University of Geosciences
No.388 Lumo Road, Hongshan District, Wuhan Hubei 430074, China
**Department of Mechanical Engineering, Boston University
Boston, MA 02215, USA
A wireless network model with the hybrid control is considered. The stability and Hopf bifurcation behavior of the wireless network model are investigated. The stability analysis and the parameter condition that the Hopf bifurcation occurs are deduced. The stability and direction of the bifurcating periodic solutions are analyzed by applying the normal form theory and the center manifold theorem. Numerical simulations demonstrate the complex behavior of the system and verify the theoretical analysis.
-  V. Jacobson, “Congestion avoidance and control,” ACM SIGCOMM Comput. Commun. Rev., Vol.18, No.4, pp. 314-329, 1998.
-  S. Floyd and V. Jacobson, “Random early detection gateways for congestion avoidance,” IEEE/ACM Trans. Netw., Vol.1, No.4, pp. 397-413, 1993.
-  V. Misra, W. B. Gong, and D. Towlsey, “Fluid-based analysis of a network of AQM routers supporting TCP flows with an application to RED,” ACM SIGCOMM., pp. 151-160, 2000.
-  J. P. Hepanha, S. Bohacek, K. Obrxzka, and J. Lee, “Hybrid modeling of TCP congestion control,” Lect. Notes Comput.Sci., Vol.2034, pp. 291-304, 2001.
-  D. Katabi, M. Handley, and C. Rohrs, “Congestion control for high bandwidth-delay product networks,” ACM SIGCOMM, 2002.
-  S. Floyd, “HighSpeed TCP for large congestion windows,” RFC Editor, pp. 3649, 2003.
-  C. Jin, D. X. Wei, and S. H. Low, “FAST TCP: motivation, architecture, algorithms, performance,” IEEE INFOCOM, 2004.
-  T. Kelly, “Scalable TCP: Improving performance in high speed wide area networks,” Computer Communication Review, Vol.32, No.2, 2003.
-  C. V. Hollot, V. Misra, D. Towsley, and W. B. Gong, “A control theoretic analysis of RED,” IEEE INFOCOM, 2001.
-  S. H. Low, F. Paganini, J. Wang, S. Adlakha, and J. C. Doyle, “Dynamics of TCP/RED and a scalable control,” IEEE INFOCOM, 2002.
-  G. Walsh, H. Ye, and L. Bushnell, “Stability analysis of network control systems,” IEEE Control Syst. Technol., Vol.10, No.3, pp. 438-446, 2002.
-  F. Zheng and J. Nelson, “An Η∞ approach to congestion control design for AQM routers supporting TCP flows in wireless access networks,” Computer Networks, Vol.51, No.6, pp. 1684-1704, 2007.
-  A. Veres and M. Boda, “The chaotic nature of TCP congestion control,” IEEE INFOCOM, Vol.3, pp. 1715-1723, 2003.
-  P. Ranjan and E. H. Abed, “Nonlinear instabilities in TCP-RED,” IEEE/ACM Trans. on Networking, Vol.12, No.6, pp. 1079-1092, 2004.
-  C. Li, G. Chen, X. Liao, and J. Yu, “Hopf bifurcation in an internet congestion control model,” Chaos Solitons Fractals, Vol.19, No.4, pp. 853-862, 2004.
-  Z. Chen and P. Yu, “Hopf bifurcation control for an internet congestion model,” Int. J. Bifur. Chaos, Vol.15, pp. 2643-2651, 2005.
-  F. Liu, H. O. Wang, and Z. H. Guan, “Hopf Bifurcation Control in the XCP for the Internet Congestion Control System,” Nonlinear Anal.: RWA., Vol.3, No.13, pp. 1466-1479, 2012.
-  F. Liu, Z. H. Guan, and H. O. Wang, “Stability and Hopf Bifurcation Analysis in a TCP Fluid Model,” Nonlinear Anal.: RWA., Vol.1, No.12, pp. 353-363, 2011.
-  F. Liu, Z. H. Guan, and H. O. Wang, “Controlling bifurcations and chaos in TCP-UDP-RED,” Nonlinear Anal.: RWA., Vol.3, No.11, pp. 1491-1501, 2010.
-  Z. Liu and K. W. Chung, “Hybrid control of bifurcation in continuous nonlinear dynamical systems,” Int. J. Bifurcation and Chaos, Vol.15, pp. 3895-3903, 2005.
-  N. Dieudonnée, “Foundations of Modern Analysis,” Academic Press, 1960.
-  K. Cooke and Z. Grossman, “Discrete delay, Distributed delay and stability switches,” J. Math. Anal. Appl., Vol.86, pp. 592-627, 1982.
-  B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, “Theory and Applications of Hopf Bifurcation” Cambridge University Press, 1981.
-  J. Hale, “Theory of Functional Differential Equations,”Spring-Verlag, 1977.
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