XU Chang-jin, TANG Xian-hua, LIAO Mao-xin. Local Hopf Bifurcation and Global Existence of Periodic Solutions in a TCP System[J]. Applied Mathematics and Mechanics, 2010, 31(6): 745-755. doi: 10.3879/j.issn.1000-0887.2010.06.012
Citation: XU Chang-jin, TANG Xian-hua, LIAO Mao-xin. Local Hopf Bifurcation and Global Existence of Periodic Solutions in a TCP System[J]. Applied Mathematics and Mechanics, 2010, 31(6): 745-755. doi: 10.3879/j.issn.1000-0887.2010.06.012

Local Hopf Bifurcation and Global Existence of Periodic Solutions in a TCP System

doi: 10.3879/j.issn.1000-0887.2010.06.012
  • Received Date: 1900-01-01
  • Rev Recd Date: 2010-04-09
  • Publish Date: 2010-06-15
  • The dynamics of a TCP system described by a firs-torder non linear delay differential equations was investigated. Byanalyzing the associated characteristic tran scendental equation, the result thata sequence of Hopf bifurcations occurat the positive equilibrium as the delay passesth rough a sequence of critical values was obtained. Explicit algorithms for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions were derived by using the normal form theory and center manifold theory. Global existence of periodic solutions was also established by using the method of Wu [Trans Amer Math Soc, 1998, 350(12):4799-38].
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  • [1]
    Hollot C V, Misra V, Towlsey D, Gong W B. A control theoretic analysis of RED[C]Proceedings of IEEE INFOCOM. Anchorage,Alaska,2001.
    [2]
    Floyd S, Jacobson V. Random early detection gateways for congestion avoidance[J]. IEEE/ACM Trans Netw, 1993, 1(4): 397-413. doi: 10.1109/90.251892
    [3]
    Hollot C V, Misra V, Towlsey D, Gong W B. On designing improved controller for AQM routes supporting TCP flows[C]Proceedings of IEEE INFOCOM. Anchorage, Alaska, 2001.
    [4]
    Kunniyur S, Srikant R. Analysis and design of an adaptive virtual queue(AVQ) algorithm for active queue management[C]Proceedings of ACM SIGCOMM. New York, USA, 2001.
    [5]
    Low S H, Paganin F, Doyle J C. Dynamics of TCP/RED and a scalable control[J].IEEE INFOCOM, 2002, 2(2): 1-10.
    [6]
    Raina G, Heckmann O. TCP: Local stability and Hopf bifurcation[J].Performance Evaluation, 2007, 64(3): 266-275. doi: 10.1016/j.peva.2006.05.005
    [7]
    Srikant R. Models and Methods for Analyzing Internet Congestion Control Algorithms[C]Abdallah C T, Chiasson J, Tarbouriech S.Advances in Communication Control Networks. Princeton, NJ: Springer-Verlag, 2004, 416-419.
    [8]
    Kelly F P. Fairness and stability of end-to-end congestion control[J]. Eur J Control, 2003, 9: 149-165.
    [9]
    RUAN Shi-gui, WEI Jun-jie. On the zero of some transcendential functions with applications to stability of delay differential equations with two delays[J]. Dyn Contin Discrete Impuls Syst, Ser A, 2003, 10(6): 863-874.
    [10]
    Kuang Y. Delay Differential Equations With Applications in Population Dynamics[M]. Boston: Academic Press, INC, 1993.
    [11]
    Hale J. Theory of Functional Differential Equation[M]. New York: Springer-Verlag, 1977.
    [12]
    Hassard B, Kazarino D, Wan Y. Theory and Applications of Hopf Bifurcation[M]. Cambridge: Cambridge University Press, 1981.
    [13]
    WU Jian-hong. Symmetric functional differential equations and neural networks with memory[J]. Trans Amer Math Soc, 1998, 350(12): 4799-4838. doi: 10.1090/S0002-9947-98-02083-2
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