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Applied Mathematics and Mechanics (English Edition) ›› 2010, Vol. 31 ›› Issue (6): 775-786.doi: https://doi.org/10.1007/s10483-010-1312-x

• Articles • 上一篇    下一篇

Local Hopf bifurcation and global existence of periodic solutions in TCP system

徐昌进1,2 唐先华1 廖茂新1,3   

  1. 1. School of Mathematical Science and Computing Technology, Central South University, Changsha 410083, P. R. China;
    2. Faculty of Science, Hunan Institute of Engineering, Xiangtan 411004, Hunan Province, P. R. China;
    3. School of Mathematics and Physics, Nanhua University, Hengyang 421001, Hunan Province, P. R. China
  • 收稿日期:2009-09-10 修回日期:2010-05-05 出版日期:2010-06-01 发布日期:2010-06-01

Local Hopf bifurcation and global existence of periodic solutions in TCP system

XU Chang-Jin1,2, TANG Xian-Hua1, LIAO Mao-Xin1,3   

  1. 1. School of Mathematical Science and Computing Technology, Central South University, Changsha 410083, P. R. China;
    2. Faculty of Science, Hunan Institute of Engineering, Xiangtan 411004, Hunan Province, P. R. China;
    3. School of Mathematics and Physics, Nanhua University, Hengyang 421001, Hunan Province, P. R. China
  • Received:2009-09-10 Revised:2010-05-05 Online:2010-06-01 Published:2010-06-01

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被引次数

4

摘要： This paper investigates the dynamics of a TCP system described by a firstorder nonlinear delay differential equation. By analyzing the associated characteristic transcendental equation, it is shown that a Hopf bifurcation sequence occurs at the positive equilibrium as the delay passes through a sequence of critical values. The explicit algorithms for determining the Hopf bifurcation direction and the stability of the bifurcating periodic solutions are derived with the normal form theory and the center manifold theory. The global existence of periodic solutions is also established with the method of Wu (Wu, J. H. Symmetric functional differential equations and neural networks with memory. Transactions of the American Mathematical Society 350(12), 4799–4838 (1998)).

Abstract: This paper investigates the dynamics of a TCP system described by a firstorder nonlinear delay differential equation. By analyzing the associated characteristic transcendental equation, it is shown that a Hopf bifurcation sequence occurs at the positive equilibrium as the delay passes through a sequence of critical values. The explicit algorithms for determining the Hopf bifurcation direction and the stability of the bifurcating periodic solutions are derived with the normal form theory and the center manifold theory. The global existence of periodic solutions is also established with the method of Wu (Wu, J. H. Symmetric functional differential equations and neural networks with memory. Transactions of the American Mathematical Society 350(12), 4799–4838 (1998)).

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