On the <i>τ</i> Decomposition Method for the Stability and Bifurcation of the TCP/AQM Networks versus Time Delay

Abstract

This paper investigates the nonlinear dynamics of Transmission Control Protocol and Active Queue Management (TCP/AQM) networks, including the local stability and periodic bifurcation. The parameter of transportation delay affects the stability of the dynamical systems. One of the purposes of our work is to determine the delay stable interval of the transportation network. It is found that there is only one critical value of network delay by the τ decomposition technique. When the delay passes the critical point, the system performs Hopf bifurcation with a pair of symmetry with purely imaginary roots (PIR). In addition, the other purpose is to consider the stability of bifurcating periodic solutions. Combining with τ-decomposition strategy and central manifold theory, the issues of delay stable interval and stability of Hopf bifurcation are all tackled. Finally, numerical examples are illustrated to show the accuracy and effectiveness of the proposed method.

Keywords

• TCP/AQM network
• time delay
• Hopf bifurcation
• τ decomposition method
• center manifold theory