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

Hui-Long Jin,Ran-Ran Zhang,Tian-Le Di,Hong Yu

doi:10.3390/sym14030463

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.

Highlights

With an increasing number of network devices in life, the quality of Internet communication becomes more important
Under the factors described above, we provide a practical and systematic procedure to test the nonlinear dynamics of the Transmission Control Protocol and Active Queue Management (TCP/active queue management (AQM)) model
An executable procedure is proposed for the linear stability analysis versus communication delay and calculation of Hopf bifurcation

Summary

Introduction

With an increasing number of network devices in life, the quality of Internet communication becomes more important. In [5], Yang and Tian consider the exhibition of Hopf bifurcation for an Internet congestion control algorithm with communication delay. Some classes of nonlinear fluid flow models with internet congestion control algorithm are investigated in [8], by using control and bifurcation theories. Wang investigated the local dynamics of a class of TCP/AQM networks with proportional feedback and communication delay. The local dynamics of FAST TCP model with internet congestion control algorithms are addressed in [12]. Both of the necessary and sufficient stability conditions and determination of bifurcating solutions are obtained. More complex analysis on large nonlinear models is provided in [15–17] Both of the analysis tools and the control algorithms are benefits for analyzing the dynamical behaviors of the TCP/AQM networks with delays. A concise algorithm is provided for the determination of the directions and stability of the bifurcation solutions

Problems Statement and Preliminaries

Direction and Stability of Bifurcating Periodic Solutions

Numerical Examples

Conclusions