Bifurcation Analysis of a Class of (n + 1)-Dimension Internet Congestion Control Systems

Abstract

This paper investigates the stability and Hopf bifurcation induced by the time delay in a class of (n + 1)-dimension Internet congestion control systems. Although there are several previous works on simplified models of Internet congestion systems with only one or two sources and such works can reflect partly dynamical behaviors of real Internet systems, some complicated problems may inevitably be overlooked. Hence, it is meaningful to study high-dimensional models which stand closer to general realistic large-scale Internet congestion networks. By analyzing the distribution of the associated characteristic roots, we can obtain conditions for keeping systems stable. When the delay increases and exceeds a critical value, the system will undergo a Hopf bifurcation. Furthermore, the explicit formulas to determine the stability and the direction of the bifurcating periodic solution are derived by applying the normal form theory and the center manifold reduction. Finally, two numerical examples are given to verify our theoretical analysis.