ARTICLE IN PRESS Nonlinear Analysis: Real World Applications ( ) – www.elsevier.com/locate/na Necessary and sufficient conditions for Hopf bifurcation in exponential RED algorithm with communication delay Songtao Guo a , ∗ , Xiaofeng Liao a , Qun Liu b , Chuandong Li a a College of Computer Science, Chongqing University, Chongqing 400044, PR China b College of Computer Science, Chongqing University of Posts andTelecommunications, Chongqing 400065, PR China Received 27 March 2007; accepted 24 May 2007 Abstract In this paper, we investigate a novel congestion control algorithm, i.e., exponential RED algorithm, with communication delay. We derive some necessary and sufficient conditions ensuring Hopf bifurcation to occur for this model. By choosing the delay as a bifurcation parameter, we demonstrated that Hopf bifurcation would occur when the delay exceeds a critical value. A formula for determining the bifurcation direction and stability of bifurcation periodic solutions is given by applying the normal form theory and the center manifold theorem. Some numerical simulations for justifying the theoretical results are also provided.  2007 Elsevier Ltd. All rights reserved. Keywords: Hopf bifurcation; Exponential RED; Congestion control; Stability 1. Introduction Congestion control in data networks usually refers to the problem of controlling the sending rate of the source hosts such that the throughput of the network is maximized under some additional constraints (e.g., fairness). The importance of congestion control schemes cannot be overemphasized, as congestion can bring even a correctly operating network to its knees [14]. When congested, the routers in the network receive more packets than they can forward, their internal buffers overflow, and packets are dropped. If the congestion control scheme is not well designed, the sources will try to push even more packets through the network in response to packet drops, thus worsening the congestion. While TCP has served its purpose exceptionally well and is partly responsible for the communication explosion of the last decade, there is a fundamental conceptual flaw. Routers have queues to enable statistical multiplexing. The goal of Floyd and Jacobson’s paper was to correct this apparent conflict through active queue management (AQM) [5]. The objective of an AQM technique is not only to maintain the queue size at desired levels but also to notify the connections to reduce their window sizes as soon as congestion is expected to occur at the gateway. The performance of AQM schema heavily depends on its dynamical behaviors. Thus, the qualitative analysis of the dynamical behaviors is a necessary step for the practical design and application of AQM schema. Recently, the stability of AQM schema with or without delays has been widely studied by many researchers, and various interesting results have been reported [15,8,26,11,19,3,1,22–24,28,18,2,6,30,13,12]. The primal algorithm for TCP rate control ∗ Corresponding author. Tel.: +86 23 65103199. E-mail address: songtao_guo@163.com (S. Guo). 1468-1218/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2007.05.014 Please cite this article as: S. Guo, et al., Necessary and sufficient conditions for Hopf bifurcation in exponential RED algorithm with communi- cation delay, Nonlinear Anal.: Real World Appl. (2007), doi: 10.1016/j.nonrwa.2007.05.014