We study the Hopf bifurcation and the chaos phenomena in a random early detection-based active queue management (RED-AQM) congestion control system with a communication delay.We prove that there is a critical value of the communication delay for the stability of the RED-AQM control system.Furthermore,we show that the system will lose its stability and Hopf bifurcations will occur when the delay exceeds the critical value.When the delay is close to its critical value,we demonstrate that typical chaos patterns may be induced by the uncontrolled stochastic traffic in the RED-AQM control system even if the system is still stable,which reveals a new route to the chaos besides the bifurcation in the network congestion control system.Numerical simulations are given to illustrate the theoretical results.
This paper uses a correlation dimension based nonlinear analysis approach to analyse the dynamics of network traffics with three different application protocols-HTTP,FTP and SMTP.First,the phase space is reconstructed and the embedding parameters are obtained by the mutual information method.Secondly,the correlation dimensions of three different traffics are calculated and the results of analysis have demonstrated that the dynamics of the three different application protocol traffics is different from each other in nature,i.e.HTTP and FTP traffics are chaotic,furthermore,the former is more complex than the later;on the other hand,SMTP traffic is stochastic.It is shown that correlation dimension approach is an efficient method to understand and to characterize the nonlinear dynamics of HTTP,FTP and SMTP protocol network traffics.This analysis provided insight into and a more accurate understanding of nonlinear dynamics of internet traffics which have a complex mixture of chaotic and stochastic components.
