Hopf Bifurcation And Chaos In Two Kinds Of AIMD/RED Network Congestion Control Models With Delay-dependent Coefficients

With the increasing number and diversification of Internet applications,network congestion has become a barrier that restricts the development and application of networks.Therefore,congestion control in the Internet is an extremely important and challenging problem.The whole Internet congestion and avoidance mechanism is a combination of the end-to-end TCP congestion control mechanism and queue management mechanism IP at router.The combination of AIMD congestion control and RED is considered to be one of the key factors for the huge success of the Internet.Since the round-trip delay appears in the coefficients of the AIMD/RED system,it leads to the generation of complex dynamics,and it is more difficult to study.Under the support of the National Natural Science Foundation of China(Nos.11372282 and 11972327),in this dissertation,mainly the dynamic behaviors of AIMD/RED system with delay-dependent coefficients and their improved system are investigated.Firstly,in the AIMD/RED network congestion system,the effects of the delay on the stability of nontrivial equilibrium are studied by analyzing the characteristic equation of the linearized system according to the geometric criterion for the stability switch of the delay-dependent coefficient DDE by Kuang.And the critical value for Hopf bifurcation is also obtained.When the delay passes through,the system loses its stability.So the system exhibits good performance only for proper delay in the stability interval.Then,MMS is performed to derive the normal form of Hopf bifurcation,and the bifurcation direction and stability of bifurcating periodic solution are also obtained.A supercritical Hopf bifurcation occurs,which induces the stable periodic solution.This reveals the possibility of periodic oscillation of the network congestion system,which is not good for maintaining the stability of the system and should be avoided.In the original system,the marking probability p(t)= kq(t)is a linear function,so it is somewhat unreasonable.Here we improve the model slightly,such that the probability function has a saturation effect,that is,p(t)= tanh(kq(t)).At the same time,we find the complex dynamics,i.e.,chaos,for delay in some interval in the improved system.The path to chaos is period doubling bifurcation.This reveals the possibility of chaotic oscillation in the network congestion control system,which must be avoided.The analysis and the stability conditions derived here can be used as a guideline to select AIMD/RED system parameters in order to maintain network stability and integrity,and to enhance system performance.The innovation and characteristics of this paper are described as follows.First,the packet loss function in AIMD/RED is reasonably improved.Then,using the geometric criterion for the stability switch and MMS,we study the Hopf bifurcation and the dynamic behavior of AIMD/RED system with delay-dependent coefficient and the more realistic improved system.And we found that supercritical Hopf bifurcation occurs in both,and the latter leads to chaos through the period doubling bifurcation,which reveals that delay has a crucial effect on the dynamics of the two systems.