Hopf Bifurcation And Control In Fractional-Order Congestion Control Systems

Past decades have witnessed the intense development of Internet.However,the huge amount of the users and the data also make congestion problem inevitable.Thus,it is of great importance and urgency to designing congestion control algorithms and investigating the dynamic behaviors of congestion control systems.Fractional calculus and its wide applications have attracted many researchers.It is shown that dynamical equations using fractional derivatives are effective and more accurate in the mathematical modeling of real world phenomena.Therefore,the study of the dynamic behaviors of these models,including stability,bifurcation and chaos,will help understand the intrinsic characteristics of these phenomena.Based on stability theorems,bifurcation theorems and bifurcation control schemes,this paper is dedicated to investigating the stability and Hopf bifurcations for a fractional-order exponential RED congestion control model and the bifurcation control for a delayed fractional-order dynamic model of dual congestion control algorithm.Main research contents are as follows.Firstly,a hybrid control strategy with time delay is presented to control the Hopf bifurcation for a delayed integer-order dynamic model of dual congestion control algorithm.By choosing the gain parameter as the bifurcation parameter and analyzing the characteristic equation,the critical value is calculated and the existence of Hopf bifurcation is proved.It is shown by simulations that the presence of hybrid controller successfully delays the onset of bifurcation and the stability domain is extended.Secondly,we propose a fractional-order exponential RED congestion control model and study the stability and the Hopf bifurcation based on stability theorems of delayed fractional-order differential equations.From the perspective of the gain parameter,the local stability is derived and the existence of Hopf bifurcations at the equilibrium is established.The critical value is also identified where a Hopf bifurcation occurs.Then,an example is given to verify the results by simulations.Thirdly,a fractional-order Proportional-Derivative(PD)feedback controller is designed to control the bifurcation generated by a delayed fractional-order dual model.By selecting the communication delay as the bifurcation parameter,we obtain some conditions for the stability of the equilibrium and the Hopf bifurcation.The critical value of delay is figured out,where a Hopf bifurcation occurs and a family of oscillations is observed to bifurcate from the equilibrium from simulations.It is also illustrated by numerical simulations that the onset of the bifurcation can be postponed or advanced by selecting proper control parameters in the fractional-order PD feedback controller.