Stability Analysis And Bifurcation Control For Several Class Of Dynamic Systems With Fractional Delays

Hopf bifurcation theory is of great theoretical value in dynamic bifurcation research and limit cycle research,and is closely related to the generation of self-excited vibration in engineering,and this phenomenon is very common in many dynamic systems,such as network systems.Power system,Goodwin model,gene regulation network model,etc.Since Hopf bifurcation sometimes affects the stability of the system,Hopf bifurcation theory has become an important tool for studying the dynamic behavior of systems.According to a large amount of data research,in the actual network,due to the interaction of information and the adjustment between nodes,it takes a certain time,so there are various time lags in the actual model.Due to the existence of time lag,the dynamic behavior of the system will become more complicated,and the appropriate time delay can also improve the stability of the system.Therefore,studying the influence of time lag on the dynamic behavior of complex systems has important theoretical and practical significance.At the same time,the fractional derivative can describe the memory and genetic characteristics of the related model more accurately.Therefore,the bifurcation dynamics of the graduated fractional time-delay system is of great significance.Based on the previous researches,based on the theory of bifurcation,this paper studies the theoretical study of applying controllers to integer-order(fractional-order)network models.The specific work is as follows:1.The bifurcation dynamics of a one-dimensional integer-order network congestion model is studied.The stability of the controlled network and the Hopf bifurcation are adjusted by applying a fractional-order PD controller.The research results show that the applied controller can effectively improve the dynamic performance of the controlled model.2.Bifurcation control of a two-dimensional fractional order single-gene regulation model with time delay is achieved by a fractional-order PD controller.According to the stability theory of fractional differential equations,the local asymptotic stability of the controlled fractional genetic model with time delay and some explicit conditions of Hopf bifurcation are proved.It is proved that by adjusting the control gain parameters,the fractional order single gene regulation model becomes controllable.In addition,the effect of fractional parameter on dynamic behavior is also shown.3.We propose a dual state feedback control method to control the stability and bifurcation of the singlegene regulatory model with the time delay.Meanwhile,the total time delay is elected as the bifurcation parameter to research the dynamic behaviors of the controlled system.Under this control mechanism,it has been obtained that Hopf bifurcation point can be effectively postponed or delay by selecting the appropriate control parameters and the stability domain can be extended.