Stability And Hopf Bifurcation Control Of Fractional-Order Differential Equations With Two Delays

Fractional calculus is an extension of traditional integral calculus.Currently,fractional calculus has been applied to neural networks,signal processing,biology,medicine and many other fields,and has gradually become the focus of scholars at home and abroad.Time delays are widespread in the real system.The researches show that the time delay has an important influence on the dynamic behavior of fractional-order system.Therefore,the stability research of fractional-order delay differential equations has theoretical significance and application value.Using the method of the stability switching curves,this thesis investigates the stability and existence of Hopf bifurcation of three kinds of fractional-order predator-prey models with two delays.The main work of this thesis is as follows:In the first part,we propose a three-dimensional fractional-order predator-prey model with two nonidentical delays.By using the method of the stability switching curves and choosing two delays as the bifurcation parameters,we investigate the stability of the positive equilibrium point and existence of Hopf bifurcation of the model.Firstly,we calculate the corresponding stability switching curves of system in the two delay planes,and determine the direction of the characteristic roots cross the imaginary axis.When two delays cross the stability switching curves,the stability of the positive equilibrium point changes from stable to unstable.Secondly,we deduce the conditions for the existence of Hopf bifurcation,and system appears to bifurcating periodic solutions near the positive equilibrium point.Finally,we give one example to verify the correctness of the theoretical analysis.It is found that with the increase of fractional-order,the stability region of the positive equilibrium point shrinks and the bifurcated periodic solutions appear earlier.In the second part,we investigate the stability and existence of Hopf bifurcation of a fractional-order two delays predator-prey model with stage structure and Holling type II functional response.In this thesis,two delays are selected as the bifurcation parameters,the critical curves for switching stability of the positive equilibrium point in two delay planes are obtained using the method of the stability switching curves.Further,we discuss the direction of the characteristic roots crossing the imaginary axis on the stability switching curves and obtain the stability region of the positive equilibrium point.Then,we give the conditions for existence of Hopf bifurcation.It is shown that when two delays cross the critical curve,the system loses stability,and Hopf bifurcation appears at the positive equilibrium point.Finally,numerical simulation verifies the correctness of theoretical analysis.In the third part,we explore the stability and Hopf bifurcation control of a class of fractional Lotka-Volterra predator-prey system with two delays.Firstly,by choosing the sum of time delay as the bifurcation parameters,we analyze the complex dynamic behaviors of the uncontrolled fractional-order system with two delays.The stability criterion and the condition of existence of Hopf bifurcation are given.Secondly,in order to control Hopf bifurcation,a delayed feedback controller with the delay-dependent coefficient is introduced to control the occurrence of Hopf bifurcation.Using the method of the stability switching curves,the stability of the positive equilibrium point and existence of Hopf bifurcation of the controlled system are further investigated.Finally,the numerical simulation results show that the stability region of the positive equilibrium point expands with the decrease of feedback gain.When the feedback gain is less than zero,the Hopf bifurcation of system is effectively controlled.