Research On Hopf Bifurcation And Stability For Two Classes Of Fractional Order Differential Equation Models With Delay

The fractional-order differential equation model is more accurate than integer order in the description of related problems in the fields of biology,physics and so on.With the development of computer technology,it gradually improves the disadvantage of complicated calculation.It has attracted widespread attention in the past half a century.Many researchers tried to use fractional-order instead of integer order to describe some phenomena,and obtained rich results in various research fields.In addition,in the field of network communication,electricity,biology,etc.,delay makes an important impact on the stability and evolution of the system.Therefore,studying the fractional-order model with delay broadens the research field partly,which has certain theoretical value and practical significance.Hopf bifurcations and stabilities for two classes of fractional-order differential equation models with delay are investigated in this thesis through theoretical analysis and numerical simulation.Initially,for the following fractional-order Bhalekar-Gejji(BG)chaotic system with delay:(?)?we have mainly done three aspects of work.Firstly,the rationality of using a linear approximation system to study the stability of the proposed system from the perspective of stability are proved by using a semi-analytical method.Secondly,the Hopf bifurcation and stability of the proposed system are analyzed.Using the linearization method and Laplace transform,one obtains the characteristic equation corresponding to the linearized system.By discussing the root and transversality condition of the characteristic equation,it is theoretically analyzed that the relationship between the delay and Hopf bifurcation of the system.Finally,a linear feedback controller with delay is introduced to study the influence of feedback gain changes on the system stability domain.Furthermore,the results of theoretical analysis are verified by numerical simulation.Next,by improving the computer virus model studied in [46],and introducing the delay and fractional-order,the following models are obtained:(?)?For the above model,the linearization method and Laplace transform are used to obtain the corresponding characteristic equations of the system.The relationship between the delay and the stability of the equilibrium point is studied by discussing the roots of the characteristic equations and the transversality condition,and the formula for calculating the critical point of delay is derived when Hopf bifurcation occurs.Moreover,the appropriate system parameters are selected for numerical simulation to verify the rationality of the theoretical analysis.Some new results are obtained in this thesis though studying fractional-order BG systems with delay and fractional-order computer virus models with delay.Those provide some theoretical support for the study of Hopf bifurcation and stability of fractional-order dynamic system with delay.On the other hand,that results help to promote the research scope and methods of these two classes of models.