Stability Research Of Two Kinds Of Fractional-order Computer Virus Propagation Models With Delay

The non-locality of fractional-order differential equation provide a good theoretical support for the material and process with ”memory” and ”heredity”,so it is often used to describe real problems in various fields of social science and natural science.Hopf bifurcation theory is a classical tool to analysize the stability of dynamical system and has been widely used to study the dynamical characteristics of system.With the vigorous development of fractional calculus,the Hopf bifurcation problem and its control of fractional systems have become a research hotspot in recent years.In addition,the time delay phenomenon of many complex systems cannot be ignored.Even a small time delay may affect the stability of the entire system.The analysis of the combination of time delay and fractional derivative is more in line with the actual state of the system.Therefore,it is of great theoretical and practical significance to study the dynamic characteristics of fractional delay systems.In this thesis,the dynamic behavior of two kinds of fractional-order delay computer virus propagation models are studied.The specific research contents are as follows:?.A delay computer virus SIR model with different fractional orders is studied::Firstly,based on the renewable matrix method,relationship between the coexistence equilibrium point and the basic reproductive number are studied.Regarding the delay as a parameter,the conditions of emergence of Hopf bifurcation near the coexistence equilibrium point are analyzed by linearization method and Laplace transform method.Moreover,threshold formula of delay is deduced.Necessary numerical simulations are made to verify the validity of the theoretical result.We show the influence of fractional-order variation on the stability region of the system.Secondly,a periodic pulse time-delay feedback controller is introduced to the model,and the semi-analytical and semi-numerical method are used to prove the equivalence between the controlled system and its linearized average system in a certain sense.Based on the theoretical analysis,the critical value of Hopf bifurcation in the controlled system is derived by the linearization method and the fractional Laplace transform,and the range of the stability domain and the variation of the stability domain with the controller parameters are obtained.The results show that the Hopf bifurcation of the system can be effectively delayed by adjusting the gain parameter value.Finally,the correctness and feasibility of the theoretical analysis of the controlled system are verified by appropriate numerical simulation,and the influence of different control gain parameters on the system stability domain is compared.?.A fractional-order delay computer virus SLBQRS model is studied:Firstly,the delay is regarded as the bifurcation parameter to analyze the stability of the system at the coexistence equilibrium point and the conditions for the occurrence of Hopf bifurcation.Secondly,the homotopy method is used to discuss the condition in detail that the characteristic equation of the linearized system of the model has pure imaginary roots.The rationality and feasibility of the theoretical analysis results are verified by numerical simulation using the Adama-Bashforth-Moulton method.This paper discuss the stability of the fractional-order delay computer virus propagation models,which is better to understand the dynamic characteristics of computer virus propagation,and the results provide a certain scientific theoretical basis for the field of network security.Further research on the periodic pulse feedback control of the model will help to intervene and eliminate computer viruses,and open up a new idea for the well maintenance of network system security.