Stability And Bifurcation Analysis Of Two Kinds Of Time-Delay Dynamical Systems

Hopf bifurcation theory has a very important theoretical value in the study of dynamic bifurcation behavior of systems,and it has made important contributions to the analysis of complex dynamics and the study of periodic oscillation.There are many kinds of time delay in most complex systems,among which distributed time delay and discrete time delay are more common.The addition of time delay changes the dynamic characteristics of the original system,such as stability,bifurcation,chaos and so on.Based on bifurcation theory,this paper studies the effects of fractional order and reaction-diffusion on the dynamic characteristics of the system.The specific work is as follows:1.The fractional enterprise competition model with time delay is established.In this model,time delay is used as the bifurcation parameter,and the influence of time delay on the dynamics of the system is analyzed and discussed by using mathematical knowledge and system stability theory.The local stability of the enterprise competition model and the existence of Hopf bifurcation are obtained.When the time delay is less than the bifurcation threshold of the system,the system is locally asymptotically stable at the equilibrium point.When the time delay is greater than the bifurcation threshold of the system,the system will become unstable,and Hopf bifurcation will occur when the time delay is equal to the bifurcation threshold.Furthermore,the influence of fractional order on the stability of the system is discussed,and the relationship between fractional order and bifurcation threshold is obtained.2.The enterprise competition model with PD controller is studied,its dynamic characteristics are explored,and the conditions required for system stability are analyzed by taking time delay as the bifurcation parameter.In order to further explore the control effect of PD controller,this paper fixed one parameter in the controller,adjusted another parameter,observed its influence on the system stability,and compared with the enterprise competition model without control,obtained the stability of the controller to improve the system.3.A time-delay rumor propagation model containing reaction-diffusion term is established,and the saturation function term is introduced to study its dynamic characteristics.The influence of diffusion term on the stability of the system without time delay is obtained.Under the appropriate diffusion coefficient,the system changes from stable to unstable,and reaction-diffusion phenomenon occurs.Secondly,the local stability of the system with time delay and the existence of Hopf bifurcation are discussed.When the time delay is less than the bifurcation threshold of the system,the system is locally asymptotically stable at the equilibrium point.When the time delay is greater than the bifurcation threshold of the system,the system will become unstable,and Hopf bifurcation will occur when the time delay is equal to the bifurcation threshold.In the range of system stability,the restraining effect of saturation function term on the propagator and the influence on the system stability are discussed.In this paper,the characteristic equation of linearized system is obtained by using nonlinear partial differential equation theory,and the condition of Hopf bifurcation is given and proved.Secondly,the stability and bifurcation direction of Hopf bifurcation are also analyzed.The theoretical criteria of bifurcation direction,bifurcation stability and periodic solution period of bifurcation can be obtained by using the gauge type theory of partial functional differential equations and the central manifold theorem.When the system parameters are determined,the related properties of Hopf bifurcation can be obtained by theoretical criteria.