Local Hopf Bifurcation And Rank One Chaos In Two Classes Of Epidemic Models With Time Delay

Delay differential equation(delay functional differential equation)is a kind of differential equation with time delay.It is mainly used to describe the system that is related to both the current state and the past state.Therefore,delay differential equations have a wide range of application prospects in solving practical problems.As bifurcation and chaos phenomena widely exist in nature,bifurcation and rank one chaos of delay differential equations have attracted many researchers' attention and achieved fruitful results.In this paper,we discuss the local Hopf bifurcation and rank one singular attractors of two kinds of delay differential equations related to infectious diseases.Firstly,we introduce the research background and current situation of delay differential equations,Hopf bifurcation theory,rank one singular attractors,the background of this paper and the research content of this paper.Secondly,The Hopf bifurcation and rank one strange attractors of a delayed epidemic model with nonlinear incidence and Holling type cure rate and a delayed epidemic model with Nonmonotone and saturated incidence are analyzed.In both models,the time delay is reduced ? As bifurcation parameters,the stability of the two Model at the positive equilibrium and the conditions for the existence of Hopf bifurcation are analyzed.The bifurcation properties of Hopf bifurcation are discussed by using Hassard method,and the condition of rank one strange attractor in perturbed system is calculated.Finally,the correctness of the theoretical analysis is proved by numerical simulation.At the end of the paper,it summarizes the work and research prospects of this paper.