Hopf Bifurcation And Rank-One Chaos Of Two Delay Differential Equations With Holling Type Functional Response Functions

In nature,time delay is unavoidable.Time-delay phenomena are widely found in chemistry,mechanics,medicine,biology,economics and control science.For dynamic systems with time delay,delay differential equations are often used to describe them.Hopf bifurcation is an important bifurcation phenomenon in the theory of delay differential equations,the occurrence of Hopf bifurcation may be accompanied by chaos phenomenon.According to the Hopf bifurcation theory and the rank-one chaos theory of delay differential equations,the existence,bifurcation direction and stability of Hopf bifurcation in two kinds of delay differential equations are studied and the rank-one chaos in disturbed system is discussed.The work of this paper is as follows:In the first part,we study a two-time delay SIR infectious disease model with Holling-IV incidence rate and Holling-? treatment rate.The following four cases are discussed in this paper:(1)?1=?2=0;(2)?1=0,?2>0;(3)?1=?2=?;(4)?1>0,?2?(0,?20).The local stability of the positive equilibrium and the existence of the Hopf bifurcation in the four cases are obtained by using the Hopf bifurcation theory.The direction and stability of the Hopf bifurcation in the fourth case are studied by using the Hassard's method,and the correctness of the theory is verified by numerical simulation.When supercritical Hopf bifurcation occurs in the system,three periodic perturbation terms are added to the model.By using the rank-one chaos theory of time-delay differential equation,the conditions for the existence of rank-one strange attractor in the perturbation system are derived.The observable rank-one strange attractor is found in numerical simulation.In the second part,we study a five dimensional plant-pest-natural enemy model with stage structure is studied.The interaction between pests and plants,and between natural enemies and pests are Holling-? and Holling-? type functional response function,respec-tively.In this paper,we discuss the case where neither time delay is zero,got the existence conditions of Hopf bifurcation and the nature of the bifurcation in ?1?(0,?10),?2>0 case.When the asymptotically stable periodic solution appears in the system,factors such as annual weather and seasonal change are taken into account,which will affect the growth of plants and the predation effect of pests and natural enemies.In this paper,a periodic perturbation is added to plants,mature pests and mature natural enemies respectively.According to the rank-one chaos theory of time-delay differential equations,the existence conditions of rank-one strange attractors in the perturbation system are obtained.Finally,the numerical simulation results agree with the theoretical results.