Local Hopf Bifurcation And Rank-one Chaos Of Two Classes Of Delay Differential Equations

Retarded differential equations refer the differential equations with time delays,which can be used to describe development systems that depend on both the current state and the past state.Due to the full consideration of the influence of history on the current state,they have many important applications in mechanics,biology,neural networks,confidential communications and economics.With the development of structural instability systems in recent years,the study of chaos of delayed system has become an important issue.In applying the rank one chaos theory to some specific differential equations with time-delay,we find that the asymptotically stable periodic solution has a rank-one chaotic attractor under the periodic external force.Since there are Hopf bifurcation periodic solutions in many time-delay systems,it is an important task to study the rank one chaotic attractors of time-delay systems.The first chapter introduces the development history,research status,main research methods and achievements of Hopf bifurcation theory and the rank one chaotic attractor of differential equations with time-delay,and introduces the basic knowledge of Hopf bifurcation theory and rank-one chaos theory for delay differential equations.In the second chapter,the stability and periodic solution of a prey-predator model with delay and square root term is studied using Hopf bifurcation theory.Taking the? as the bifurcation parameter,the stability of the equilibrium change and a periodic solution generate when the delay passes a certain critical point.Then using the Hassard method,obtain the conditions for determining the bifurcation direction and stability of the periodic solution.Numerical simulations and theoretical analysis are consistent.In the third chapter,we use the center manifold theorem and normal form method to study the Yang-Chen system with time delay,and derive the condition of the supercritical Hopf bifurcation of the system.After adding periodic excitation term to a Yang-Chen system with delay when it has the supercritical Hopf bifurcation,rank one chaotic attractors can be observed.Numerical simulations and theoretical analysis are consistent.