Stability And Hopf Bifurcation Analysis Of Two Kinds Of Nonlinear Delay Differential Equations

There are many problems in different aspects such as economics,ecological economics,ecology,social sciences that can be explained more accurately by delay differential equations.In recent years,more and more researchers use of timedelay differential force range to describe nonlinear models.Based on existing models,this paper proposes two types of nonlinear time-delay models: the price model in timedelay economics and the ecological epidemic model.By using of functional differential equation stability theory,bifurcation theory,central manifold and canonical form theory,the stability of the equilibrium point of these two types of models and the dynamic properties of Hopf bifurcation have been studied in detail.In first chapter,we introduce some research about delay differential equations and Hopf bifurcation,as well as the research status of differential equations and delay phenomena at home and abroad.At the same time,the stability of the equation and the basic concepts related to Hopf bifurcation theory are introduced.The second chapter studies the price model in time-delay economics,discusses the stability of the equilibrium point and the condition of the existence of Hopf branch when the model has time-delay,and verify the theoretical analysis results by simulation experiment with Matlab.In third chapter,we study the ecological infectious disease model with double time delay,discusses the stability of the equilibrium point of the system and the conditions for the existence of Hopf bifurcation,and discusses the Hopf bifurcation direction of the system and the stability of the periodic solution.Then,MATLAB is used to conduct numerical simulation to verify the theoretical results.In this paper,the stability of equilibria of two kinds of time delay models is studied,and numerical simulation experiments and formal proofs are carried out.At the same time,the research methods and contents in this paper are also of reference significance for the further research of high-dimensional time-delay models.