Bifurcation Analysis Of A Business Cycle Model With Time Delay

The study of the dynamic properties of time-delay systems has always been a field with rich practical background and wide application.Whether in ecology,infectious disease,chemistry,physics,or economics,many processes can be characterized and analyzed by time-delay systems.In fact,in the field of differential equations and dynamic systems,the study of this subject has always been paid attention to by researchers.The bifurcation analysis is an important way to study the dynamics of the system areas of differential equations,topology bifurcation refers to the type of system through the bifurcation value(threshold)when parameters change,specifically the phenomenon of the bifurcation system for certain parameters Research on the impact of whether there will be generated,can be more profound,more widely applied practice guidance that can effectively combine theory and practice.The main contents of this paper is to study a class of business cycle model,for example,investigate the effect of time delay in the class model,as follows:In the first chapter,Introduction.It mainly introduces the background and significance of the research,and lists several common business cycle models.And the main problems and innovations of the model studied in this paper are briefly explained,and the corresponding chapters are arranged.The second chapter is about preliminary knowledge.Some basic theorems and definitions and research methods used are elaborated.In the third chapter,Based on the Kaldor-Kalecki economic cycle model with time-delay feedback control,the stability of the model and the generation of Hopf branches under the spatial diffusion effect are discussed.Whether it is the spatial diffusion effect or the influence of time-delay feedback,it may affect the stability of the equilibrium point of the original system(so that the equilibrium point becomes stable from unstable state,and vice versa).This chapter uses branch theory to make a detailed analysis of the Kaldor-Kalecki economic cycle model.As a branching parameter,the Kaldor-Kalecki economic cycle model with spatial diffusion effect is taken as the research object.By analyzing the distribution of the corresponding characteristic equation roots of the linear part of the system at the equilibrium point,the stability of the system and the Hopf branch are obtained.Sufficient conditions.Combined with the central manifold theorem(dimension reduction)and the application of normalization theory(downward parameter)to obtain the relevant parameter values to prove the correlation properties of Hopf bifurcation,finally the theoretical analysis results are verified by numerical simulation in this part,Specifically,Numerical simulations are made for systems without diffusion(3.4.1)and systems with diffusion terms(3.4.3),verify the stability range before theory push and generation of the correctness of Hopf bifurcation region,andfrom the corresponding time history diagram,phase diagram and space wave map can be seen,the size of time delay and the introduction of diffusion coefficient have really changed the dynamic characteristics of the system.Whether it is the spatial diffusion effect or the time-delay feedback effect,it is possible to affect the stability of the original system equilibrium point(so that the equilibrium point becomes stable by the unstable state,and vice versa).In the fourth chapter,The main study is the case of the Simple-Zero and Double-Zero branches of the Kaldorian model with time-delay in the case of fixed exchange rates.Firstly,the discussion of the distribution of eigenvalues is carried out.Secondly,the central manifold theorem and the application of normalization theory are respectively used to reduce the two branches and draw corresponding conclusions.In the fifth chapter,Make necessary summary and prospect to the conclusion of the third and fourth chapter.