Double Hopf Bifurcation And Dynamical Analysis Of Time-delayed Fiscal Dynamical Model And Cournot Duopoly Model

In the face of the continuous spread of the new crown pneumonia epidemic and its impact on the domestic and foreign economies,China has adopted an active economic regulation and control policy to maintain positive economic growth at the macro level and help enterprises resume work and production at the micro level.In this context,we study a macroscopic fiscal dynamics model with tax and decisionmaking time delays and a microscopic Cournot duopoly model with production time delays.We have studied the influence of time delays on the dynamics of these two systems,and found the double Hopf bifurcation and complex dynamic behavior.So we can control the range of time delays to keep the system in a stable state,thus on the one hand,we can ensure the proportion of employment rate and wages in national income on the macro level,and on the other hand,we can keep the output balance between the two companies on the micro level.Supported by the National Natural Science Foundation of China(Nos.11972327 and 11372282),in this dissertation the multi-scale method(MMS)is mainly employed to study the double Hopf bifurcation and complex dynamical behaviors of these two models.Firstly,we use DDE-BIFTOOL software package in Matlab to draw Hopf bifurcation critical curves,and their intersection points may be double Hopf bifurcation points.Then,we use DDE-BIFTOOL to draw the eigenvalues distribution of these points,and find out the points that satisfy the double Hopf bifurcation conditions.Secondly,We use the multi-scale method to get the amplitude equations of the local double Hopf bifurcations near the double Hopf bifurcation points.Through the normal analysis of these amplitude equations,we get the unfolding and classification of these double Hopf bifurcation points.Finally,the numerical verification with Win PP is executed,and the numerical simulations results are in good agreement with the theoretical analysis results,which shows that the multi-scale method is effective and accurate.The results of this dissertation are helpful for decision makers to choose the appropriate time delay range to maintain the proportion of employment rate and wage in national income and the balance of enterprise production,so as to ensure the stability of the fiscal system and maximize the interests of enterprises.The features and innovations of this dissertation are the following:(1)By using multi-scale method and normal analysis method,we classify and unfold the double Hopf bifurcation points of the fiscal dynamical model with the tax and decision-making time delays,which is improved from Wolfstetter classical growth cycle model.And we found three types of double Hopf bifurcation and the complex dynamical behaviors including quasi-periodic and phase-locked solutions.(2)In the Cournot duopoly model with production delays,we found one kind of double-Hopf bifurcation type and two routes to chaos i.e.,Periodic Doubling and Intermittency to chaos.(3)When generating bifurcation diagram by 4)9)(8(6′0)section for the Cournot duopoly with production delays,the selected 4)9)(8(6′0)section is not a straight line or plane in the past,but a parabola that is more in line with the trajectory trend.