Complex Dynamic Analysis For Two-stage Cournot Duopoly Game With Semi-collusion

Accompanied by the continuous development and progress of science and technology,nonlinear dynamics also obtained swift and violent development,chaos and bifurcation phenomena are the main forms of complex dynamics in nonlinear dynamical systems.At present,chaos and bifurcation can be found in discrete systems and continuous systems.However,proving mathematical bifurcation and the existence of chaos of chaotic systems are all technically difficult problems.The dynamic model established in the field of economics play an important role on the study of chaos and bifurcation.This paper studies two-stage duopoly game under semi-collusion models.Two two-stage duopoly game models are established about semi-collusion in production and R&D.The existence and local stability of equilibrium of the discrete-time models are represented.It has been strictly proved that the Flip bifurcation at the Nash equilibrium.The systems cannot experience Neimark-Sacker bifurcation is proven.Local dynamics of the model are analyzed by numerical simulation and the chaos is controlled.Finally,the synchronous model of two-stage duopoly game model is studied.The model of irreversibility,invariant set,global dynamic behavior and symmetry are broken under the basin of attraction is researched.In this paper,the main research has the following four parts.In Chapter 1,the preliminaries of dynamic systems are presented,including the research background and significance of the thesis.This paper briefly reviews the development history of Cournot model in game theory and the application of nonlinear dynamics in economic field.The development of research and development,dynamic model,two-stage model and synchronization in economic model has been briefly described.Specifically,the basic stability theory,bifurcation and chaos theory and central manifold theorem are introduced.The second chapter established a two-stage Cournot duopoly game of semi-collusion in production.In the first stage,all companies compete in R&D.In the second stage,all enterprises should coordinate production activities to maximize the joint profits.Using the norm form theory and the center manifold theorem,the local stability of the nonlinear discrete model is studied,and the local stability of four equilibrium points and the parameter conditions for the existence of these equilibriums are investigated.We found that the model could show very complex dynamics,but the system cannot embody Neimark-Sacker bifurcation.Moreover our results are illustrated by the numerical simulations.We have presented the two-dimensional bifurcation diagram,single parameters bifurcation diagram,the maximal Lyapunov exponent,phase portraits,the attractor coexistence curve and the basins of attraction.Finally,the chaotic motion of the model is controlled by the delayfeedback control method.The third chapter established a two-stage Cournot duopoly game of semi-collusion in R&D.In the first stage,two enterprises collaborate in R&D and coordinate R&D investment to reduce cost of production and maximize profits.In the second stage,the enterprise carries out Cournot competition and selects its own output to maximize its profit.Next,the local stability of the nonlinear discrete model is studied,and the existence,stability and direction of the model are discussed.Through numerical simulation two-dimensional bifurcation diagram,single parameters bifurcation diagram,the maximal Lyapunov exponent,phase portraits,initial sensitivity about the different adjustment rate of R&D and R&D spillover coefficient,qualitative analysis the dynamic characteristics of the system,the bifurcation of different single parameters under different initial values of the same parameters is given,which shows that the attractors coexist.The intermittent chaos of the system is studied,and the two groups of attractors in the phase space are caused by intermittent chaos instead of attractors.The third chapter established a synchronization model of a two-stage Cournot duopoly game of semi-collusion in production,the final model has been described by a two-dimensional non-reversible discrete time dynamic system,showing that the synchronization dynamics occur on the diagonal.When considering lateral stability,intermittent phenomena is point out.In addition,the transition of kinetics of from simple to complex,and critical line technology is used to describe the structure of attractor,explains the global bifurcation phenomenon caused in attracting basin are discussed.In this paper,the coexistence of attractors under different parameters and the attracting basin are studied.Finally gives a model of break under synchronous symmetry and basins of attraction under different R&D spillover coefficient,illustrates the symmetry broken cases attractor coexistence still occurs,but the symmetry of the diagonal is lost.