Dynamic Evolution And Chaos Control Of Duopoly Game Model With Corporate Social Responsibility

This paper studies the dynamic evolution and chaos control of several duopoly Cournot models with corporate social responsibility,in which corporate social responsibility is interpreted as consumer surplus or social welfare.The effects of adjustment speed,the weight of consumer surplus and the weight of social welfare on the dynamic behavior of the system are analyzed.Meanwhile,the chaotic phenomena of the system are controlled by means of control methods.The main contents and results of this paper are as follows:1.Based on the assumption of bounded rationality,a duopoly model that ragards the objective function of the firms as the weighted sum of consumer surplus and firm's own profits and takes output as the decision variable is established.The 2-D bifurcation diagram is used to analyze the path of the system to chaos.It is concluded that the area with clutter on the two parameters plane is caused by the coexistence of multiple attractors in the system.The phenomenon is explained by the attraction basin.Through the symmetry of the system when two firms have the same economic environment,it is proved that the diagonal line is the one-dimensional invariant manifold of the system.The existence of Milnor attractor in the system is obtained by the transverse stability,which verifies the occurrence of synchronization.2.We focus on discussing the different weights of consumer surplus between two firms.The bifurcation at the Nash equilibrium is analyzed by bifurcation theory and central manifold theorem.The conditions of Flip bifurcation and Neimark-Sacker bifurcation are given.By numerical simulation,the evolution of chaotic attractor generated by Flip bifurcation and limit cycle generated by Neimark-Sacker bifurcation is described by phase diagram.According to the Delayed Feedback Control,the chaotic phenomena of the system is controlled.The output,profit and objective function of the firms before and after the control are compared.It is found that after adding chaos control,the output,profit and objective function of the two firms tend to be stable from the sharp change before.3.Duopoly model considering social welfare is studied.From 2-D bifurcation diagrams,we can find that the dynamic phenomena caused by the change of adjustment speed are more abundant than that caused by the change of the weight of social welfare.Through the critical curve of the system,the topological structure of attraction basin is analyzed,and the evolution process of the ‘holes' of attraction basin are simulated.The critical curve divides the phase space into regions with different first-order preimages.The global bifurcation occurs when the critical curve contacts the boundary of the attraction basin.The sequence of holes will be generated in the attraction basin with the iteration of the system.Finally,the chaos of the system is controlled by using state feedback and parameter adjustment control methods.Afteradding control,the occurrence of bifurcation is delayed and the stability region of the system is enlarged.4.A duopoly model is studied in which one firm considers consumer surplus and the other considers social welfare.The stability region and bifurcation curve of Nash equilibrium are given by numerical simulation.The bifurcation of the system is analyzed by bifurcation diagram,largest Lyapunov exponential diagram and phase diagram.The coexistence of multiple attractors is proved by using multi-initial bifurcation diagrams.The numerical simulation shows another global bifurcation of the system,that is,the bifurcation occurring when the attractor contacts the boundary of the attraction basin.It is found that the global bifurcation will destroy the attractor and its basin.