Dynamical Analysis And Chaos Control Of Three Oligopoly Game Models With Time Delays And Nonlinear Cost Functions

With the rapid development of economy and the deepening of global integration,the characteristics of the supply and demand of the market have changed significantly.In order to meet the needs of the customers,the manufacturing industry constantly innovating and producing more new products.With the emergence of a variety of substitutes,there is a great impact on the original product market.The oligopoly of the industry is gradually appearing,and the phenomenon that several enterprises monopolize an industry can be seen everywhere.However,most of the current researches are aimed at duopoly market.For three oligopoly and more oligopoly market,there are few studies.Therefore,it is necessary to analyze the three oligarch market.At present,the most of the current production game research is to introduce one-phase delay in the game model.Although the state of the next time depends not only on the current state and the state at the first time(one-phase delay),but also on the state at the earlier time(the two period of time delays)in the real game environment,the effect of the two time delays on the stability of the system is not much.It is time to analyze the dynamic behavior of the game model with two time delay.Considering the fact that the cost of enterprises in the real world does not always increase linearly with the increase of output,so based on theory,system stability theory and chaos theory,this paper introduces time-delay and nonlinear cost function into the three oligarch output game model,then deduces and analyzes the model,and completes the numerical simulation analysis with Matlab.The main contents of the paper are as follows.(1)The output game model of the three oligopoly market is built.In order to make the model more practical,we integrate two factors such as time lag,nonlinear cost function and bounded rationality into the model.On this basis,we analyze the complex dynamic behavior of the model,and deduce the Nash equilibrium point and the stability range of the system by using mathematical knowledge.We also study the influence of output adjustment,delayed decision coefficient and product differentiation degree on output,profit and system equilibrium by numerical simulation.The results of the study are as follows.First,the oversize of the outputadjustment will accelerate the system into chaos,and can not achieve the steady growth of the profits of the enterprises in the system.Secondly,the increase of the delay parameter will increase the stability area of the Nash equilibrium point of the system,but the random change will also make the system unstable.Last,the change of product substitution coefficient will not only affect the stability area of the system,but also affect the Nash equilibrium point of the system,the higher the substitution coefficient is,the lower the profit of the enterprise in the oligarchy,which indicates that the enterprise should pay attention to the differentiated products.(2)Chaos is not all harmful.A reasonable use of chaos can increase the profit of the enterprise to a certain extent,but we should pay more attention to the way of controlling chaos.This paper will build a three oligopoly output game model with time delay and nonlinear cost function as the driving system,and then construct a similar response system.Based on Lyapunov stability theory,we add a non autonomous controller to the response system by using the Active Control method,which can makes the drive system and response system synchronization.The numerical simulation shows that this method can effectively achieve the chaos synchronization between the driving system and the response system,thus realizing the purpose of chaos control.