Study On A Dual Oligopoly Bertrand Price Game Model With Quadratic Cost Functions And Time Delays

With the rapid development of Internet technology in the world economy,e-commerce has been widely used and popularized;consumers can know all the price information of the same kind of products.Therefore,under the premise of guaranteeing the product quality,the price advantage competition is becoming more and more obvious and intense in the market competition.In the rapid development of social economy and frequent merger and reorganization in various industries,more and more markets have entered the era of oligopoly.In the past,most researches on oligopoly competition in the market were carried out around the production game,and the research on price game basically unified the cost function into a linear one.There is less research on price game with nonlinear cost function.However,in the real economy the cost does not always increase linearly with the increase in output.Therefore,based on dual oligopoly Bertrand model,system stability theory and chaos theory,this paper studies the dynamic characteristics and chaos control of enterprises in the dual oligopoly market competition with quadratic cost functions.It also analyzes the factors that affect the decision of enterprise price,and how to restrain and eliminate the negative effect of these factors.This paper mainly completes the following three aspects of work.(1)This paper considers the product difference factors,so as to avoid "Bertrand paradox",and consider the enterprise produces in the process of pricing decision time delay factor,built a quadratic cost functions and delays the Bertrand price game model of dual oligopoly.On this basis,the paper makes a complex analysis of the model and calculates the stable domain of Nash equilibrium point.(2)Through the numerical simulation of Matlab,the bifurcation diagram,chaos diagram and profit function of each enterprise were analyzed,and the stability of the system at the Nash equilibrium point was explored.Studied business adjustment parameters,product differentiation,price level and the weight coefficients of delay on the system stability and the influence of corporate profits,for enterprises to better understand and use delay to provide valuable theoretical reference and basis.The numerical simulation results based on Matlab show that the increase of the enterpriseprice adjustment speed will cause the bifurcation of the price evolution of the two enterprises,and finally the chaotic behavior.The dynamic system with delay has the same Nash equilibrium as the non-delay system,and the delay increases the local stability of the system.(3)The chaotic state not only affects the stability of the nonlinear dynamical system,but also affects the price decision of the enterprise,which causes the enterprise profit to fluctuate violently and becomes unpredictable.Therefore,in order to synchronize and eliminate the influence of chaotic state,this paper firstly adopts the active-control method to synchronize the chaotic state of the reaction system with different initial values and the driving system.Then the chaotic state of the system is controlled by the delay feedback control method.And proposes several can help enterprises to implement feedback control measures,such as control product price adjustment speed,reduce the product replacement rate,choosing the appropriate delay weight coefficient,and so on,these measures are chaotic systems can help quickly restored to meet the steady state has been a steady rise in profits.