Dynamical Analysis And Chaotic Control For Bertrand Pricing Game Models With Time Delays

In recent years,consumers can have all price information of the similar products just at home by Internet and e-commerce.On the premise of ensuring product quality,the price advantages become more obvious in the market competition.Previous studies are mostly about output game,rather than the pricing game.Therefore,the dynamic characteristics of the model and its chaotic control are studied on the basis of Bertrand game model,stability theory and chaos theory in this thesis.Then the factors which will influence the company's pricing strategy and take effective measures to reduce and inhibit the chaos when we are faced with it are analyzed.The main content of this dissertation is as follows.(1)On the basis of Bertrand game models,product differentiation is introduced to avoid "the Bertrand paradox" in this dissertation.The time-delay factor which is produced in the process of enterprise pricing is also considered to build a Bertrand pricing game model.Based on the delayed bounded rationality and the stability method of the chaotic discrete system,the dynamic characteristics of the model and its stability are studied.Then through the bifurcation diagram,chaos graphics and the variations of various corporate profits,the stability of equilibrium is analyzed.Finally the impact of price adjustment parameters,product differentiation level and time delays on the system is investigated,which can provide the effective reference for the enterprise to better understand and to use time delays.(2)Based on the bounded rationality and the stability method of the chaotic discrete system,a master-slave Bertrand duopoly price game model with time delay is addressed.The dynamic characteristics of the model are studied.Through the bifurcation diagram and the variations of various corporate profits,the stability of equilibrium points is analyzed,and the impact of delays on the system is investigated which can provide the effective reference for the enterprise to better understand and to use time delays.Meanwhile,the simulation results based on MATLAB illustrates that the increase of price adjustment speed(strength)will generate two bifurcation phenomenon of enterprise price evolution,eventually produce the chaotic behaviors.The delayed system has the same Nash equilibrium point with the non-delay system,and the delay can increase the local stability region.(3)Chaos not only affects pricing decisions,but also enables the corporate profits become volatility and harder to be predicted.In order to inhibit and eliminate the effects of chaos,this paper uses time-delayed feedback control method and straight-line stabilization method to control chaos in two models.Then it gives several measures to help enterprises control the chaos:controlling the speed of adjustment of product prices,lowering the alternative rate of the product,reducing the marginal cost of the product and selecting the appropriate time-delay weight,which can help system back to a steady growth in corporate profits.