Two-parameter Bifurcation Characteristics Of Dynamic Competition Among Nonlinear Oligopolies Under Advertising Externalities

Since the reform and opening up,the advertising industry has developed rapidly in China,which has promoted the rapid development of China's economy.The advertisement has a crucial influence on promoting the enterprises to win an advantage in market competition.The nonlinear dynamic models of the oligopoly competition are established in this research with considering the factors that bounded rationality and the externality of advertisement.This paper mainly investigate the influences of the parameters,like the speed of adjustment,the externality of advertisement and the advertising share,on the dynamical behaviors of the system.Firstly,basing on the assumption of bounded rationality,a duopoly game model with the advertising spillover effects(namely,the externality of advertisement)is built by simultaneously considering that these two manufacturers produce complementary products.By calculating,it is obtained that the system has three boundary equilibrium points and a unique Nash equilibrium point,and stability conditions of equilibrium points are also analyzed.The dynamic evolution processes of the system with the changes of the parameters(such as spillover effect,adjustment speed and marginal cost)are discussed by the help of numerical simulations.There are two distinct paths for the system entering into chaos,that is,flip bifurcation and Neimark-Sacker bifurcation.Especially,by simulating the dynamic behaviors of the system in two-parameter space,it can be found that the regularity of periodic number of Arnold's tongues is in line with the generating rule of the Stern-Brocot tree.In addition,the phenomenon of multiple groups of attractors can be also found in the two-parameter space.Through simulating the dynamical evolution of the basins of attraction,it can be concluded that there is global bifurcation,like contact bifurcation,will occur when the border of the attractors contact with the boundary of the basin of attraction.A nonlinear duopoly dynamic model with the externality of advertisement is established,where these two manufacturers produce homogeneous products.The existence of equilibrium points are verified by using the analytical method.Furthermore,the local stabilities of equilibrium points are analyzed by using the stability theory,and then their stability conditions are given.Through numerical simulation,the complex dynamical behaviors of the system at the equilibrium points after losing their stability is discussed,it can be observed that the system will enter into chaos via flip bifurcation with increasing the values of the speed of adjustment.The parameters that the speed of adjustment and the externality of advertisement have no influence on the bifurcating type of the system,while they will affect the dynamical behaviors of the system.A special case is worth noticing that when all parameters of two manufacturers are the same,their long-term dynamical behaviors will tend to converge though there is a great gap between their initial states.That is to say,the decision of these two manufactures will have synchronization.A two-stage duopoly dynamical game model with semi-collusion is built.These two manufacturers who produce homogeneous products cooperate with each other during the advertising stage,while they compete in production state.This system has four equilibrium points,and their existence conditions can be easily obtained.In addition,the local stability of the equilibrium points is analyzed.Through numerical simulation,we can observe that(1)Although the advertising sharing rates of two enterprises are different,the system loses stability in almost the same way;(2)The advertising cooperation between the manufacturers can improve their sales and joint profits,and also promote their joint development;(3)There are two typical intermittent chaos appearing in the system,that is,the PM-III intermittent chaos and the crisis-induced intermittent chaos.The occurrence of intermittent chaos implies that the system has weak self-regulation ability.However,this kind of self-regulation capacity can't maintain long-term stability of the system.