Stability,multi-stability And Instability Of R&D Competition Between Nonlinear Duopoly

Nowadays,the interest competition among enterprises is becoming much fiercer.They not only compete in product quality,but also in function,shape,service and other aspects.Technological innovation enables enterprises to gain greater advantages in the process of competition.Technological innovation ability not only determines the scientific and technological strength of a country,but also is an important driving force to promote economic development.As the main body of national economy,enterprises also need to carry out innovation and Research and Development activities(R&D in short).Based on the nonlinear inverse demand function and the linear inverse demand function,two competition models have been established in this research,one is the Cournot duopoly game model with R&D competition and nonlinear inverse demand function and another is the Stackelberg duopoly game model with R&D competition and linear inverse demand function.The stability of equilibrium point of the two models is studied by nonlinear theory,and the complex dynamics behavior of the two models under different parameters is numerically simulated.In addition,in order to better simulate the R&D competition process of enterprises in the real market,the author establishes a Counot model of duopoly R&D competition with constraint conditions,and carries out theoretical analysis and numerical research on this model.The main contents of the study are as follows:First,a Cournot model of duopoly R&D competition with a nonlinear inverse demand function is established,and the stability type of the system boundary equilibrium point through a theoretical form are analyzed.Since the analytical solution of the Nash equilibrium point cannot be obtained,but through its characteristic equation system of discriminant can prove that Neimark-Sacker bifurcation cannot occur at Nash equilibrium.In addition,the dynamic behavior of the system under different parameters is simulated.It is found that the system will enter into chaos through flip bifurcation and the Neimark-Sacker bifurcation which occurs in 2-period.This result can also be proved by the bifurcation diagram of single parameter.In addition,a special fractal structure,that is period-doubling bifurcation and period-halving bifurcation sequence,is also found through the two-parameter bifurcation diagram: period 7→ period 14→ period 28→ chaos → period 28→ period 14→ period 7.Second,a Stackelberg model of duopoly R&D competition is established.Through theoretical analysis,it is found that the stability of boundary equilibrium point is different under different parameters.The Jury criterion is used to obtain that the system will have Neimark-Sacker points at the Nash equilibrium point.The results of theoretical analysis are verified by numerical simulation.In addition,numerical simulations are used to study the boundary curve of stability region of the system.It is found that the increase of overflow parameter will reduce the range of stability region.Once the Nash equilibrium loses its stability,the system will pass through the Neimark-Sack bifurcation and flip bifurcation into chaos.Finally,the multi-stability phenomenon of the system is studied by the basin of attraction and critical curves.The results indicate that the type and number of coexisting attractors change with the change of parameters,and the internal structure of the corresponding basin also changes.Finally,a constraint condition is added on the basis of Cournot model studied in Chapter3,establishes a duopoly R&D competition model with constraint condition.First,the stability of its equilibrium point is analyzed.Second,the system’s R&D investment situation under the long-term evolution are studied,which shows that the R&D investment of the enterprise will tend to be balanced or one party will withdraw from the market under the long-term evolution.Finally,the two-parameter bifurcation diagram and the single-parameter bifurcation diagram of the system are drawn by numerical simulation,and compared with the results of numerical analysis in Chapter 3,the results indicate that the dynamics of the non-smooth dynamic system in the two-parameter space are more abundant and diversified.In addition,by observing the single-parameter bifurcation diagram,it can be found that the system also has some bifurcation phenomena peculiar to non-smooth systems,such as periodic bifurcation phenomenon.