Research On The Evolution Of R&D Competition Of Duopoly Under Dynamical Adjustment Mechanism

Nonlinear game models,which depict a kind of Cournot game behavior that duopoly com-petes by two stages of R&D,are researched.The discussion of system parameters in dynamic adjustment mechanism,including the speeds of adjustment,the parameter of R&D spillover effect induced by factors such as the technical personal's fluxion and knowledge sharing,the cost parameter of technical innovation,and so forth,reveal the influence of evolution law on R&D competition.By using the Game Theory,Dynamics Economics Theory and Nonlinear Dynamics Theory,all models are researched by theoretical and numerical simulation,in which the local and global properties are in-depth analyzed in two-parameter space.What's more,the complex evolution phenomena,including bifurcation analysis,attractor,fractal structure,multi-stability and so on,are studied.1.Research on complexity of the nonlinear game model based on two stages of R&D,the study of speeds of adjustment reflects evolution behaviors of the complex model.On the one hand,the local stability of equilibriums and the type of bifurcation are illustrated using the Local Bifurcation theory.On the other hand,the dynamics behavior of the complex model are characterized by numerical simulation.With the speeds of adjustment increases,the route of Flip bifurcation,which is the unique route that the system from Nash equilibrium enters chaos,is discussed in two-parameter space.Not only that,the change of speeds of adjustment is accompanied by evolution of strange attractor and the change of multi-stability,manifested on the change of number of coexisting,the critical bifurcation of attractor and the global bifurcation of basin of attraction.What's more,special coexistence phenomenon,which manifests on the coexistence of boundary attractor and attractor with hierarchical structures,is illustrated by multi-stability,and the largest Lyapunov exponent corresponding to the evolution of attractor displays oscillatory wave through increased speeds of adjustment.2.Evaluation behaviors of another model,in which the R&D spillover effect is taken into account,are researched.Theoretically,the local stability of equilibriums,the type of bifurca-tion and the direction of Flip bifurcation,are illustrated using the Local Bifurcation theory.On the other hand,some luxuriant and complex dynamics evolution phenomena are characterized by two-parameter space.It is found that there are two routes with the speeds of adjustment increases,which is Flip bifurcation and Neimark-Sacker at quadratic mapping respectively.In addition,the increase of speeds of adjustment,on the one hand,leads to the formation of "fre-quency and homoclinic cycle,on the other hand,brings about the intermittent chaos phenomena,including PM-III intermittent chaos and intermittent chaos induced by crisis,and multi-stability induced by non-standard bifurcation.Additionally,another parameter named as spillover effect also causes the change of multi-stability,represented as the number of coexisting attractors and the structure of basin of attraction.3.A two-dimensional symmetric noninvertible map is obtained by the transformation of the model with R&D spillover effect,afterwards,invariant one-dimension submanifolds of the map is discussed by the theoretical analysis.By numerical simulation,the synchronous phe-nomenon,which is a counterintuitive but interesting phenomenon,is discovered.While the transverse Lyapunov exponents less than zero,the synchronous phenomenon occurs.However,the further increase of speed of adjustment not only destroys the synchronous phenomenon,and leads to the occurrence of Blowout bifurcation,and eventually forms riddle basin.But also gives rise to global bifurcation of basin of attraction,which causes by the contact critical line LC and the boundary of basin of attraction,and finally forms "hole" appeared in basin of attraction.However,the increase of cost parameters of technical innovation leads to critical bifurcation of attractor,which causes by the contact critical line LC and the boundary of attractor,and finally forms "ghost" scattered in the original basin of attraction.More than this,the increase of cost parameter of technical innovation not only results in the formation of fractal structure of"con-tinent" and "island" in basin of attraction,but also causes the coexistence phenomenon,which is about Nash equilibrium and the evolution of homoclinic cycle.