The Chaotic Dynamics Analysis Of Two-stage Game With R&D Spillovers

With the increase of market competitiveness,R&D activities have become a significant method for enhancing the firms' core competitiveness and technological strength.A lot of firms choose cost-reducing R&D activities,making them more aggressive in seeking cheaper technologies to improve market competitiveness.However,under the condition of R&D spillovers,the firms' R&D activities are not only influenced by market environment such as R&D methods(cooperation or competition)among firms and uncertainty of firms' innovation period,but also influenced by the ability of firms to obtain decision-making information in market competition.As a result,how to make the best R&D decision in market competition is the main issue in this paper.In this paper,under the condition of “R&D spillovers”,based on the theory of nonlinear dynamic and game theory,a series of two-stage duopoly dynamic game models with R&D spillovers are considered.The long-term complex dynamic behaviors of the system are analyzed and summarized through numerical simulation,and corresponding economic explanations are given.The main research results of this paper are as follows:Firstly,based on the background of “R&D cooperation”,this paper proposes a two-dimensional dynamic Cournot model with homogenous agents in a two-stage game.In the first stage,two firms choose R&D cooperation to optimize their investment in cost-reducing R&D;then,in the second stage,firms compete in a Cournot duopoly game.By using the center manifold theorem and bifurcation theory,the strict mathematical proofs of Flip bifurcation and Neimark-Sacker bifurcation are obtained.The two-parameter bifurcation diagram,two-parameter largest Lyapunov exponent,and one-parameter bifurcation diagram of the system are obtained by numerical simulation to visually demonstrate the phenomenon of various periodic windows in the system from bifurcation to chaos.This paper analyzes how all the relevant parameters such as the adjustment speed of R&D investment,learning ability,and technology spillovers affect the R&D investment and profits of the firms.The global dynamics of the system is studied by analyzing the basin of attraction with coexistence of several attractors under different initial value conditions,and the corresponding economic explanations are given.Secondly,Based on the background of “R&D competition”,this paper proposes a two-dimensional dynamic Bertrand model with heterogeneous agents in a two-stage game,in the first stage two firms choose R&D competition;then,in the second stage,firms choose price competition.The existence and local stability of the equilibrium point of the system are analyzed by using bifurcation theory.During numerical simulation,the R&D investment of the firm is used as the decision variable to analyze the dynamic evolution of system with different parameters.The two-parameter bifurcation diagram,one-parameter bifurcation diagram,and phase diagram are used to show the local dynamic behaviors of the system such as bifurcation,period,and chaos.Finally,the global dynamics of the system are analyzed by using the basin of attraction with coexistence of several attractors.Thirdly,On the basis of the two-stage game model with price competition,the concept of synchronization is introduced,and a two-stage game model is established in which synchronous trajectories can occur when producers have different initial values.The method of critical curves is used to study the global dynamics of a discrete dynamical system with an invariant one-dimensional submanifold.The synchronization trajectory and intermittent phenomena of the system are examined by numerical simulation.Thanks to the fold action of critical curve,the structure of the attractor is depicted,which shows that the global bifurcation of multi-attractor coexistence leads to the occurrence of the fractal of the basin of attraction.Finally,this paper analyzes the impact of R&D investment,learning ability,and technology spillovers on the system in the two-stage game when firms adopt production and price competition in the context of “R&D spillovers”.The analysis and comparison are complemented to find different local and global dynamic phenomena with different models,which provide a theoretical basis and reference for firms on how to make the optimal R&D decision in the market competition.