Global dynamics of a two-stage duopoly game incorporating R&D spillover effects

Within the framework of bounded rationality expectations, a nonlinear dynamical model of a two-stage duopolistic R&D game incorporating spillover effects is constructed. A stability analysis of the proposed model is conducted, and the stable region of the Nash equilibrium is determined by the Jury criterion. Through numerical simulations, the effects of R&D effort adjustment speed and technology spillover rate on system stability are examined, leading to intermittent chaos. Since small perturbations in the initial values will have a huge impact on the long-term evolutionary behavior of the system, it is necessary to grasp the global properties of the system. Subsequently, a global analysis is performed using the composite cell coordinate system method. Three types of crises are revealed when the R&D effort adjustment speed varies: interior, boundary, and merging crises, providing insights into the system’s global properties. It is discovered that changes in the technology spillover rate significantly influence the basin structure, exhibiting strong fractal characteristics. Furthermore, chaotic phenomena are controlled by the time-delay feedback control method, ensuring that the R&D efforts of both firms remain in a stable state. The findings of this study guide firms in making decisions, helping to mitigate losses resulting from irrational strategic choices and preventing market instability.