Abstract
In this paper, a dynamic two-stage Cournot duopoly game with R&D efforts is built. Then, the local stability of the equilibrium points are discussed, and the stability condition of the Nash equilibrium point is also deduced through Jury criterion. The complex dynamical behaviors of the built model are investigated by numerical simulations. We found that the unique route to chaos is flip bifurcation, and the increase of adjusting speed will cause the system to lose stability and produce more complex dynamic behavior. In addition, we also found the phenomenon of multistability in the given model. Several kinds of coexistence of attractors are shown. In particular, we found that boundary attractors can coexist with internal attractors, which also aggravates the complexity of the system. At last, the chaotic state in the built system has been successfully controlled.
Highlights
In recent years, the market competition mode among firms are no longer subjected to price or quantity, but more towards soft power, including technical level, talent resources, and corporate culture
Luckraz [6] established a Cournot duopoly game model with isoelastic demand function and considered the case without Research and Development (R&D) spillovers. e work of Chatterjee [7] showed that the firms in the market undertake R&D activity and studied that R&D incentive may depend on the information structures they
A new way of development for firms will generate when the economy develops to a certain stage, and there is only one way for firms to further develop the economy and they will face a dilemma in the choice of the new and old way
Summary
The market competition mode among firms are no longer subjected to price or quantity, but more towards soft power, including technical level, talent resources, and corporate culture. Zhou et al [3] studied the types of bifurcation, intermittent chaos, and coexistences in a dynamical two-stage Cournot duopoly game with R&D spillover effect. Askar et al [15] investigated the properties in a dynamic Cournot duopoly game model with nonlinear demand function. Zhang et al [20] proposed a two-stage Cournot duopoly game of semicollusion in production and analyzed its complex dynamics, such as local stabilities and existence. Fanti et al [36] provided local and global dynamic characteristics in a duopoly game model with price-setting competition and market share delegation. The influence of firm’s adjustment speed on the development of firms is studied through combining a two-stage dynamic game with the method of numerical simulation and the properties of nonlinear dynamics.
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