Global Analysis of a Duopoly Game with Privatization of State-Owned Enterprises

This paper constructs a nonlinear dynamical model of a mixed ownership market duopoly game for environmental tax collection within the framework of bounded rationality expectations. A stability analysis of the proposed model is conducted, and a Nash equilibrium stability region is estimated using the Jury criterion. The effects of the degree of privatization and environmental tax rate on the stability are investigated through numerical simulations. Subsequently, the comprehensive global analysis is carried out using the composite cell coordinate system method. Two types of crisis phenomena, namely, interior and boundary crises, have been identified. The former is caused by the collision between chaotic or periodic attractors and interior chaotic saddle within the basin of attraction, and the latter is caused by the collision between chaotic attractor and periodic saddle on the basin boundary. Meanwhile, the global dynamic behavior of two private enterprises engaged in synchronized output adjustment is investigated. When the output adjustment speed varies, the structure of the basin of attraction undergoes a transformation as changes are made to the number of coexisting attractors. The sudden disappearance of one attractor is due to the collision of the chaotic attractor with the saddle, on the basin boundary, merging into a larger saddle and resulting in a boundary crisis. The emergence of numerous “holes” can also be observed. It is found that the presence of periodic or chaotic saddles enriches the dynamical phenomena. Finally, the chaotic phenomenon occurring in the mixed ownership market is controlled using the time-delay feedback control method, stabilizing the system in a stable state, which implies a steady-state of total market output. The study of this model can provide theoretical guidance for the decision-making process of enterprises, helping minimize profit losses while stabilizing the market.