Complex dynamics investigations of a mixed Bertrand duopoly game: synchronization and global analysis

In this paper, an economic competition between two firms that want to maximize the weighted-average social welfare and own profits is proposed. This kind of competition is eligible in the complex economic market. The competition is described by a nonlinear discrete dynamic map whose variables are prices of quantities produced by firms. Complex dynamic characteristics such as global dynamic behavior, multi-stability and synchronization are investigated for the competition’s map. The map’s fixed points are calculated, and their stability conditions are discussed. The obtained results show that the map’s Nash point can be destabilized through flip bifurcation. The global bifurcation of the game is analyzed using a 2D map, which corresponds to the model via critical curves. The results show that the map’s synchronization is equivalent to the standard logistic map. Through numerical simulation, some attracting sets of the synchronized map are given. Finally, we calculate the critical curves of the map and prove that it belongs to Z₄- Z₂- Z type, and hence, it is noninvertible.