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

In this paper, an economic competition between two firms thatwant to maximize theweightedaverage 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, multistability 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 themodel via critical curves. The results showthat themap'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 Z(4) - Z(2) - Z(0) type, and hence, it is noninvertible.

Automated summary

This paper proposes an economic competition between two firms aiming to maximize the weighted average social welfare and own profits. The competition is described by a nonlinear discrete dynamic map, and its complex dynamic characteristics such as global dynamic behavior, multistability, and synchronization are investigated. The results show that the Nash equilibrium point of the competition can be destabilized through flip bifurcation.