Complex dynamics analysis for a two-stage Cournot duopoly game of semi-collusion in production

In this paper, a two-stage oligopoly game of semi-collusion in production is analyzed and expounded, where at first stage all firms compete in R&D and at second stage all firms coordinate the production activities in order to make their joint profit maximized. Not only the local stability of equilibriums, but also the existence, stability and direction of flip bifurcation of the discrete nonlinear model are investigated by using the normal form method and the center manifold theory. Then, the validity of the theoretical analysis is justified through numerical simulation. We find that the model we built can exhibit very complex dynamical behaviors, but it cannot undergo the Neimark-Sacker bifurcation. Also the coexistence of attractors is found through numerical simulation, and their basins of attraction are simulated. At last of this paper, the chaotic motion of the proposed model is controlled by delayed feedback control method.