Global Analysis, Multi-stability and Synchronization in a Competition Model of Public Enterprises with Consumer Surplus

We research global dynamic behaviors, multi-stability and synchronization in a dynamic duopoly Cournotian model with consumer surplus on the basis of nonlinear demand function and bounded rationality. This game generalizes the traditional dynamic duopoly Cournot game. The aim of enterprises is to optimize their own profits and the social welfare. This game is characterized by discrete difference equations incorporated in the competition model's optimization problem. The stability of equilibrium points and complicated dynamical phenomena are studied, such as flip bifurcation for unique Nash equilibrium point. The global bifurcation of non-invertible two-dimensional map is analyzed through critical curves, which is an important method to further study global properties. Multi-stability is observed due to the presence of multiple complex attractors. An inspection of the basins of attraction is provided for the situation of several attractors coexisting. Since this game has symmetry, it can be stated that a one-dimensional invariant sub-manifold is the diagonal of the system by analyzing the nonlinear map x(t+1)=μx(t)(1−x(t)). The properties of this nonlinear map is completely different from the logistic map although both of them have similar form. Along the invariant diagonal, the synchronization phenomenon of the game is discussed in the end.