Simulation and Control of a Duopoly Model

This paper studies the dynamics of a duopoly model with bounded rationality and nonlinear demand function. Based on the stability theorem and Jurys criterions, we prove that the model has two unstable boundary fixed points and a local stable Nash equilibrium. Then we depict the stability region of Nash equilibrium, and investigate the effects of output adjustment speed on the players profit respectively. Theoretical analysis and simulations show that higher output adjustment speed can result in chaotic variation of outputs, and that the Nash equilibrium is the optimal result of duopoly game. To improve the profitability of each player and achieve the optimal game result, we put forth a new scheme combined with the time-delayed feedback control and the limiter control to stabilize the output to Nash equilibrium. Finally, the numerical simulation is adopted to verify the effectiveness and feasibility of the above control scheme.