Bifurcation and chaos control in a heterogeneous Cournot-Bertrand duopoly game model

In this study, we investigate a Cournot-Bertrand duopoly model with heterogeneous players, where the first player uses bounded rationality and the second player is naive. We examine the existence and stability of all fixed points in the model and use center manifold and bifurcation theory to analyze the occurrence and direction of period-doubling and Neimark-Sacker bifurcations at the positive fixed point. To control bifurcation and chaos, feedback control and hybrid control methods are applied. Numerical examples confirm our theoretical results and reveal the model's complex dynamics. Notably, we show that increasing the speed adjustment parameter alpha in the first player's bounded rationality mechanism can drive the system from stable equilibrium through periodic oscillations to chaos, highlighting the parameter's critical role in market complexity and instability.