Dynamical analysis of a fractional-order Cournot–Bertrand duopoly model with time delays

This paper investigates a fractional-order Cournot–Bertrand duopoly model with time delays. Employing stability theory for fractional-order delayed dynamical systems and Hopf bifurcation (HB) analysis, we rigorously derive stability criteria for equilibrium points and HB thresholds across six distinct scenarios. Theoretical and numerical results demonstrate that both fractional order and delay length significantly influence the model's dynamical properties. Enterprises should account for memory effects and decision delays in market information to construct monitoring mechanisms, while regulators must track corporate decision-making strategies and market dynamics to establish early-warning systems. Such systems can prevent market imbalance risks through real-time monitoring of key parameters.