Quantum-coupled dual-channel pricing dynamics with reference-price memory

Abstract

This paper studies nonlinear dynamics and chaos control in a dual-channel pricing game with quantum-inspired price coupling and reference-price memory. A manufacturer sells through a direct channel and supplies a retailer under a wholesale contract. Observable prices are generated by a strategic coupling mapping with intensity parameter \(\gamma \), while consumers form channel-specific reference prices through exponential smoothing. The static Stackelberg model determines the contractual wholesale benchmark, whereas, conditional on this fixed wholesale price, multiplicative bounded-rationality learning generates a four-dimensional nonlinear discrete-time system with an independently computed dynamic fixed point. We analyze local stability through the full Jacobian and reveal global phenomena including period-doubling routes to chaos, multi-band attractors, and irregular bounded oscillations. Wholesale-price sensitivity analysis shows that the qualitative flip-bifurcation mechanism is preserved around the contractual benchmark. We further compare local state-feedback and delayed-feedback controllers. Both achieve convergence to the dynamic fixed point in the chaotic baseline regime, as confirmed by tracking errors that remain below the prescribed threshold, decaying control inputs, and controlled spectral radii below one.