A Neimark–Sacker Bifurcation Analysis of a Decision Delay Duopoly Model and Its Control Using Improved Impulsive Control

This paper investigates the nonlinear dynamics of a decision delay duopoly model that characterizes the competitive landscape within China’s Low-Temperature Milk market. These dynamics include equilibrium points, period-doubling bifurcations, complex torus-like motions, and closed invariant curves induced by Neimark–Sacker bifurcation. Then the Neimark–Sacker bifurcation phenomenon is analyzed. Next, to further stabilize the nonlinear dynamics, an improved impulsive control strategy is introduced. Sufficient conditions for the asymptotic stability of the controlled system are derived using Lyapunov stability theory. Numerical simulations demonstrate that, under appropriate impulsive control, an originally divergent system can converge to equilibrium, effectively stabilizing market profits. This research provides a theoretical reference for oligopolistic firms to optimize their marketing rhythm and for policymakers to maintain market stability.