<p>Advertising and promotion strategies are crucial marketing tools for increasing sales. In this paper, we primarily investigate the mathematical mechanisms for increasing sales by inducing a single-parameter sales promotion strategy into a differential advertising model. Based on the continuous model, we derive the discrete governing system for sales. By leveraging the existence, stability, and bifurcation and chaotic behavior of fixed points of the discrete system, we have elucidated the dynamic behavior of sales in the continuous model. The specific parameter ranges for the existence of a T-period solution and its stability conditions are given. Furthermore, we perform a flip bifurcation analysis of the positive fixed point. This analysis helps us to obtain the existence and stability conditions for nT-period solutions. Interestingly, for the same model, when we take different parameter values, flip bifurcation and inverse flip bifurcation can coexist. The bifurcation provides a route to chaos. In the simulations, we find that in some situations, there exists a pathway for the system to enter into chaos from a stable state through flip bifurcation, and then enter into a stable state through inverse flip bifurcation, while in other situations, there exists no such pathway. We propose an effective control strategy that serves to suppress flip bifurcation and promote inverse flip bifurcation to eliminate chaos. These findings have significant theoretical implications and practical applications in relevant markets.</p>