Dynamic analysis of a generalized fractional‐order model under a variable‐order integral–derivative controller with delayed feedback

Abstract This paper address the bifurcation control problem for generalized fractional‐order systems with time delays. Unlike traditional proportional‐derivative(PD) control, the proposed controller combines integral and differential terms with time‐delay feedback characteristics, utilizing cumulative features and time‐delay signals to adjust system dynamics. It provides greater flexibility in dynamic response tuning, making the controller more adaptable to complex systems. First, a variable‐order integral–derivative controller with delayed feedback (IDCF) was designed, where the order and time delay of the controller are consistent with the system. Subsequently, using time delay as the bifurcation parameter, the stability conditions and Hopf bifurcation characteristics of generalized fractional‐order systems under uncontrolled and controlled conditions were analyzed using Hopf bifurcation theory, and fully evaluated the control parameter range of controlled systems. Finally, the dynamic characteristics of congestion control systems and small‐world networks under the controller were studied, and the results showed that adjusting the controller parameters and can effectively control the bifurcation critical point of the system and provide sensitivity analysis of the system to parameters.