Primary resonance and multi-strategy time-delayed feedback control for fractional-order economic fluctuation systems

Fractional-order modeling has become a standard approach to characterize long-memory and hereditary behaviors in macroeconomic and financial fluctuation systems. This paper constructed a fractional-order Duffing-Holmes economic fluctuation model subjected to constant and periodic compound excitations. Constant excitation characterized long-term fiscal inertia and steady investment trends, whereas periodic excitation represented business cycles and seasonal economic shocks. The multiple scale method was adopted to derive the primary resonance amplitude-frequency relation. The influences of fractional order, damping, and stiffness parameters on resonant intensity and system stability were systematically examined. To suppress the excessive oscillation, jump, and hysteresis of economic cycles, four nonlinear time-delayed feedback schemes including displacement, velocity, acceleration, and hybrid velocity-acceleration feedback were proposed. Analytical amplitude-frequency solutions under each control strategy were derived. Numerical results from amplitude-frequency curves and time-domain trajectories demonstrated that all controllers can effectively reduce resonance magnitude and eliminate multivalued bifurcation. In particular, the hybrid feedback scheme exhibited the strongest suppression effect. The findings provided theoretical support and policy reference for countercyclical regulation of fractional-order macroeconomic fluctuation systems.