<p>The maglev train is affected by cogging effects, which, under time-delay control, cause complex nonlinear behaviors. To study the impact of cogging and time-delay control on system stability and vibration characteristics, track unevenness is modeled as harmonic excitation. Nonlinear displacement–velocity dual-time-delay feedback control is considered, and the corresponding maglev dynamics equations are established. The system’s relative displacement amplitude–frequency response is solved using the method of multiple scales, while the perturbation method identifies unstable regions. The effects of time delay and feedback parameters on the system’s response are analyzed. Singular theory is applied to study static bifurcations, and dynamic bifurcations of frequency and control parameters are examined. A comparison of chaotic conditions under fractional-order and integer-order time-delay control is also made. The results show that time delay, position, and velocity feedback parameters affect the system’s resonance frequencies, resonance peaks, and unstable regions. Time delay can lead to closed-loop frequency islands under certain conditions. The system’s transition set divides the bifurcation parameter space into three regions, each with different bifurcation topologies. The presence of time delay significantly impacts the system’s stability, with velocity feedback reducing stability and position feedback improving it. Under certain conditions, fractional-order time-delay control is more effective in reducing system amplitude than integer-order control.</p>