Dynamics analysis of a two-degree-of-freedom fractional-order maglev system under slotting effect and time-delay control

To improve the operational stability of high-speed maglev trains under complex multi-source excitations, this study aims to clarify the nonlinear dynamic mechanisms induced by controller time delay, slotting disturbance, and suspension-guidance coupling-phenomena that remain insufficiently understood in existing research. A two-degree-of-freedom maglev model is established incorporating suspension–guidance coupling and the fractional viscoelasticity of air springs. A time-delay PD controller is introduced, and the slotting effect is modeled as a harmonic track excitation. The primary resonance response is derived using the multi-scale and harmonic balance methods, and the stability conditions are obtained via Lyapunov's first method and the Routh-Hurwitz criterion. A systematic parametric investigation is conducted to evaluate how time delay, structural nonlinearity, fractional order, and control gains modulate the amplitude-frequency characteristics, stability boundaries, and bifurcation evolution. Time-domain simulations further reveal the transitions among periodic, quasiperiodic, and chaotic responses. The results show that these parameters jointly reshape the resonance peak, shift the instability region, and generate closed frequency islands and multistable behaviors. Time delay is identified as the dominant factor that amplifies nonlinear effects and induces chaotic instability. Increasing the delay leads to frequent alternations between periodic and chaotic states as system parameters vary, highlighting the heightened complexity of the maglev system's dynamics.