Making use of the numerical simulation method, the phenomenon of vibrational resonance and electrical activity behavior of a fractional-order FitzHugh–Nagumo neuron system excited by two-frequency periodic signals are investigated. Based on the definition and properties of the Caputo fractional derivative, the fractional L1 algorithm is applied to numerically simulate the phenomenon of vibrational resonance in the neuron system. Compared with the integer-order neuron model, the fractional-order neuron model can relax the requirement for the amplitude of the high-frequency signal and induce the phenomenon of vibrational resonance by selecting the appropriate fractional exponent. By introducing the time-delay feedback, it can be found that the vibrational resonance will occur with periods in the fractional-order neuron system, i.e., the amplitude of the low-frequency response periodically changes with the time-delay feedback. The weak low-frequency signal in the system can be significantly enhanced by selecting the appropriate time-delay parameter and the fractional exponent. In addition, the original integer-order model is extended to the fractional-order model, and the neuron system will exhibit rich dynamical behaviors, which provide a broader understanding of the neuron system.

Magnetorheological (MR) dampers have emerged as prominent semi-active actuators in the field of structural vibration control, owing to their millisecond-level responsiveness and continuously adjustable damping characteristics. However, their inherent strong nonlinearity, dynamic hysteretic behavior, and path dependency pose significant challenges for precise dynamic modeling. To address these issues, this paper introduces a non-parametric modeling approach based on the Fourier Neural Operator (FNO). By leveraging operator learning, the proposed method establishes an infinite-dimensional function-space mapping from excitation inputs to damping force outputs. Unlike traditional neural network models, FNO parameterizes the integral kernel in the frequency domain. This enables the model to capture the underlying continuous dynamics of the physical system while maintaining discretization invariance and high computational efficiency. Validated on a comprehensive public experimental dataset, the proposed FNO model achieves a Root Mean Square Error (RMSE) of 10.39 N and a coefficient of determination of 0.9998 on the test set, significantly outperforming baseline models such as LSTM and CNN-LSTM. Furthermore, the study reveals that the proposed model exhibits distinct advantages in characterizing the highly nonlinear roll-off region of force-velocity curves, handling complex multi-frequency excitations, and demonstrating superior extrapolation capabilities across unseen operating conditions.