Multistable responses and Wada basin boundaries in the fractional-order viscoelastic column

To address the coexistence of steady-state responses in the viscoelastic column subjected to multifrequency excitation, this study for the first time introduces the generalized fractional-order constitutive model into the dynamic modeling of viscoelastic columns subjected to multi-frequency excitation, and establishes the governing equation that is more consistent with the actual mechanical behavior of materials. By performing dimensional reduction through the single mode approximation and the Galerkin method, and subsequently applying the method of multiple scales, the steady-state solutions of the system are derived. The existence conditions of these steady-states are then determined by the Routh-Hurwitz criterion. Through analyses of steady-state response curves, basins of attraction, time series, and phase diagram, a novel multi-stability phenomenon characterized by the long-term coexistence of four stable periodic attractors in the system is revealed. Furthermore, combining the weighted truncated Shannon entropy based on Wada index and the grid method, the partial Wada basin boundaries are quantitatively characterized. The modulation effects of damping and external forcing amplitude on the multistable structures are also examined. Overall, the results break through the limitations of traditional integer-order constitutive relations in the dynamic analysis of viscoelastic materials, and provide theoretical insight for the dynamical analysis of fractional-order viscoelastic columns and offer guidance for engineering vibration control.