Global dynamics in the lateral oscillation model of pedestrian walking on a vibrating surface

Abstract

This paper studies the lateral oscillations of pedestrian walking on a vibrating ground with a known motion, which can be modeled by a hybrid Rayleigh–van der Pol–Duffing oscillator with quintic nonlinearity and dual parametric excitations. The focus of the work is on the global dynamics of the oscillator, including chaos and subharmonic bifurcations. It reveals that the system can be subdivided into three categories in the undisturbed case: single well, double hump, and triple well. Specifically, the exact solutions for homoclinic, heteroclinic and subharmonic orbits in triple-well case are obtained analytically. The Melnikov method is employed to investigate the chaotic phenomena resulting from different orbits. Compared to a single self-excited oscillator, this hybrid oscillator exhibits higher sensitivity to external excitation and strong nonlinear terms. By adjusting the system parameters, the peak value of the chaos threshold can be controlled to avoid the occurrence of chaos. Based on the subharmonic Melnikov method, the subharmonic bifurcations of the system are examined and the extreme case is discussed. Some nonlinear phenomena are discovered. The system only exhibits chaotic behavior when there is a strong resonance, that is, when there is an integer-order subharmonic bifurcation. Furthermore, we find the pathways to chaos though subharmonic bifurcations encompass two distinct mechanisms: odd and even finite bifurcation sequences. The numerical simulation serves to verify the findings of the preceding analysis, while simultaneously elucidating a number of additional dynamic phenomena, including multi-stable state motion, bursting oscillations, and the coexistence of attractors.