Torus T2 and its locking, doubling, chaos of a vibro-impact system

A three-degree-of-freedom vibro-impact system is considered. The nonlinear dynamical model and the six-dimensional Poincaré map are established and the dynamical behaviors of the system, including double Neimark–Sacker bifurcation, torus T2 and its routes to chaos, is investigated by numerical simulations. As the control parameters vary, the torus T2 changes into multi-circle torus T1 via one-frequency phase locking on its position, which are divided into longitude circles and latitude circles, and the system keeps quasi-periodic motion. Further the impact motion settles into periodic orbit via two-frequency phase locking, then the system leads eventually to chaos. The second route to chaos shows, by establishing the secondary Poincaré section, that the torus T2 leads to chaos via torus doubling bifurcation and there may exist torus doubling cascade.