Hopf bifurcation, tori bifurcation and chaos of a vibratory system impacting a stop

Dynamics of a two-degree-of-freedom vibro-impact system is studied by numerical and analytical method. The maximum displacement of one of the vibratory masses is limited to a threshold value by a rigid wall, which gives rise to a non-linearity in the system. The Hopf bifurcations of periodic motions in non-resonance and weak resonance cases will be studied by establishing the Poincare map of the vibro-impact system with a proportional damping property. The stability of periodic motions with one impact is analyzed. Hopf bifurcation values and intersecting condition of the periodic motion with one impact are determined. The routes of quasi-periodic impacts to chaos; via. tori bifurcation,are obtained by numerical simulations.