A model of a rub-impact rotor system under axial thrust is established based on the classic impact theory and analyzed by the Lagrangian dynamics and bifurcation theories. The rubbing condition is modeled using the impact-contact idealization, which consists of normal and tangential forces at the rotor-to-stator contact point. The effects of lateral damping and rotor rotating speed on the rotor system response are investigated in detail. It is demonstrated that the system goes through an extraordinary route to bifurcation. Periodic, quasi-periodic and chaotic motions are found as the system parameters change.