In this paper, a numerical method is presented to determine the periodic response of hydrodynamic bearing-rotor system. The observed state information of the system is used to solve inversely the Jacobian matrix, and to trace the periodic response with the change of the control parameter. Jacobian matrix obtained is used to calculate the Floquet multiplier, so the stability of the periodic response can be determined by Floquet theory. The proposed method is applied to a rotor system with the elliptical bearing supports to solve the periodic response and determine its nonlinear stability. Validity of this method is illustrated by comparing numerical results with the traditional method.