This paper deals with a non-linear mathematical model for simulation and analysis of a dynamic behaviour of a rotor-bearing system. The model is used for the simulation of plane motion of a centre of rotor’s cross-section which makes the model convenient for the analysis of vibrations generated in a rolling bearing as well as for the analysis of accuracy of the revolution of rotor supported on a rolling bearing. The model takes into account the following quantities: internal radial clearance, external radial load and unbalanced load. Differential equations of a motion are derived using Lagrange’s equations. The contacts between the balls and the rings are considered to be non-linear with a stiffness derived by the Hertzian theory of an elastic contact. The proposed model enables direct determination of local contact deformations which significantly reduces needed computations and CPU time. The results obtained by the model are used to reconstruct phase-space trajectories and Poincaré maps and to calculate the largest Lyapunov exponent in order to establish the stability of rotor-bearing system motion. A computer program is developed based on the mathematical model for the simulation and analysis of a dynamic behaviour of a rotor-bearing system.