The paper considers specialized original computer software for calculating and analyzing parameters of rotor dynamics taking into account various types of magnetic bearings (MBs). The object of research is the dynamics of rotors in magnetic bearings, taking into account interconnected electromagnetic and mechanical controlled processes. The main aim is to simulate and visualize the non-linear phenomena of rotor dynamics with presentation of the results in graphical and textual form, which is adapted for determination of dynamic characteristics. A mathematical description uses an analytical model. It is based on the Lagrange-Maxwell differential equations in the form similar to the Routh equations in mechanics. Implementation of computer modeling and visualization of nonlinear phenomena of dynamics of a rotor in a magnetic field was performed using a computer mathematics system in the form of a nonlinear simulation computational model of dynamics of rotors in passive and active magnetic bearings (SCM-DRMB-N). This simulation model has an ability to construct vibrograms, spectrograms, motion paths, phase trajectories and stroboscopic Poincare sections, as well as three-dimensional spectra, amplitude-frequency characteristics and dependences of amplitudes of sub- and superharmonics on rotation speed for selected rotor points. Verification of SCM-DRMB-N was performed by comparing the results with calculated and experimental data for existing rotor systems.