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In this paper, we present a novel approach to quantify regular or chaotic dynamics of either smooth or non-smooth dynamical systems. The introduced method is applied to trace regular and chaotic stick-slip and slip-slip dynamics. Stick-slip and slip-slip periodic and chaotic trajectories are analyzed (for the investigated parameters, a stick-slip dynamics dominates). Advantages of the proposed numerical technique are given.

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2-DOF nonlinear dynamics of the rotor supported by the magneto-hydrodynamic bearing is investigated using the perturbation analysis. The two modes corresponding to the vertical and horizontal vibrations of the rotor are coupled. The nonresonant case and the various resonant cases (with and without an internal resonance) are considered. The frequency-response curves are obtained. When the amplitude of the external harmonic excitation is near to one of the natural frequency of the vibrations and the system experiences conditions of an internal resonance, a saturation phenomenon occurs. When the amplitude of the external excitation increases, after some critical value the energy pumping between various submotions of the rotor occurs for each mode. Further, it was shown, that in the case of rigid magnetic materials, hysteresis may be a cause of chaotic vibrations of the rotor. Chaotic regions and the amplitude level contours of the rotor vibrations have been obtained in various control parameter planes.

In the present work hysteresis is simulated by means of internal variables. The analytical models of different types of hysteresis loops allow the reproduction of major and minor loops and provide a high degree of correspondence with experimental data. In models of this type adding an external periodic excitation or increasing the number of dimensions can lead to the occurrence of chaotic behaviour. Using an effective algorithm based on numerical analysis of the wandering trajectories [1-7], the evolution of the chaotic behaviour regions of oscillators with hysteresis is presented in various parametric planes. The substantial influence of a hysteretic dissipation value on the form and location of these regions, as well as the restraining and generating effects of hysteretic dissipation on the occurrence of chaos, are ascertained. Conditions for pinched hysteresis are defined. Furthermore, autonomous coupled hysteretic oscillators under sliding friction are investigated. Conditions for the occurrence of chaotic behaviour in a two-degree-of-freedom (two-DOF) hysteretic system are found in the plane of maximal static friction forces of both oscillators versus belt velocity.