# Regular and Chaotic Vibrations of a Nonlinear
## 1 Introduction  
The complete elimination of transverse vibration of rotors is practically impossible, primarily due to residual static and dynamic unbalancing [1; 2; 3; 4; 5]. In the case of rotors operating at supercritical speeds, there is a possibility that some elements will clash during transient processes. Such impacts are possible, e.g., between the rotor and stator (i.e., limiter of motion), the case which has been studied in references [6; 7; 8]. As a result of these impact forces, it is possible under some conditions that periodic, almost periodic or chaotic vibrations get excited in the system [9; 10; 11]. Undesirable rotor impacts and the friction forces occurring between the elements in rotating machinery can lead to the destruction of system components, which negatively affects the efficiency of the device's operation [12; 13; 14; 15; 16]. Many studies describe the impact of collisions between rotating elements of various systems on the type of induced vibrations, including chaotic vibrations. For example, in the work [17], the influence of the parameters of a vibration-isolated hand grinder on the nature of the induced vibrations is examined. In the literature dealing with impacts in systems including a rotating element, a simplified model of a Jeffcott rotor is usually used, which neglects the gyroscopic effects and the influence of internal friction. The analysis of the simplified model is capable of explaining a number of interesting phenomena brought about by impacts between the elements of the system. Using this model, [PERSON] et al. [18; 19] and [PERSON], [PERSON] and [PERSON] [20] have shown that for the static deflection, which is large compared to the clearance between the rotor and the limiter of motion or for a big amplitude of inertial excitation, subharmonic or chaotic vibrations set in. [PERSON] and [PERSON], in their work [21], developed a new algorithm (NNMs) for rotor balancing. Using a Jeffcott rotor, authors demonstrated that the NNMs method is significantly more effective than the linear modal method. [PERSON] et al. [22] analysed the modified Jeffcott rotor, taking into account torsional vibrations and rotor-stator contact. Using numerical research, the authors determined the vibration areas duringwhich the rotor and stator come into contact. A new method to suppress the resonant vibrations of the non-linear Jeffcott-rotor system has been presented by [PERSON] et al. in their works [23; 24]. In order to suppress the vibrations in the paper [23], a radial Proportional Derivative (PD-) controller was used along with the eight-pole electro-magnetic actuator, while in the paper [24] they employed a proportional integral resonant controller (PIRC). In references [18; 19], the influence of the excitation frequency and the stiffness of the limiter of motion is discussed. [PERSON] and [PERSON] [25] analysed the influence of the gravity force, [PERSON] et al. [26] studied the effect of the torsional rigidity of the shaft, whereas [PERSON] and [PERSON] [27] paid attention to the mode shapes of excited vibrations. [PERSON] et al. [3] also analysed the influence of a rotor-to-stator contact on the lateral-torsional coupled vibrations. In some works, attempts were made to study systems using analytical methods [28; 29], e.g., the Krylov-Bogolyubov or Van der Pol methods, which allow for a more precise explanation of certain physical phenomena but are less accurate in the case of strongly nonlinear systems. For example, [PERSON] et al. [28] studied the rotor-stator system for an elastically mounted limiter. They compared the results obtained by the elliptic functions method with the results obtained by the numerical [PERSON]'s 4-order method and the approximate analytical Van der Pol method. In [30], [PERSON] et al. proposed a new phase-shift empirical mode decomposition integration (PSEMDI) method to fault diagnose an unbalanced rotor subjected to a friction load. [PERSON] [31] analyses the vibrations of rotors supported on magnetic bearings, accounting for the impact on the retainer bearings.