## 1 Introduction
### The Fundamental Theoretical Considerations  
Although part of the discussed literature directly concerns stability issues, the fundamentals of stability are given below.  
The physical explanation for practical problems with rail vehicle stability is the self-exciting vibrations that appear in vehicle-track systems. This phenomenon has been recognised for a long time [25]. Some vehicle parts undergo self-exciting vibrations under certain motion conditions (parameter values). The wheelsets oscillate laterally and around the vertical axis (yawing). Wheelange-rail contact, noise, and a higher risk of derailment accompany this. Although a vehicle's motion along the track is usually possible under such conditions, self-exciting vibrations cause obvious adverse phenomena in rail vehicle operation in straight track (ST) and circular curve (CC) sections. The self-exciting vibrations observed in real vehicle-track systems [25] can be identified using vehicle-track models with bifurcation of the solutions [26; 27]. Bifurcation analysis is an effective method for testing modelled objects [26; 27; 28; 29] and is thus applied also in the presented research. Numerical simulation studies on vehicle-track systems enable us to observe whether the model solutions are stable. Stable stationary solutions (with constant values of the model's observed coordinates on STs and CCs) describe the normal exploitation conditions for a real vehicle. This means that in the real system, each disturbance, for example, disturbance caused by track irregularities, is effectively damped, while the system tends toward equilibrium. A properly designed vehicle should ensure running stability in the expected range of exploitation parameter changes, such as velocity (0 to \(v_{\text{max}}\)), load (empty or partially to fully loaded), load asymmetry, track shape (CC, transition curve (TC), ST), track maintenance, etc. Stable stationary solutions for a modelled vehicle-track system on STs and CCs are expected under typical operation conditions. In contrast, due to the constant change in the curve radius and superelevation on TCs, stationary solutions for TCs are neither possible nor expected.  
The model's stable periodic solutions (limit cycles) are attributed to self-exciting vibrations in the real system. Due to them, the lateral displacements in the wheelset change periodically during the main vehicle motion (along the track). Calling such solutions \"stable periodic\" is justified because they are limited and observable and can last any length of time. On the other hand, stable periodic solutions represent the highest capacity of the modelled system in terms of stable behaviour and also bring other valuable information. The key parameter in stability analysis is nonlinear critical velocity \(v_{n}\), as well as in the present research. The value of \(v_{n}\) separates the range of vehicle velocity \(v\) for which stable stationary solutions exist (\(v<v_{n}\)) from the range for which stable periodic solutions may appear (\(v\geq v_{n}\)). The value of \(v_{n}\) is the minimum vehicle velocity \(v\) starting from which stable periodic solutions can exist. For a real object, this means self-exciting vibrations may occur, and a vibrational system state is reached. Further increase in the system output (for example, with a rise in \(v\)) leads to an increase in the self-exciting vibration amplitude. Finally, the amplitude values are no longer constant, which causes unbounded growth of the solutions (a loss of stability), which could lead to an accident due to the derailment in the real system. The research on periodic solutions for vehicle models under increased system input (vehicle velocity) in an above-critical system state is interesting. It is conducted to ascertain how much the system parameters (velocity) can be exceeded and the solutions still kept stable (periodic or stationary). A wide range of system velocities between the values corresponding to the appearance of periodic solutions and a loss of stability is always desirable for a real vehicle.