## 1 Introduction
### The Literature Survey  
The papers in which problems on railway vehicles' stability, including determination of their nonlinear critical velocity \(v_{n}\), are discussed by the present authors are [5; 6; 7]. Broad reference to the corresponding literature by other authors is provided in these publications, too. The newest examples of papers discussing the stability and dynamics of rail vehicles, also with their motion on CCs and TCs taken into account, are [8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24]. In [8], its authors study the stability of high-speed trains through a focus on bogie behaviour. A novel method for stability determination, based on root loci methods, is successfully tested. A sensitivity analysis was performed on the selected vehicle parameters, too. In [9], one can see the interesting differences in stability issues for rail and road vehicles. The important influence of the vehicle roll angle on stability is considered. The results on the stability region are presented as a three-dimensional stability domain. Paper [10] describes a novel measuring system that monitors vehicles' hunting motion. Based on the monitoring results, the system can predict the lateral and yaw displacements of the wheelset and the contact relationships for the wheel-rail pair in real time. The accuracy and efficiency of the whole system were validated by comparing the predicted results with simulated and experimental results. Paper [11] is truly interesting and of high intellectual value. It reveals and verifies experimentally (on a roller rig) the possible existence of parametric vibrations in a wheelset-track system. They appear parallel to self-exciting vibrations (hunting motion) and are caused by wheel load fluctuations. Systems particularly prone to this type of resonance are those with large tread angles, which also decrease the critical (hunting) velocity, favouring the simultaneous existence of self-exciting and parametric vibrations. Publication [12] represents a study of the chaos in a mechanical impacting system. The novelty of this paper lies in the extension of previous results for a one-degree-of-freedom (DoF) system to a two-DoFs system. The existence of phenomena such as narrow-band chaos, finger-shaped attractors, etc., was demonstrated numerically and then experimentally verified. In [13], the authors propose a new method for stability determination based on data instead of equations. With this method, they managed to determine the long-term statistics of the chaotic state, the covariant Lyapunov vector, the Lyapunov spectrum, the finite-time Lyapunov exponents, and the angles between the stable, neutral, and unstable splitting of the tangent space [13]. Publication [14] presents and uses one more nonlinear wheelset model to perform bifurcation analysis. The nonlinearity in the wheel-rail contact is of primary interest. The analysis comprises comparisons between the results for linearised and nonlinear models. The results for different field-measured wheel profiles are also compared. Finally, the conclusion is that a greater suspension stiffness increases the stability under wheel wear conditions. Publication [15] is to some extent similar to [14]. A nonlinear wheelset model undergoes bifurcation analysis. The nonlinear equivalent conicity in wheel-rail contact is of interest. The results for linear and nonlinear approaches to equivalent conicity and contact forces are compared. The authors show the possible coexistence of stable and unstable limit cycles and expand on the consequences of Chinese high-speed trains' observed unfavourable behaviour. In paper [16], the authors  
Figure 1: Y25 bogie side view. study the influence of the curved track parameters, such as their radii, superelevation, and TC and CC lengths, on vehicle-track interactions in the case of a side-frame cross-braced two-axle railway bogie.