Multiple attractors and critical parameters and how to find them numerically: the right,the wrong and the gambling way

In recent years, several authors have proposed ‘easier numerical methods’ to find the critical speed in railway dynamical problems. Actually, the methods do function in some cases, but in most cases it is really a gamble. In this article, the methods are discussed and the pros and contras are commented upon. I also address the questions when a linearisation is allowed and the curious fact that the hunting motion is more robust than the ideal stationary-state motion on the track. Concepts such as ‘multiple attractors’, ‘subcritical and supercritical bifurcations’, ‘permitted linearisation’, ‘the danger of running at supercritical speeds’ and ‘chaotic motion’ are addressed.