There are many mathematical models of drilling systems Despite, huge efforts in constructing models that would allow for precise analysis, drilling systems, still experience breakdowns. Due to complexity of systems, engineers mostly use numerical analysis, which may lead to unreliable results. Nowadays, advances in computer engineering allow for simulations of complex dynamical systems in order to obtain information on the behavior of their trajectories. However, this simple approach based on construction of trajectories using numerical integration of differential equations describing dynamical systems turned out to be quite limited for investigation of stability and oscillations of these systems. This issue is very crucial in applied research; for example, as stated in Lauvdal et al. (Proceedings of the IEEE control and decision conference, 1997) the following phrase: “Since stability in simulations does not imply stability of the physical control system (an example is the crash of the YF22) stronger theoretical understanding is required”. In this work, firstly a mathematical model of a drilling system developed by a group of scientists from the University of Eindhoven will be considered. Then a mathematical model of a drilling system with perfectly rigid drill-string actuated by induction motor will be analytically and numerically studied. A modification of the first two models will be considered and it will be shown that even in such simple models of drilling systems complex effects such as hidden oscillations may appear, which are hard to find by standard computational procedures.