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This thesis concerns on dynamical systems whose behavior can be described as a deterministic in time thus based on initial conditions it is possible to predict how the system will behave. The main goal of the work is to analyze deterministic chaos with special attention to usability of this phenomenon in mechanical systems in everyday life, moreover the work indicate the typical symptoms, circumstances that germinate and support their occurrence. The project is an attempt to correctly identify and describe definitions of: deterministic chaos, chaos quantifiers, attractors, fractals and of course chaos in conservative systems. Experimental part of my work represents the basic chaotic event that can be noticed in daily life

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In this article we examine the lateral vibrations in rotating machinery can lead to rotor unstable behaviour. In some types of rotating machinery, the angular speed is subjected to wide ranges of change. The problem is to find the range of angular speeds for which a machine exhibits stable behaviour in the presence of the coupling of dynamical modes. For determine of mathematical model for vibration we will use Föppl-Jeffcott rotor model. We will examine a undamped and damped vibration on the Föppl-Jeffcott rotor model.