Mathematical modelling and numerical analysis of a vibrating system with dry friction is presented. Three qualitatively different friction characteristics are considered. One of them is an example of so-called sensitive friction characteristic. Their influence on the dynamics on the attractor of the friction oscillator is investigated through bifurcational analysis. This analysis is supported by Lyapunov exponents estimated using approach for the systems with discontinuities. Theoretical background for such a synchronisation-based method of determining the largest Lyapunov exponent is explained. The results obtained through the proposed approach approximate the LLE with a good precision.