As noise reduction tends to be part of environmental directives, predicting squeal noise generated by disc brakes is an important industrial issue. It involves both the transient and stationary nonlinear dynamics of self-excited systems with frictional contact. Time simulation of the phenomenon is an attractive option for reducing experiment costs. However, since such computations using full finite element models of industrial disc brake systems is time-consuming, model reduction has to be performed. In this paper, both the transient and stationary nonlinear behaviors of the friction destabilized system and the effect of dynamical reduction on the nonlinear response of a simple friction destabilized system are carried out. The first part provides a description of the general modeling retained for friction destabilized systems. Then, discretization and solving processes for the stability analysis and the temporal evolution are presented. The third part presents an analysis of a sliding elastic layer for different operating conditions, in order to better understand the nonlinear behavior of such systems. Finally, spatial model reduction is performed with different kinds of reduction bases in order to analyze the different effects of modal reductions. This clearly shows the necessity of including static modes in the reduction basis and that nonlinear interactions between unstable modes are very difficult to represent with reduced bases. Finally, the proposed model and the associated studies are intended to be the benchmark cases for future comparison.