Analysis of discontinuous bifurcations in nonsmooth dynamical systems

Dynamical systems with discontinuous right-hand sides are considered. It is well known that the trajectories of such systems are nonsmooth and the fundamental solution matrix is discontinuous. This implies the presence of the so-called discontinuous bifurcations, resulting in a discontinuous change in the multipliers. A method of stepwise smoothing is proposed allowing the reduction of discontinuous bifurcations to a sequence of typical bifurcations: saddle-node, period doubling and Hopf bifurcations. The results obtained are applied to the analysis of the well-known system with friction a block on the moving belt, which serves as a popular model for the description of selfexcited frictional oscillations of a brake shoe. Numerical techniques used in previous investigations of this model did not allow general conclusions to be drawn as to the presence of self-excited oscillations. The new method makes it possible to carry out a complete qualitative investigation of possible types of discontinuous bifurcations in this system and to point out the regions of parameters which correspond to stable periodic regimes.