Bifurcation for Non-smooth Dynamical Systems via Reduction Methods (original) (raw)

Due to the presence of discontinuities, non-smooth dynamical systems (PWS) present a wide variety of bifurcations, which cannot be explained by the classical theory, for instance, transition from sticking to sliding due to friction and sudden loss of stability as typically observed in mechanics. These phenomena are due to interactions between the boundaries and the phase trajectories that cross them from one region to another. In the present work, we review the concept of invariant sets given as cone-like objects which has turned out as an appropriate generalization of the notion of center manifolds. The existence of invariant cones containing a segment of sliding orbits and stability properties of those cones are also investigated. Based on these results we present new bifurcation phenomena in a class of 3D-PWS concerning sliding modes. Further we show that the dynamics within the sliding motion area is described by a simple one-dimensional equation. We illustrate various forms of bifurcation, stick-slip motion, and the reduction procedure by a sixdimensional brake system given by three coupled oscillators.