Classification of sliding bifurcations in dry-friction oscillators

Recent investigations of non-smooth dynamical systems led to the study of a class of novel bifurcations termed as sliding bifurcations. These bifurcations are a characteristic feature of so-called Filippov systems, i.e. systems of ODEs with discontinuous right-hand side. In this paper we show that sliding bifurcations also play an important role in organising the dynamics of dry friction oscillators, which constitute a subclass of non-smooth systems; non-smoothness being brought about by the character of friction law. After introducing the possible codimension-1 sliding bifurcations, we show that these bifurcations organise different types of "slip to stick-slip" transitions in the dry friction oscillators. In particular, we show both numerically and analytically that a sliding bifurcation is the mechanism causing the sudden jump to chaos often reported in the literature on friction systems. To analyse such bifurcation we make use of a new analytical method based on the study of appropriate normal form maps describing the sliding bifurcation. In so doing we explain under what circumstances the theory of so-called border-collision bifurcations can be used to explain the onset of complex behaviour in stick-slip systems.