The motion of a single-degree-of-freedom, periodically forced oscillator subjected to a rigid amplitude constraint is considered. Using analytical methods, the singularities caused by grazing impact are studied. It is shown that as a stable periodic orbit comes to grazing impact under the control of a single parameter, a special type of bifurcation occurs. The motion after the bifurcation may be non-periodic, and a criterion for this based on orientation and eigenvalues is given.

Abstract.A review is presented of the one-parameter, nonsmooth bifurcations that occur in a variety of continuous-time piecewise-smooth dynamical systems. Motivated by applications, a pragmatic approach is taken to defining a discontinuity-induced bifurcation (DIB) as a nontrivial interaction of a limit set with respect to a codimension-one discontinuity boundary in phase space. Only DIBs that are local are considered, that is, bifurcations involving equilibria or a single point of boundary interaction along a limit cycle for flows. Three classes of systems are considered, involving either state jumps, jumps in the vector field, or jumps in some derivative of the vector field. A rich array of dynamics are revealed, involving the sudden creation or disappearance of attractors, jumps to chaos, bifurcation diagrams with sharp corners, and cascades of period adding. For each kind of bifurcation identified, where possible, a kind of "normal form" or discontinuity mapping (DM) is given, together with a canonical example and an application. The goal is always to explain dynamics that may be observed in simulations of systems which include friction oscillators, impact oscillators, DC-DC converters, and problems in control theory.

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Recent investigations of nonsmooth dynamical systems have resulted in the study of a class of novel bifurcations termed as sliding bifurcations. These bifurcations are a characteristic feature of so-called Filippov systems, that is, systems of ordinary differential equations (ODEs) with discontinuous right-hand sides. In this paper we show that sliding bifurcations also play an important role in organizing the dynamics of dry friction oscillators, which are a subclass of nonsmooth systems. After introducing the possible codimension-1 sliding bifurcations of limit cycles, we show that these bifurcations organize different types of "slip to stick-slip" transitions in dry friction oscillators. In particular, we show both numerically and analytically that a sliding bifurcation is an important mechanism causing the sudden jump to chaos previously unexplained in the literature on friction systems. To analyze such bifurcations we make use of a new analytical method based on the study of appropriate normal form maps describing sliding bifurcations. Also, we explain the circumstances under which the theory of so-called border-collision bifurcations can be used in order to explain the onset of complex behavior in stick-slip systems.