  Common experience suggests that attracting invariant sets in nonlinear
dynamical systems are generally stable. Contrary to this intuition, we present
a dynamical system, a network of pulse-coupled oscillators, in which
\textit{unstable attractors} arise naturally. From random initial conditions,
groups of synchronized oscillators (clusters) are formed that send pulses
alternately, resulting in a periodic dynamics of the network. Under the
influence of arbitrarily weak noise, this synchronization is followed by a
desynchronization of clusters, a phenomenon induced by attractors that are
unstable. Perpetual synchronization and desynchronization lead to a switching
among attractors. This is explained by the geometrical fact, that these
unstable attractors are surrounded by basins of attraction of other attractors,
whereas the full measure of their own basin is located remote from the
attractor. Unstable attractors do not only exist in these systems, but moreover
dominate the dynamics for large networks and a wide range of parameters.
