  We investigate the scattering properties of one-dimensional, periodically and
non-periodically forced oscillators. The pattern of singularities of the
scattering function, in the periodic case, shows a characteristic hierarchical
structure where the number Nc of zeros of the solutions plays the role of an
order parameter marking the level of the observed self-similar structure. The
behavior is understood both in terms of the return map and of the intersections
pattern of the invariant manifolds of the outermost fixed points. In the
non-periodic case the scattering function does not provide a complete
development of the hierarchical structure. The singularities pattern of the
outgoing energy as a function of the driver amplitude is connected to the
arrangement of gaps in the fundamental regions. The survival probability
distribution of temporarily bound orbits is shown to decay asymptotically as a
power law. The "stickiness" of regular regions of phase space, given by KAM
surfaces and remnant of KAM curves, is responsible for this observation.
