  We investigate the post-bounce background dynamics in a certain class of
single bounce scenarios studied in the literature, in which the cosmic bounce
is driven by a scalar field with negative exponential potential such as the
ekpyrotic potential. We show that those models can actually lead to cyclic
evolutions with repeated bounces. These cyclic evolutions, however, do not
account for the currently observed late-time accelerated expansion and hence
are not cosmologically viable. In this respect we consider a new kind of cyclic
model proposed recently and derive some cosmological constraints on this model.
