  Positively-curved, oscillatory universes have recently been shown to have
important consequences for the pre-inflationary dynamics of the early universe.
In particular, they may allow a self-interacting scalar field to climb up its
potential during a very large number of these cycles. The cycles are naturally
broken when the potential reaches a critical value and the universe begins to
inflate, thereby providing a `graceful entrance' to early universe inflation.
We study the dynamics of this behaviour within the context of braneworld
scenarios which exhibit a bounce from a collapsing phase to an expanding one.
The dynamics can be understood by studying a general class of braneworld models
that are sourced by a scalar field with a constant potential. Within this
context, we determine the conditions a given model must satisfy for a graceful
entrance to be possible in principle. We consider the bouncing braneworld model
proposed by Shtanov and Sahni and show that it exhibits the features needed to
realise a graceful entrance to inflation for a wide region of parameter space.
