Most of the "bounce" cosmologies I've read about have been quantum mechanical, but a favorite of mine, Aguirre & Gratton's with arrows of time pointing in opposite directions, is relativistic. Vilenkin, who wrote a critique of it (available free on the web, and called "Arrows of Time and the Beginning of the Universe") and also influenced its formulation, refers to the three-spherical boundary between the arrows (which isn't quite a "singularity") as a "bounce surface".
As you'll see on its p.3, he also describes "the universe" as "bouncing" at that surface. (I've also skimmed through AG's original paper, linked to the bibliography entry of it in Vilenkin's critique, and, surprisingly, Aguirre & Gratton themselves do not use the "bounce" terminology!)
I'm rather lame in physics, and was very surprised to find out, just a couple of days ago, that the de Sitter space that most of the cosmologies I've seen described are supposed to occur in actually requires a contracting phase, which is supposed to precede an expanding phase! The expanding phase, and the inflation or near-exponential expansion, is all that most of the relativistic cosmologists I've read ever talk about, in English at least. (I've mostly stuck to them, because, although I can't follow much of their math, I know even less about QM notation.)
I believe that the entropy problem Chronos was mentioning just has to do with repeatedly-bouncing cosmologies, and you can zero in on them, in web searches, by googling "oscillating universes". The density of entropy increases with each bounce, so that, if that was the way things had always been, tracing them back in time would reach a point where the smallest density traceable would have left their size near the Planck limit. (I've never been able to pry a straight answer out of anyone as to why the Planck limit is such a sacred cow, but it's been occurring to me lately that it may have something to do with differences between Euclidean and elliptical geometries that prevent a simple increase in scale between a figure drawn in one and a figure drawn in the other, although the angles can be preserved through a "conformal diagram".) In its effects, this limitation is similar to the more general problem formulated in the Borde-Guth-Vilenkin Theorem of 2003, which requires a beginning for universes with an average expansion rate greater than zero.