  We examine the dynamics of scalar perturbations in closed
Friedmann-Lema\^itre-Robertson- Walker (FLRW) universes in the framework of
Brane World theory with a timelike extra dimension. In this scenario, the
unperturbed Friedmann equations contain additional terms arising from the
bulk-brane interaction that implement non-singular bounces in the models with a
cosmological constant and non-interacting perfect fluids. The structure of the
phase-space of the models allows for two basic configurations, namely, one
bounce solutions or eternal universes. Assuming that the matter content of the
model is given by dust and radiation, we derive the dynamical field equations
for scalar hydrodynamical perturbations considering either a conformally flat
(de Sitter) bulk or a perturbed bulk. We perform a numerical analysis which can
shed some light on the study of cosmological scalar perturbations in bouncing
brane world models. From a mathematical point of view we show that although the
bounce enhances the amplitudes of scalar perturbations for one bounce models in
the case of a de Sitter bulk, the amplitudes of the perturbations remain
sufficiently small and bounded relative to the background values up to a
certain scale. For one bounce models in the case of a perturbed bulk the
amplitudes of all perturbations (apart from the Weyl fluid energy density)
remain sufficiently small and bounded relative to the background values for any
scale of the perturbations. We also discuss and compare the stability and
bounded behaviour of the perturbations in the late accelerated phase of one
bounce solutions. For eternal universes we argue that some of these features
are maintained only for early times (typically of the order of the first
bounce). In this sense we show that eternal solutions are highly unstable
configurations considering the background model of this paper.
