  Five-dimensional cosmological models with two 3-branes and with a buck
cosmological constant are studied. It is found that for all the three cases
($\Lambda =0$, $\Lambda >0$, and $\Lambda <0$), the conventional space-time
singularity ``big bang'' could be replaced by a matter singularity ``big
bounce'', at which the ``size'' of the universe and the energy density are
finite while the pressure diverges, and across which the universe evolves from
a pre-existing contracting phase to the present expanding phase. It is also
found that for the $\Lambda >0$ case the brane solutions could give an
oscillating universe model in which the universe oscillates with each cosmic
cycle begins from a ``big bounce'' and ends to a ``big crunch'', with a
distinctive characteristic that in each subsequent cycle the universe expands
to a larger size and then contracts to a smaller (but non-zero) size. By
studying the gravitational force acted on a test particle in the bulk, a
gravitational stability condition is derived and then is used to analyze those
brane models. It predicts that if dark energy takes over ordinary matter,
particles on the brane may become unstable in the sense that they may escape
from our 4D-world and dissolve in the bulk due to the repulsive force of dark
energy.
