  We study the accelerating expansion and the induced dark energy of the $5D$
Ricci-flat universe which is characterized by having a big bounce as opposed to
a big bang. We show that the arbitrary function $\mu(t)$ contained in the $5D$
solutions can be rewritten in terms of the redshift $z$ as a new arbitrary
function $f(z)$, and we find that there is a correspondence between this $f(z)$
and the potential $V(\phi)$ of the 4D quintessence models. Using this
correspondence, the arbitrary function $f(z)$ and the $5D$ solution could be
specified for a given form of the potential $V(\phi)$.
