  We review the theoretical model and experimental realization of the atom
optics $\delta-$kicked rotor (AOKR), a paradigm of classical and quantum chaos.
We have performed a number of experiments with an all-optical Bose-Einstein
condensate (BEC) in a periodic standing wave potential in an AOKR system. We
discuss results of the investigation of the phenomena of quantum resonances in
the AOKR. An interesting feature of the momentum distribution of the atoms
obtained as a result of short pulses of light, is the variance of the momentum
distribution or the kinetic energy $<p^{2}>/2m$ in units of the recoil energy
$E_{rec} = \hbar \omega_{rec}$. The energy of the system is examined as a
function of pulse period for a range of kicks that allow the observation of
quantum resonances. In particular we study the behavior of these resonances for
a large number of kicks. Higher order quantum resonant effects corresponding to
the fractional Talbot time of (1/4)$T_{T}$ and (1/5)$T_{T}$ for five and ten
kicks have been observed. Moreover, we describe the effect of the initial
momentum of the atoms on quantum resonances in the AOKR.
