  Measurements on magnetization and relaxation have been carried out on an
optimally doped Ba$_{0.59}$K$_{0.47}$BiO$_{3+\delta}$ single crystal with $T_c$
= 31.3 K. Detailed analysis is undertaken on the data. Both the dynamical
relaxation and conventional relaxation have been measured leading to the
self-consistent determination of the magnetization relaxation rate. It is found
that the data are well described by the collective pinning model leading to the
glassy exponent of about $\mu\approx$ 1.64 to 1.68 with the magnetic fields of
1 and 3 T. The analysis based on Maley's method combining with the conventional
relaxation data allows us to determine the current dependent activation energy
$U$ which yields a $\mu$ value of about 1.23 to 1.29 for the magnetic fields of
1 and 3 T. The second magnetization peaks appear in wide temperature region
from 2 K to 24 K. The separation between the second peak field and the
irreversibility field becomes narrow when temperature is increased. When the
two fields are close to each other, we find that the second peak evolves into a
step like transition of magnetization, suggesting a crossover from the second
order to first order transition. Finally, we present a vortex phase diagram and
demonstrate that the vortex dynamics in Ba$_{1-x}$K$_x$BiO$_3$ can be used as a
model system for studying the collective vortex pining.
