  A systematic study is made on the time-dependent dynamic transport
characteristics of the side-coupled double quantum-impurity system based on the
hierarchical equations of motion. It is found that the transport current
behaves like a single quantum dot when the coupling strength is low during
tunneling or coulomb coupling. The dynamic current oscillates due to the
temporal coherence of the electron tunneling device only when the tunneling
transition is coupled. The oscillation frequency of the transport current is
related to the step voltage applied by the lead, while the $T$, e-e interaction
$U$ and the bandwidth $W$ have little influence. The amplitude of the current
oscillation exists in positive correlation with $W$ and negative correlation
with $U$. With the increase in coupling $t_{12}$ between impurities, the ground
state of the system changes from a Kondo singlet of one impurity to a
spin-singlet of two impurities. Moreover, lowering the temperature could
promote the Kondo effect to intensify the oscillation of the dynamic current.
When only the coulomb transition is coupled, it is found that the two split-off
Hubbard peaks move upward and have different interference effects on the Kondo
peak at the Fermi surface with the increase in $U_{12}$, from the dynamics
point of view.
