  We analyze the effect of the barrier motion on the Bose-Hubbard Hamiltonian
of a ring-shaped Bose-Einstein condensate interrupted by a pair of Josephson
junctions, a configuration which is the cold atom analog of the well-known dc
superconducting quantum interference device (SQUID). Such an effect is also
shown to modify the Heisenberg equation of motion of the boson field operator
in the two-mode approximation, where a hysteretic contribution that could
affect the dynamics for accelerated or overlapping barriers is identified. By
studying the energy landscape as a function of order and control parameters, we
determine the diagram with the location of the dc and ac Josephson regimes,
along with the critical points that are shown to depend on the junctions
position. We analyze the dc to ac Josephson transition for adiabatic barrier
trajectories that lead to a final uniform velocity, or which perform symmetric
velocity paths. We show that such symmetric trajectories may induce, when
reaching the critical point, highly hysteretic oscillating return paths within
the dc regime, similar to the underdamped hysteresis loops arising from the
action of a resistive flow in the ac regime. We also consider nonequilibrium
initial conditions resulting from a finite phase difference on either side of
the junctions, along with the critical features of such a parameter. An
excellent agreement between the Gross-Pitaevskii simulations and the two-mode
results is found in all cases.
