  We consider a binary repulsive Bose-Einstein condensate in a harmonic trap in
one spatial dimension and investigate particular solutions consisting of two
dark-bright (DB) solitons. There are two different stationary solutions
characterized by the phase difference in the bright component, in-phase and
out-of-phase states. We show that above a critical particle number in the
bright component, a symmetry breaking bifurcation of the pitchfork type occurs
that leads to a new asymmetric solution whereas the parental branch, i.e., the
out-of-phase state becomes unstable. These three different states support
different small amplitude oscillations, characterized by an almost stationary
density of the dark component and a tunneling of the bright component between
the two dark solitons. Within a suitable effective double-well picture, these
can be understood as the characteristic features of a Bosonic Josephson
Junction (BJJ), and we show within a two-mode approach that all characteristic
features of the BJJ phase space are recovered. For larger deviations from the
stationary states, the simplifying double-well description breaks down due to
the feedback of the bright component onto the dark one, causing the solitons to
move. In this regime we observe intricate anharmonic and aperiodic dynamics,
exhibiting remnants of the BJJ phase space.
