  We present the phase diagram of the mean-field driven-dissipative
Bose-Hubbard dimer model. For a dimer with repulsive on-site interactions
($U>0$) and coherent driving we prove that $\mathbb{Z}_2$-symmetry breaking,
via pitchfork bifurcations with sizable extensions of the asymmetric solutions,
require a negative tunneling parameter ($J<0$). In addition, we show that the
model exhibits deterministic dissipative chaos. The chaotic attractor emerges
from a Shilnikov mechanism of a periodic orbit born in a Hopf bifurcation and,
depending on its symmetry properties, it is either localized or not.
