  Asymptotic state of an open quantum system can undergo qualitative changes
upon small variation of system parameters. We demonstrate it that such 'quantum
bifurcations' can be appropriately defined and made visible as changes in the
structure of the asymptotic density matrix. By using an $N$-boson open quantum
dimer, we present quantum diagrams for the pitchfork and saddle-node
bifurcations in the stationary case and visualize a period-doubling transition
to chaos for the periodically modulated dimer. In the latter case, we also
identify a specific bifurcation of purely quantum nature.
