Asymptotic state of an open quantum system may change substantially upon variations of system parameters. These changes can be often identified as bifurcation transitions in the classical mean‐filed equations describing evolution of the relevant observables. We demonstrate that these bifurcations can be made visible as changes in the structure of the asymptotic density matrix of the original quantum system. By using an N‐boson dimer, we present quantum ‘bifurcation diagrams’ for the pitchfork and saddle‐node bifurcations (in the stationary dimer) and visualize a period‐doubling transition to chaos (in a periodically modulated dimer).