  It is shown that bifurcations of the mean-field dynamics of a Bose-Einstein
condensate can be related with the quantum phase transitions of the original
many-body system. As an example we explore the intra-band tunneling in the
two-dimensional optical lattice. Such a system allows for easy control by the
lattice depth as well as for macroscopic visualization of the phase transition.
The system manifests switching between two selftrapping states or from a
selftrapping state to a superposition of the macroscopically populated
selftrapping states with the step-like variation of the control parameter about
the bifurcation point. We have also observed the magnification of the
microscopic difference between the even and odd number of atoms to a
macroscopically distinguishable dynamics of the system.
