  We examine an effectively repulsive Bose-Einstein condensate of atoms, that
rotates in a quadratic-plus-quartic trapping potential. We investigate the
phase diagram of the system as a function of the angular frequency of rotation
and of the coupling constant, demonstrating that there are phase transitions
between multiply- and singly-quantized vortex states. The derived phase diagram
is shown to be universal and exact in the limits of small anharmonicity and
weak coupling constant.
