  We use a free-energy minimization approach to describe the secular and
dynamical instabilities as well as the bifurcations along equilibrium sequences
of rotating, self-gravitating fluid systems. Our approach is fully nonlinear
and stems from the Landau-Ginzburg theory of phase transitions. Here we examine
higher than 2nd-harmonic disturbances applied to Maclaurin spheroids, the
corresponding bifurcating sequences, and their relation to nonlinear fission
processes. The triangle and ammonite sequences bifurcate from the two
3rd-harmonic neutral points on the Maclaurin sequence while the square and
one-ring sequences bifurcate from two of the three known 4th harmonic neutral
points. In the other three cases, secular instability does not set in at the
corresponding bifurcation points because the sequences stand and terminate at
higher energies relative to the Maclaurin sequence. There is no known
bifurcating sequence at the point of 3rd-harmonic dynamical instability. Our
nonlinear approach easily identifies resonances between the Maclaurin sequence
and various multi-fluid-body sequences that cannot be detected by linear
stability analyses. Resonances appear as first-order phase transitions at
points where the energies of the two sequences are nearly equal but the lower
energy state belongs to one of the multi-fluid-body sequences.
