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We use a free-energy minimization approach to describe the secular anddynamical instabilities as well as the bifurcations along equilibrium sequencesof rotating, self-gravitating fluid systems. Our approach is fully nonlinearand stems from the Landau-Ginzburg theory of phase transitions. Here we examinehigher than 2nd-harmonic disturbances applied to Maclaurin spheroids, thecorresponding bifurcating sequences, and their relation to nonlinear fissionprocesses. The triangle and ammonite sequences bifurcate from the two3rd-harmonic neutral points on the Maclaurin sequence while the square andone-ring sequences bifurcate from two of the three known 4th harmonic neutralpoints. In the other three cases, secular instability does not set in at thecorresponding bifurcation points because the sequences stand and terminate athigher energies relative to the Maclaurin sequence. There is no knownbifurcating sequence at the point of 3rd-harmonic dynamical instability. Ournonlinear approach easily identifies resonances between the Maclaurin sequenceand various multi-fluid-body sequences that cannot be detected by linearstability analyses. Resonances appear as first-order phase transitions atpoints where the energies of the two sequences are nearly equal but the lowerenergy state belongs to one of the multi-fluid-body sequences.Comment: 23 pages, postscript, compressed, uuencoded. Figs. (6) available by  anonymous ftp from ftp://asta.pa.uky.edu/shlosman/paper4/ , get *.ps.Z). To  appear in Ap

Phase-Transition Theory of Instabilities. IV. Critical Points on the Maclaurin Sequence and Nonlinear Fission Processes