We use a free-energy minimization approach to describe in simple and clear physical terms 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 Ginzburg-Landau theory of phase transitions. In this paper, we examine fourth-harmonic axisymmetric disturbances in Maclaurin spheroids and fourth-harmonic nonaxisymmetric disturbances in Jacobi ellipsoids. These two cases are very similar in the framework of phase transitions. It has been conjectured (Hachisu and Eriguchi 1983) that third-order phase transitions, manifested as smooth bifurcations in the angular momentum-rotation frequency plane, may occur on the Maclaurin sequence at the bifurcation point of the axisymmetric one-ring sequence and on the Jacobi sequence at the bifurcation point of the dumbbell-binary sequence. We show that these transitions are forbidden when viscosity maintains uniform rotation. The uniformly rotating one-ring/dumbbell equilibria close to each bifurcation point and their neighboring uniformly rotating nonequilibrium states have higher free energies than the Maclaurin/Jacobi equilibria of the same mass and angular momentum. These high-energy states act as free-energy barriers preventing the transition of spheroids/ellipsoids from their local minima to the free-energy minima that exist on the low rotation frequency branch of the one-ring/binary sequence. At a critical point, the two minima of the free-energy function are equal, signaling the appearance of a first-order phase transition. This transition can take place beyond the critical point only nonlinearly if the applied perturbations contribute enough energy to send the system over the top of the barrier (and if, in addition, viscosity maintains uniform rotation). In the angular momentum-rotation frequency plane, the one-ring and dumbbell-binary sequences have the shape of an ''inverted S'' and two corresponding turning points each. Because of this shape, the free-energy barrier disappears suddenly past the higher turning point, leaving the spheroid/ellipsoid on a saddle point but also causing a ''catastrophe'' by permitting a ''secular'' transition toward a one-ring/binary minimum energy state. This transition appears as a typical second-order phase transition, although there is no associated sequence bifurcating at the transition point (cf. Christodoulou et al. 1995a). Irrespective of whether a nonlinear first-order phase transition occurs between the critical point and the higher turning point or an apparent second-order phase transition occurs beyond the higher turning point, the result is fission (i.e., ''spontaneous breaking'' of the topology) of the original object on a secular timescale: the Maclaurin spheroid becomes a uniformly rotating axisymmetric torus, and the Jacobi ellipsoid becomes a binary. The presence of viscosity is crucial since angular momentum needs to be redistributed for uniform rotation to be maintained. We strongly suspect that the ''secular catastrophe'' is the dynamical analog of the notorious A-transition of liquid He-4 because it appears as a ''second-order'' phase transition with infinite ''specific heat'' at the point where the free-energy barrier disappears suddenly. This transition is not an elementary catastrophe. In contrast to this case, a ''dynamical catastrophe'' takes place from the bifurcation point to the lower branch of the Maclaurin toroid sequence because all conservation laws are automatically satisfied between the two equilibrium states. Furthermore, the free-energy barrier disappears gradually, and this transition is part of the elementary cusp catastrophe. This type of ''lambda-transition'' is the dynamical analog of the Bose-Einstein condensation of an ideal Bose gas. The phase transitions of the dynamical systems are briefly discussed in relation to previous numerical simulations of the formation and evolution of protostellar systems. Some technical discussions concerning related results obtained from linear stability analyses, the breaking of topology, and the nonlinear theories of structural stability and catastrophic morphogenesis are included in an appendix.
