We investigate the occurrence of bifurcations in the dynamical trajectories depicting nuclear collisions at Fermi energies. The quantitative description of the reaction dynamics is obtained within a novel one-body transport approach, based on the solution of the Boltzmann-Langevin equation in three dimensions, with a broad applicability for dissipative fermionic dynamics. Dilute systems formed in central collisions are shown to fluctuate between two energetically favourable mechanisms: reverting to a compact shape or rather disintegrating into several fragments. The latter result can be connected to the recent observation of bimodal distributions for quantities characterising fragmentation processes.