Everyday experience shows that twisted elastic filaments spontaneously form
loops. We model the dynamics of this looping process as a sequence of bifu
rcations of the solutions to the Kirchhoff equation describing the evolutio
n of thin elastic filaments. The control parameter is taken to be the initi
al twist density in a straight rod. The first bifurcation occurs when the t
wisted straight rod deforms into a helix. This helix is an exact solution o
f the Kirchhoff equations, whose stability can be studied. The secondary bi
furcation is reached when the helix itself becomes unstable and the localiz
ation of the post-bifurcation modes is demonstrated for these solutions. Fi
nally, the tertiary bifurcation takes place when a loop forms at the middle
of the rod and the looping becomes ineluctable. Emphasis is put on the dyn
amical character of the phenomena by studying the dispersion relation and d
eriving amplitude equations for the different configurations.