  Initially straight slender elastic rods with geometrically constrained ends
buckle and form stable two-dimensional shapes when compressed by bringing the
ends together. It is also known that beyond a critical value of the pre-stress,
clamped rods transition to bent, twisted three-dimensional equilibrium shapes.
Recently, we showed that pre-stressed planar shapes when immersed in a
dissipative fluid and animated by nonconservative follower forces exhibit
stable large-amplitude flapping oscillations. Here, we use time-stepper methods
to analyze the three-dimensional instabilities and dynamics of pre-stressed
planar and non-planar filament configurations when subject to active follower
forces and dissipative fluid drag. First, we find that type of boundary
constraint determines the nature of the non-linear patterns following
instability. When the filament is clamped at one end and pinned at the other
with follower forces directed towards the clamped end, we observe only stable
planar (flapping) oscillations termed flapping result. When both ends are
clamped however, we observe a secondary instability wherein planar oscillations
are destabilized by off-planar perturbations and result in fully
three-dimensional swirling patterns characterized by two distinct time-scales.
The first time scale characterizes continuous and unidirectional swirling
rotation around the end-to-end axis. The second time scale captures the rate at
which the direction of swirling reverses or flips. The overall time over which
the direction of swirling flips is very short compared to the long times over
which the filament swirls in the same direction. Computations indicate that the
reversal of swirling oscillations resembles relaxation oscillations with each
cycle initiated by a sudden jump in torsional deformation and then followed by
a period of gradual decrease in net torsion until the next cycle of variations.
