  Cilia and flagella are highly conserved slender organelles that exhibit a
variety of rhythmic beating patterns from non-planar cone-like motions to
planar wave-like deformations. Although their internal structure, composed of a
microtubule-based axoneme driven by dynein motors, is known, the mechanism
responsible for these beating patterns remains elusive. Existing theories
suggest that the dynein activity is dynamically regulated, via a geometric
feedback from the cilium's mechanical deformation to the dynein force. An
alternative, open-loop mechanism based on a 'flutter' instability was recently
proven to lead to planar oscillations of elastic filaments under follower
forces. Here, we show that an elastic filament in viscous fluid, clamped at one
end and acted on by an external distribution of compressive axial forces,
exhibits a Hopf bifurcation that leads to non-planar spinning of the buckled
filament at a locked curvature. We also show the existence of a second
bifurcation, at larger force values, that induces a transition from non-planar
spinning to planar wave-like oscillations. We elucidate the nature of these
instabilities using a combination of nonlinear numerical analysis, linear
stability theory, and low-order bead-spring models. Our results show that away
from the transition thresholds, these beating patterns are robust to
perturbations in the distribution of axial forces and in the filament
configuration. These findings support the theory that an open-loop,
instability-driven mechanism could explain both the sustained oscillations and
the wide variety of periodic beating patterns observed in cilia and flagella.
