  We investigate the dynamics of a charged particle confined to move on a
toroidal helix while being driven by an external time-dependent electric field.
The underlying phase space is analyzed for linearly and circularly polarized
fields. For small driving amplitudes and a linearly polarized field, we find a
split-up of the chaotic part of the phase space which prevents the particle
from inverting its direction of motion. This allows for a non-zero average
velocity of chaotic trajectories without breaking the well-known symmetries
commonly responsible for directed transport. Within our chosen normalized
units, the resulting average transport velocity is constant and does not change
significantly with the driving amplitude. A very similar effect is found in
case of the circularly polarized field and low driving amplitudes. Furthermore,
when driving with a circularly polarized field, we unravel a second mechanism
of the split-up of the chaotic phase space region for very large driving
amplitudes. There exists a wide range of parameter values for which
trajectories may travel between the two chaotic regions by crossing a permeable
cantorus. The limitations of these phenomena, as well as their implication on
manipulating directed transport in helical geometries are discussed.
