  Ratchet dynamics of topological solitons of the forced and damped discrete
double sine-Gordon system are studied. Directed transport occurring both in
regular and in chaotic regions of the phase space and its dependence on
damping, amplitude and frequency of the driving, asymmetry parameter, coupling
constant, has been extensively investigated. We show that the passage from
ratchet phase-locked regime to chaotic ratchets occurs via a period doubling
route to chaos and that, quite surprisingly, pinned states can exist inside
phase-locking and chaotic transport regions for intermediate values of the
coupling constant. The possibility to control chaotic discrete soliton ratchets
by means of both small subharmonic signals and more general periodic drivings,
has also been investigated.
