  We study nonlinear response of spinning dark matter (DM) halos to dynamic and
secular evolution of stellar bars in the embedded galactic disks, using
high-resolution numerical simulations. For a sequence of halos with the
cosmological spin parameter lambda=0-0.09, and a representative angular
momentum distribution, we analyze evolution of induced DM bars amplitude and
quantify parameters of the response as well as trapping of DM orbits and
angular momentum transfer by the main and secondary resonances. We find that
(1) maximal amplitude of DM bars depends strongly on `lambda', while that of
the stellar bars is indifferent to `lambda'; (2) Efficiency of resonance
trapping of DM orbits by the bar increases with `lambda', and so is the mass
and the volume of DM bars; (3) Contribution of resonance transfer of angular
momentum to the DM halo increases with `lambda', and for larger spin, the DM
halo `talks' to itself, by moving the angular momentum to larger radii --- this
process is maintained by resonances; (4) Prograde and retrograde DM orbits play
different roles in angular momentum transfer. The `active' part of the halo
extends well beyond the bar region, up to few times the bar length in
equatorial plane and away from this plane. (5) We model evolution of diskless
DM halos and halos with frozen disks, and found them to be perfectly stable to
any Fourier modes. Finally, further studies adopting a range of mass and
specific angular momentum distributions of the DM halo will generalize the
dependence of DM response on the halo spin and important implications for
direct detection of DM and that of the associated stellar tracers, such as
streamers.
