  This study examines the effect of dark matter halos on the orbital and escape
dynamics of stars in the central region of barred galaxies. A three-dimensional
gravitational model with a central bulge, bar, disc, and dark matter halo (or
simply dark halo) has been set up and analyzed from the viewpoint of escape in
open Hamiltonian systems for this purpose. Additionally, this model has been
examined separately for the dark halo profiles: oblate and NFW. In both
circumstances, an escape mechanism has been identified near the saddle points
of the phase space, which correspond to the bar ends. The escaping motion of
stars is seen using orbital maps and Poincar\'e surface section maps generated
in various phase planes. Finally, the relationship between chaos and dark halo
parameters such as mass, size, circular velocity, and nature has been studied.
Our findings suggest that oblate dark halos are preferred over NFW dark halos
for justifying the formation of full-fledged spiral arms and extended
distribution of dark halos in giant spiral galaxies with supermassive black
holes (SMBHs) at their centers. Again, the oblate dark halos well justify the
emergence of less prominent or poor spiral arms and the core-dominated
distribution of dark halos in dwarf and LSB galaxies in the absence of central
SMBHs. On the other hand, extreme central baryonic feedback is required for the
NFW halos to generate spiral patterns, and such dark halos should be preferred
for galaxies with extremely energetic centers.
