  Dissipationless N-body models of rotating galaxies, iso-energetic to a
non-rotating model, are examined as regards the mass in regular and in chaotic
motion. The values of their spin parameters $\lambda$ are near the value
$\lambda=0.22$ of our Galaxy.
  We obtain the distinction between the sets of particles moving in regular and
in chaotic orbits and we show that the spatial distribution of these two sets
of particles is much different. The rotating models are characterized by larger
fractions of mass in chaotic motion ($\thickapprox 65%$) compared with the
fraction of mass in chaotic motion in the non-rotating iso-energetic model
($\thickapprox 32%$). Furthermore, the Lyapunov numbers of the chaotic orbits
in the rotating models become by about one order of magnitude larger than in
the non-rotating model. Chaotic orbits are concentrated preferably in values of
the Jacobi integral around the value of the effective potential at the
corotation radius.
  We find that density waves form a central rotating bar embedded in a thin and
a thick disc with exponential surface density profile. A surprising new result
is that long living spiral arms are exited on the disc, composed almost
completely by chaotic orbits.
  The bar excites an $m=2$ mode of spiral waves on the surface density of the
disc, emanating from the corotation radius. These spiral waves are deformed,
fade, or disappear temporarily, but they grow again re-forming a well developed
spiral pattern. Spiral arms are discernible up to 20 or 30 rotations of the bar
(lasting for about a Hubble time).
