Orbital structure in barred spiral galaxies
  We study the orbital structure in a series of self-consistent $N$-body
configurations simulating rotating barred galaxies with spiral and ring
structures. We perform frequency analysis in order to measure the angular and
the radial frequencies of the orbits at two different time snapshots during the
evolution of each $N$-body system. The analysis is done separately for the
regular and the chaotic orbits. We thereby identify the various types of
orbits, determine the shape and percentages of the orbits supporting the bar
and the ring/spiral structures, and study how the latter quantities change
during the secular evolution of each system. Although the frequency maps of the
chaotic orbits are scattered, we can still identify concentrations around
resonances. We give the distributions of frequencies of the most important
populations of orbits. We explore the phase space structure of each system
using projections of the 4D surfaces of section. These are obtained via the
numerical integration of the orbits of test particles, but also of the real
$N$-body particles. We thus identify which domains of the phase space are
preferred and which are avoided by the real particles. The chaotic orbits are
found to play a major role in supporting the shape of the outer envelope of the
bar as well as the rings and the spiral arms formed outside corotation.
