Orbital and escape dynamics in barred galaxies - III. The 3D system:
  Correlations between the basins of escape and the NHIMs
  The escape dynamics of the stars in a barred galaxy composed of a spherically
symmetric central nucleus, a bar, a flat thin disk and a dark matter halo
component is investigated by using a realistic three degrees of freedom (3-dof)
dynamical model. Modern colour-coded diagrams are used for distinguishing
between bounded and escaping motion. In addition, the smaller alignment index
(SALI) method is deployed for determining the regular, sticky or chaotic nature
of bounded orbits. We reveal the basins of escape corresponding to the escape
through the two symmetrical escape channels around the Lagrange points $L_2$
and $L_3$ and also we relate them with the corresponding distribution of the
escape times of the orbits. Furthermore, we demonstrate how the stable
manifolds, around the index-1 saddle points, accurately define the fractal
basin boundaries observed in the colour-coded diagrams. The development
scenario of the fundamental vertical Lyapunov periodic orbit is thoroughly
explored for obtaining a more complete view of the unfolding of the singular
behaviour of the dynamics at the cusp values of the parameters. Finally, we
examine how the combination of the most important parameters of the bar (such
as the semi-major axis and the angular velocity) influences the observed
stellar structures (rings and spirals) which are formed by escaping stars
guided by the invariant manifolds near the saddle points.
