Orbital and escape dynamics in barred galaxies - II. The 3D system:
  Exploring the role of the normally hyperbolic invariant manifolds
  A three degrees of freedom (3-dof) barred galaxy model composed of a
spherically symmetric nucleus, a bar, a flat disc and a spherically symmetric
dark matter halo is used for investigating the dynamics of the system. We use
colour-coded plots to demonstrate how the value of the semi-major axis of the
bar influences the regular or chaotic dynamics of the 3-dof system. For
distinguishing between ordered and chaotic motion we use the Smaller ALingment
Index (SALI) method, a fast yet very accurate tool. Undoubtedly, the most
important elements of the dynamics are the normally hyperbolic invariant
manifolds (NHIMs) located in the vicinity of the index 1 Lagrange points $L_2$
and $L_3$. These manifolds direct the flow of stars over the saddle points,
while they also trigger the formation of rings and spirals. The dynamics in the
neighbourhood of the saddle points is visualized by bifurcation diagrams of the
Lyapunov orbits as well as by the restriction of the Poincar\'e map to the
NHIMs. In addition, we reveal how the semi-major axis of the bar influences the
structure of these manifolds which determine the final stellar structure (rings
or spirals). Our numerical simulations suggest that in galaxies with weak bars
the formation of $R_1$ rings or $R_1'$ pseudo-rings is favoured. In the case of
galaxies with intermediate and strong bars the invariant manifolds seem to give
rise to $R_1R_2$ rings and twin spiral formations, respectively. We also
compare our numerical outcomes with earlier related work and with observational
data.
