Role of galactic bars in the formation of spiral arms: A study through
  orbital and escape dynamics -- I
  In the present work we have developed a three-dimensional gravitational model
of barred galaxies, in order to study orbital and escape dynamics of the stars
inside their central barred region. Our gravitational model is composed of four
components, central nucleus, bar, disc and dark matter halo. Furthermore we
have analysed the model for two different types of bar potentials. The study
has been carried out for a Hamiltonian system and thorough numerical studies
have been done in order to categorize regular and chaotic motions of stars. We
have seen that escape mechanism has only seen near saddle points ($L_2$, $L_4$
and $L_2^{'}$, $L_4^{'}$) of the Hamiltonian system. Orbital structures in $x$
- $y$ plane indicate that this escaping motion corresponds to the two ends of
the bar. Classifications of orbits are found by calculating maximal Lyapunov
exponent of the stellar trajectories corresponding to a specific initial
condition vector. Poincar\'e surface section maps are studied in both $x$ - $y$
and $x$ - $p_x$ ($p_x$ is the momentum along $x$ - direction) plane to get a
complete view of the escape properties of the system in the phase space. Also
we studied in detail how the chaotic dynamics varies with mass, length and
nature of the bar. We found that under suitable physical conditions the chaos
plays a pivotal role behind the formation of grand design or poor spiral
pattern for stronger bars and ring structures for weaker bars.
