  In this paper we use the two-dimensional (2D) version of a new analytical
gravitational model in order to explore the orbital as well as the escape
dynamics of the stars in a barred galaxy composed of a spherically symmetric
central nucleus, a bar, a flat disk and a dark matter halo component. A
thorough numerical investigation is conducted for distinguishing between
bounded and escaping motion. Furthermore bounded orbits are further classified
into non-escaping regular and trapped chaotic using the Smaller ALingment Index
(SALI) method. Our aim is to determine the basins of escape through the two
symmetrical escape channels around the Lagrange points $L_2$ and $L_3$ and also
to relate them with the corresponding distribution of the escape rates of the
orbits. We integrate initial conditions of orbits in several types of planes so
as to obtain a more complete view of the overall orbital properties of the
dynamical system. We also present evidence that the unstable manifolds which
guide the orbits in and out the interior region are directly related with the
formation of spiral and ring stellar structures observed in barred galaxies. In
particular, we examine how the bar's semi-major axis determines the resulting
morphologies. Our numerical simulations indicate that weak barred structures
favour the formation of $R_1$ rings or $R_1'$ pseudo-rings, while strong bars
on the other hand, give rise to $R_1R_2$ and open spiral morphologies. Our
results are compared with earlier related work. The escape dynamics and the
properties of the manifolds of the full three-dimensional (3D) galactic system
will be given in an accompanying paper.
