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In this paper we use the two-dimensional (2D) version of a new analyticalgravitational model in order to explore the orbital as well as the escapedynamics of the stars in a barred galaxy composed of a spherically symmetriccentral nucleus, a bar, a flat disk and a dark matter halo component. Athorough numerical investigation is conducted for distinguishing betweenbounded and escaping motion. Furthermore bounded orbits are further classifiedinto non-escaping regular and trapped chaotic using the Smaller ALingment Index(SALI) method. Our aim is to determine the basins of escape through the twosymmetrical escape channels around the Lagrange points $L_2$ and $L_3$ and alsoto relate them with the corresponding distribution of the escape rates of theorbits. We integrate initial conditions of orbits in several types of planes soas to obtain a more complete view of the overall orbital properties of thedynamical system. We also present evidence that the unstable manifolds whichguide the orbits in and out the interior region are directly related with theformation of spiral and ring stellar structures observed in barred galaxies. Inparticular, we examine how the bar's semi-major axis determines the resultingmorphologies. Our numerical simulations indicate that weak barred structuresfavour the formation of $R_1$ rings or $R_1'$ pseudo-rings, while strong barson the other hand, give rise to $R_1R_2$ and open spiral morphologies. Ourresults are compared with earlier related work. The escape dynamics and theproperties of the manifolds of the full three-dimensional (3D) galactic systemwill be given in an accompanying paper.

Orbital and escape dynamics in barred galaxies - I. The 2D system