  The escape dynamics in a simple analytical gravitational model which
describes the motion of stars in a Seyfert galaxy is investigated in detail. We
conduct a thorough numerical analysis distinguishing between regular and
chaotic orbits as well as between trapped and escaping orbits, considering only
unbounded motion for several energy levels. In order to distinguish safely and
with certainty between ordered and chaotic motion, we apply the Smaller
ALingment Index (SALI) method. It is of particular interest to locate the
escape basins through the openings around the collinear Lagrangian points $L_1$
and $L_2$ and relate them with the corresponding spatial distribution of the
escape times of the orbits. Our exploration takes place both in the physical
$(x,y)$ and in the phase $(x,\dot{x})$ space in order to elucidate the escape
process as well as the overall orbital properties of the galactic system. Our
numerical analysis reveals the strong dependence of the properties of the
considered escape basins with the total orbital energy, with a remarkable
presence of fractal basin boundaries along all the escape regimes. It was also
observed, that for energy levels close to the critical escape energy the escape
rates of orbits are large, while for much higher values of energy most of the
orbits have low escape periods or they escape immediately to infinity. We also
present evidence obtained through numerical simulations that our model can
describe the formation and the evolution of the observed spiral structure in
Seyfert galaxies. We hope our outcomes to be useful for a further understanding
of the escape mechanism in galaxies with active nuclei.
