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The escape dynamics of the stars in a barred galaxy composed of a sphericallysymmetric central nucleus, a bar, a flat thin disk and a dark matter halocomponent is investigated by using a realistic three degrees of freedom (3-dof)dynamical model. Modern colour-coded diagrams are used for distinguishingbetween bounded and escaping motion. In addition, the smaller alignment index(SALI) method is deployed for determining the regular, sticky or chaotic natureof bounded orbits. We reveal the basins of escape corresponding to the escapethrough the two symmetrical escape channels around the Lagrange points $L_2$and $L_3$ and also we relate them with the corresponding distribution of theescape times of the orbits. Furthermore, we demonstrate how the stablemanifolds, around the index-1 saddle points, accurately define the fractalbasin boundaries observed in the colour-coded diagrams. The developmentscenario of the fundamental vertical Lyapunov periodic orbit is thoroughlyexplored for obtaining a more complete view of the unfolding of the singularbehaviour of the dynamics at the cusp values of the parameters. Finally, weexamine how the combination of the most important parameters of the bar (suchas the semi-major axis and the angular velocity) influences the observedstellar structures (rings and spirals) which are formed by escaping starsguided by the invariant manifolds near the saddle points.

Orbital and escape dynamics in barred galaxies - III. The 3D system: Correlations between the basins of escape and the NHIMs