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A three degrees of freedom (3-dof) barred galaxy model composed of aspherically symmetric nucleus, a bar, a flat disc and a spherically symmetricdark matter halo is used for investigating the dynamics of the system. We usecolour-coded plots to demonstrate how the value of the semi-major axis of thebar influences the regular or chaotic dynamics of the 3-dof system. Fordistinguishing between ordered and chaotic motion we use the Smaller ALingmentIndex (SALI) method, a fast yet very accurate tool. Undoubtedly, the mostimportant elements of the dynamics are the normally hyperbolic invariantmanifolds (NHIMs) located in the vicinity of the index 1 Lagrange points $L_2$and $L_3$. These manifolds direct the flow of stars over the saddle points,while they also trigger the formation of rings and spirals. The dynamics in theneighbourhood of the saddle points is visualized by bifurcation diagrams of theLyapunov orbits as well as by the restriction of the Poincar\'e map to theNHIMs. In addition, we reveal how the semi-major axis of the bar influences thestructure of these manifolds which determine the final stellar structure (ringsor spirals). Our numerical simulations suggest that in galaxies with weak barsthe formation of $R_1$ rings or $R_1'$ pseudo-rings is favoured. In the case ofgalaxies with intermediate and strong bars the invariant manifolds seem to giverise to $R_1R_2$ rings and twin spiral formations, respectively. We alsocompare our numerical outcomes with earlier related work and with observationaldata.

Orbital and escape dynamics in barred galaxies - II. The 3D system: Exploring the role of the normally hyperbolic invariant manifolds