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We present and discuss the orbital content of a rather unusual rotatingbarred galaxy model, in which the three-dimensional (3D) family, bifurcatingfrom x1 at the 2:1 vertical resonance with the known "frown-smile" side-onmorphology, is unstable. Our goal is to study the differences that occur in thephase space structure at the vertical 2:1 resonance region in this case, withrespect to the known, well studied, standard case, in which the families withthe frown-smile profiles are stable and support an X-shaped morphology. Thepotential used in the study originates in a frozen snapshot of an $N$-bodysimulation in which a fast bar has evolved. We follow the evolution of thevertical stability of the central family of periodic orbits as a function ofthe energy (Jacobi constant) and we investigate the phase space content bymeans of spaces of section. The two bifurcating families at the vertical 2:1resonance region of the new model change their stability with respect to thatof most studied analytic potentials. The structure in the side-on view that isdirectly supported by the trapping of quasi-periodic orbits around 3D stableperiodic orbits has now an infinity symbol (i.e. $\infty$-type) profile.However, the available sticky orbits can reinforce other types of side-onmorphologies as well. In the new model, the dynamical mechanism of trappingquasi-periodic orbits around the 3D stable periodic orbits that build thepeanut, supports the $\infty$-type profile. The same mechanism in the standardcase supports the X shape with the frown-smile orbits. Nevertheless, in bothcases (i.e. in the new and in the standard model) a combination of 3Dquasi-periodic orbits around the stable x1 family with sticky orbits cansupport a profile reminiscent of the shape of the orbits of the 3D unstablefamily existing in each model.

astronomy and astrophysics

Building CX peanut-shaped disk galaxy profiles. The relative importance  of the 3D families of periodic orbits bifurcating at the vertical 2:1  resonance