Conclusions

The previous description of periodic orbits in a warped analytical potential allows us to draw the following conclusions.

1)

The circular orbit family with an axisymmetric potential survives the warp and becomes quasi-annular with a tilt corresponding to the warped galactic disk. Thus, this work confirms the intuitive idea that the orbits follow the warp. Without this confirmation, the ring model of warps would have remained baseless. Note that the ring model has been used in the past in different contexts, but if the underlying circular orbits do not exist, the ring model is nonphysical.

2)

The existing transverse resonances in the axisymmetric case are also preserved in the warped case but occur at slightly lower energies. Subfamilies with the same period as the quasi-annular family are generated and most of them keep their stability. In our model, the perturbation induced by the warp is not strong enough to generate observable chaotic regions.

3)

Since no bifurcation with $\nu=\Omega\pm\Omega_p$ is generated by the warp, it is not possible in our model to excite a 1:1 normal mode, which is often invoked to explain the galactic warps. Other modes are not favored either because the stable regions around their corresponding subfamilies remain very thin.

4)

The study of the consistency of the density distribution underlines the importance of the global rotation. A direct rotation of the warp pattern generates orbits destroying the density distribution, but on the contrary, orbits can locally reinforce the density in the presence of a retrograde rotation. This could help maintaining external warped regions.

The behavior of the discussed periodic orbits will be examined in N-body models and the results will be the object of a future paper.