  Recent observational studies of X-shaped structures revealed that values of
their opening angles lie in a narrow range: from 20 deg to 43 deg with smaller
X-shaped structures appearing to follow a characteristic opening angle ~ 27 deg
- 31 deg. We use self-consistent numerical simulations to uncover what
parameters of host galaxies govern the opening angle spread. We constructed a
series of equilibrium models of galaxies with high spatial resolution, varying
the dark halo and bulge contribution in the overall gravitational potential,
the initial disc thickness of models and the Toomre parameter Q and followed
their evolution for almost 8 Gyrs. Each model demonstrated the formation of
clear X-structures with different flatness. We have obtained that opening
angles lie in the range from 25 deg to 42 deg throughout the entire evolution.
These values are roughly consistent with observational data. The greatest
variation in the opening angles is obtained by varying the mass of the dark
halo. The initial thickness of the disc and the Toomre parameter are
responsible for smaller variations of the angle and shorter X-structures. An
increase of both parameters changes the morphology of in-plane bars and
X-structures. In some cases we observed even double X-structures. The main
effect of the bulge is to prevent buckling at intermediate and late stages of
the disc evolution. Comparison of models with different halo masses indicates
that the smallest observable values of opening angles can be associated with
the presence of a heavy dark halo (more than 3 masses of the disc within the
optical radius).
