The phase-space of boxy-peanut and X-shaped bulges in galaxies II. The   relation between face-on and edge-on boxiness
  We study the dynamical mechanisms that reinforce the formation of boxy structures in the \textit{inner} regions, roughly in the middle, of bars observed nearly \textit{face-on}. Outer boxiness, at the ends of the bars, is usually associated with orbits at the inner, radial 4:1 resonance region and can be studied with 2D dynamics. However, in the middle of the bar dominate 3D orbits that give boxy/peanut bulges in the edge-on views of the models. In the present paper we show that 3D quasi-periodic, as well as 3D chaotic orbits sticky to the x1v1 and x1v1$^{\prime}$ tori, especially from the Inner Lindblad Resonance (ILR) region, have boxy projections on the equatorial plane of the bar. The majority of vertically perturbed 2D orbits, initially on the equatorial plane in the ILR resonance region, enhance boxy features in face-on bars. Orbits that build a bar by supporting sharp "{\sf X}" features in their side-on views at energies \textit{beyond} the ILR, may also have a double boxy character. If populated, the extent of the inner boxiness along the major axis is about the same with that of the peanut supporting orbits in the side-on views. At any rate these orbits do not obscure the observation of the boxy orbits of the ILR region in the face-on views, as they contribute more to the surface density at the sides of the bar than to their central parts. 
P.A.Patsis and M. Katsanikas
astro-ph.GA math.DS nlin.CD