The phase-space of boxy-peanut and X-shaped bulges in galaxies I.
  Properties of non-periodic orbits
  The investigation of the phase-space properties of structures encountered in
a dynamical system is essential for understanding their formation and
enhancement. In the present paper we explore the phase space in energy
intervals where we have orbits that act as building blocks for boxy-peanut
(b/p) and "{\sf X}-shaped" structures in rotating potentials of galactic type.
We underline the significance of the rotational tori around the 3D families
x1v1 and x1v1$^{\prime}$ that have been bifurcated from the planar x1 family.
These tori play a multiple role: (i) They belong to quasi-periodic orbits that
reinforce the local density. (ii) They act as obstacles for the diffusion of
chaotic orbits and (iii) they attract a large number of chaotic orbits that
become sticky to them. There are also bifurcations of unstable families (x1v2,
x1v2$^{\prime}$). Their unstable asymptotic curves wind around the x1v1 and
x1v1$^{\prime}$ tori generating orbits with hybrid morphologies between that of
x1v1 and x1v2. In addition, a new family of multiplicity 2, called x1mul2, is
found to be important for the peanut construction. Our work shows also that
there are peanut-supporting orbits before the vertical ILR. Non-periodic orbits
associated with the x1 family secure this contribution as well as the support
of b/p structures at several other energy intervals. Non-linear phenomena
associated with complex instability of single and double multiplicity families
of periodic orbits show that these structures are not interrupted in regions
where such orbits prevail. Depending on the main mechanism behind their
formation, boxy bulges exhibit different morphological features. Finally our
analysis indicates that "X" features shaped by orbits in the neighbourhood of
x1v1 and x1v1$^{\prime}$ periodic orbits are pronounced only in side-on or
nearly end-on views of the bar.
