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 this paper, we explore the phase space in energy intervals where we have orbits that act as building blocks for boxy–peanut (b/p) and ‘X-shaped’ structures in rotating potentials of galactic type. We underline the significance of the rotational tori around the 3D families x1v1 and x1v1′ 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′). Their unstable asymptotic curves wind around the x1v1 and x1v1′ 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. This family produces stickiness phenomena in the critical area of the radial and vertical inner Lindblad resonances (ILRs) and reinforces b/p bulges. 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′ periodic orbits are pronounced only in side-on or nearly end-on views of the bar.