Peanut/X-shaped bulges: Returning to the vertical resonance and separatrix

We investigate the vertical structure and evolution of an N-body galactic bar by analyzing pure resonant orbits that form the skeleton of boxy/peanut/X-shaped (BP/X) bulges in Cartesian and action-angle spaces (þeta_ =þeta_z - þeta_R, J_z). We show that the main backbone of this skeleton is made up of separatrix orbits associated with unstable figure-of-eight resonant orbits. For the first time, we examine the long-term behavior of the separatrix, showing how it sinks to the bottom of phase space and shrinks. As a result, ``fresh'' orbits involved in a flat bar are captured into vertical resonance, passing through it near the separatrix and creating orbital crowding. This crowding traces the thickest part of the BP/X bulge and outlines the ends of its X-structure rays. The rise of orbits leads to changes in the potential and geometry of phase space in these regions. Over time, when the rise ceases, evolution slows and stops. res