The reflection in the center occurs only if a wave can travel there without being damped at the inner Lindblad resonance. The problem of a possible Landau damping of waves at the inner resonance has long suggested that bars can only develop without this resonance. The pattern speed should then be high enough to prevent the resonance. This appears to be verified in N-body simulations in the linear regime, at the beginning of bar formation. Another point of view to better understand the N-body problem is in terms of stellar orbits, and families of periodic orbits as will be described in the next section.
Periodic orbits are closed orbits in the frame rotating with the bar.
The stable ones trap regular orbits around them. They are thus the skeleton of the orbital structure of the disk. Periodic orbits are the fixed points that depend essentially on the symmetry of the potential, and not on the detailed mass distribution. In the potential of a rotating bar, the main family of orbits is elongated along the bar, supporting it, and we can understand under which conditions a self-gravitating system will become barred. When the mass concentration towards the center is strong enough, the elongated orbits will be replaced by periodic orbits perpendicular to the bar, and we can predict the dissolution of the bar.