“Double section” apocentric manifolds (blue) and (red) are plotted at four different times t_s as indicated in each panel. The black-dotted curves mark the local maxima of the surface density σ(ρ, ϕ, t), corresponding to the potential V = V_d(ρ)+V_b(ρ, ϕ)+V_sp(ρ, ϕ, t) at the times t = t_s. These maxima are plotted in the domain 6 kpc < ρ <  15 Kpc, where the imposed spirals have a significant amplitude.