Phase-space portraits (ξ, P_ξ) for the model of Eq. (8) with pattern speed Ω_sp = 15 km s⁻¹ kpc⁻¹ and density of the spiral potential ρ₀ = 5 × 10⁷ M_⊙ kpc⁻³ for 12 different values of the radius r_c namely r_c = 1, 2, ..., 12 kpc. Precessing ellipses responsible for the spiral density waves are periodic orbits of the x₁ family that correspond to radii between approximately 5 kpc (second ILR) and 11 kpc (4:1 resonance).