Temporal evolution of the Iso-NBody simulation in the range R ∈ [5.5, 6.5] kpc. Top: relative amplitude of the Fourier mode (Σ₂/Σ₀), i.e. strength of the spiral arms for the stellar (solid ochre) and DM (dashed grey) components. We see that both spirals evolve in a correlated way, although the relative amplitude of the DM spirals is one order of magnitude smaller (notice the auxiliary axis, which refers to the amplitude of the DM spiral arms). Middle: pattern speed (Ω_s) of the mode m = 2 structures in stars (solid ochre) and DM (dashed grey). Bottom: phase difference between the mode m = 2 of the stellar and DM structures (long-dashed red line). Shaded regions shows the dispersion of the phase lag. Once the spirals are strong enough (t ≳ 4.5 Gyr, vertical lines), we detect a constant pattern speed of Ω_s ∼ 20 km s⁻¹ kpc⁻¹ for the stellar and DM spirals, and a constant phase lag of Δ ϕ₂ ∼ 10^∘, showing a strong correlation between the two spiral arms.