Evolution of the binary system in a secular timescale with dissipation introduced. Each column corresponds to a given initial condition from Table 2, chosen in a neighbourhood of an equilibrium point. On the top row: the trajectory of each spin axis ŝ_ℓ is plotted against the unit sphere, in blue for the primary and green for the secondary. Colours are interpolated towards red from initial time to final time of convergence. Initial conditions for the spin axes are portrayed as dashed lines from the origin to ŝ_ℓ (0). The orbital plane is represented in a darker shade with black contours, and the axis ê_z as a vertical black arrow. On the middle row: the secular magnetic interaction energy. On the bottom row: the angular distance of each spin axis with respect to the stable equilibrium point (ê_z, − ê_z).