Evolution of ξ and the σ_j (in degrees) as a function of time for system 1 (see Table 2) as integrated by the simplified model F (Eqs. (25) and (56)) (top) and the direct N-body simulation (Eq. (E.1) (bottom). In the direct simulation horseshoe-shaped orbits are reached after 6.42 τ_hs, and the co-orbital motion is destroyed shortly after that (chaoticity in Fig. 6). The model reaches the horseshoe-shaped orbits at t = 13.1 τ_hs. Here, the presence of the chain increases the lifetime of the co-orbitals by a factor of 6.42. The thickness of the lines in the bottom plot (see e.g. σ₃) is due to the short-period oscillations that were averaged out in the model. The grey-shaded area is the linearly stable region.