Position of the linearly stable region for the resonance chain 1:1:2 in the plane (m₃, δ). For every point in this plane the eigenvalues of the linearised system associated with the vector field F are computed, and the point is plotted only if all the real parts are negative. The co-orbital masses are m₁ = m₂ = 10⁻⁴ and the tidal parameters are those of the system 0 in Table 2. The position of the linearly stable region weakly depends on m₁/m₂ and on the tidal parameters. The colour gives n₁/n₃ and shows that the chain stabilises the dynamics far from the Keplerian resonance (for which n₁/n₃ = 2). The dashed yellow line plots the secular 1 : 1 resonance between ν and ν₃, computed with Eqs. (42) and (33), respectively. For m₃ > 18 (m₁ + m₂), the linearly stable region disappears, while for m₃ < 0.29 (m₁ + m₂), two distinct linearly stable regions exist, whose widths tend to 0 with m₃ (see Appendix F for a discussion of the impact of m₃ on the dynamics).