Values of the eccentricities and fundamental frequencies given by the analytical results. Left: value of e₁ at the fixed points of the resonance chain 1:1:2 as predicted by Eq. (35). Right: values of ν/η and ν₃/η along the main branch as a function of δ, for the same resonance chain, predicted by Eqs. (42) and (33). In both panels the planetary masses are (m₁ + m₂) /2 = m₃ = 10⁻⁴ m₀ and m₂/m₁ = 10. According to Eq. (32), e₁ = e₂, and for this choice of masses and resonance chain e₃/e₁ = 0.5471. In Table 1, which gathers the fixed points at second order in eccentricity, the three branches visible in the left plot are branches 1 (main branch), 2, and 6. The secular resonances between ν and ν₃ are shown on the right, and are also easy to spot in Fig. 6.