Final period ratio of GJ 163’s inner planets (3:1 MMR) as a function of the dissipation balance between both planets (Δt₂/κΔt₁, bottom x-axis) and initial amplitude of libration (M₁). The coefficient κ = k_2,1/k_2,2(R₁/R₂)⁵ is unknown, but can be estimated using a mass-radius power law. The top x-axis (Δt₂/ Δt₁) is scaled using the value κ ≈ 2.2 obtained in Eq. (86). When M₁ is set to 0°, the system begins at the libration center, whereas when M₁ is set to 180° it starts at the separatrix. We fixed Δt₁ = 10⁷ s in order to speed up the simulations and integrated the system during 0.1 Myr which roughly corresponds to 10 Gyr for a realistic value of Δt₁ (100 s). The horizontal gray lines correspond to both sets of initial conditions used in Fig. 4. The vertical white lines mark the range of Δt₂/κΔt₁ that may conduct to internal circulation according to our analytical model.