Damped, modified Laplace–Lagrange secular solution of a two-planet system with and e ₂ = 0.1. Three solutions are presented corresponding to the initial conditions e ₁ = 0.1 (blue), e ₁ = 0.03 (red), and e ₁ = 0 (green), with randomly chosen longitudes of perihelia. The black line shows the eccentricity of the outer planet. The apsidal angles initially circulate, but switch to libration at t ≈ Q × 6 × 10 ⁵ years. The system reaches a fixed point as the anti-aligned ( g ₁) mode decays away completely at t ≈ Q × 1.3 × 10 ⁶ years. Note that the system loses memory of its initial conditions as it approaches the fixed point.