Poincare surface of section of the 2:1 mean motion resonance. The values of energy and angular momentum correspond to the dashed white line shown in the HD 82943 global dynamical map i.e. Fig. 4. Chaotic trajectories are shown with small red points while regular trajectories are depicted as purple and black curves. Conservation of the adiabatic invariant is ensured by the separation of timescales along the shown regular trajectories. The orbits shown in purple are characterized by libration in the (Ξ,ξ) plane, while the black orbits imply circulation. Unsurprisingly, the (Υ,υ) phase-space occupied by orbits that entail libration in the (Ξ,ξ) plane are separated from those that entail circulation by the projection of the (Ξ,ξ) separatrix, shown as thick brown line. Additionally, as in Fig. 10, the unperturbed orbit derived from the integrable Hamiltonian (48) is shown as a dashed line.