Geometrical representation of the resonant dynamics. As in Fig. 3, the mass-ratio and angular momentum are chosen to be equivalent to that of the 2:1 resonant HD 82943 system (see Table 1). The two panels depict the surfaces of section of the dynamical evolution with respect to Δγ, taken at Δγ = 0, ± π. As described in the text, showing both e₁ and e₂ is redundant and from the figures it can be understood that the two surfaces of section are a simple rotation away from each-other. In both panels, the permissible region is defined by a combination of the constants of motion & Ω and is in essence dictated by the angular momentum of a given system. Contours of Ψ₂ are shown as black lines and the background color is indicative of the value of Ψ₂. Namely, the dark blue region corresponds to a Ψ₂ minimum. For each plotted value of Ψ₂, the separatrix is mapped onto the figure using white dots. Maximal libration amplitudes for Ψ₂ levels that allow for the existence of a critical curve are traced along the corresponding contours with thick black lines. Maximal pseudo-resonant libration widths (i.e. those coresponding to Ψ₂ levels that do not allow for the existence of separatricies) are shown as thick orange lines and are bounded by yellow (caution: these points may appear green on some monitors and/or printers), rather than white dots. The Ψ₂ contours labeled A B C D parallel the dynamical portraits depicted in Fig. 3. The green vertical lines show constant energy levels. The approximate location of the separatrix of the neighboring 3:2 mean motion resonance is shown as a red curve. The (Lee et al. 2006) fit to the radial velocity data is shown with a gray line. The dashed white line depicts the initial condition of the system considered in Sect. 3.2.