Representative phase portraits of resonances with orders j=1, 2, 3, 4, and 5, from top to bottom, respectively. Each phase portrait shows level curves of the Hamiltonians ℋ(X, Y) given in Eq. (D.5), where the mixed variables X and Y (Eq. (D.3)) define the eccentricity vector, e (Eq. (D.4)). The fixed elliptic points away from the origins correspond to maxima of ℋ(X, Y). For each resonance, four representative values of ∆J decreasing from left to right have been considered to illustrate the varying topologies of the phase portraits. The homoclinic trajectories are drawn in red. In all the plots, the value of the parameter ϵ appearing in Eq. (D.5) is taken as negative, as is the case for outer resonances. The topology of resonances with orders j > 5 are similar to the case j=5, except that there are j islands instead of five, with widths that decrease as j increases.