Distinguishing homoclinic and heteroclinic connections in the full phase space

Clarify the full six-dimensional phase-space connectivity of the observed momentum-space trajectories, including whether figure-eight and related trajectories are homoclinic or heteroclinic when relative phase and body symmetries are taken into account.

Background

The paper represents the dynamics using separate linear-momentum and angular-momentum spaces. This projection can conceal the angular-invariant hypersurfaces and the relative phase information needed to determine whether a trajectory returns to the same full state. Body symmetries further complicate the issue because identifying symmetry-related points can change the effective connectivity of the phase space. The authors therefore leave unresolved the classification of these connections and the appropriate phase-space identification, suggesting that Poincaré sections may help.