Indefinite excursions and non-visibility of the one-dimensional RPSSL network under aperiodic dynamics

Show that, in the aperiodic parameter regime for the Rock–Paper–Scissors–Spock–Lizard system described in Section 3.3.4, trajectories exhibit excursions away from the one-dimensional set of heteroclinic connections indefinitely, implying that the trajectories do not satisfy any definition of visibility for the one-dimensional network.

Background

Section 3.3.4 presents a regime where trajectories repeatedly visit all one-dimensional connections but sporadically traverse the interior of the two-dimensional unstable manifolds, suggesting non-visibility of the one-dimensional network.

In Section 4.2, the authors explicitly conjecture that such excursions persist indefinitely, which if proven would confirm non-visibility of the one-dimensional network in this regime.