Full visitation of the two-dimensional RPSSL network under aperiodic dynamics

Establish whether, for the Rock–Paper–Scissors–Spock–Lizard system in the aperiodic regime described in Section 3.3.4, trajectories eventually come arbitrarily close to every point of the two-dimensional unstable manifolds forming the network, i.e., whether the entire two-dimensional RPSSL network is visited by sufficiently long trajectories.

Background

The authors report a parameter regime where trajectories follow an aperiodic sequence of A and B edges and occasionally leave the one-dimensional set of connections to pass through interiors of two-dimensional unstable manifolds.

They explicitly state that, although numerical evidence suggests widespread exploration, they have not proved that long trajectories are dense in the two-dimensional manifolds; resolving this would clarify whether the two-dimensional network is fully visited.