Other attracting dynamics in the RPSSL AAB-cycle regime

Determine whether the Rock–Paper–Scissors–Spock–Lizard dynamical system admits any attracting invariant sets besides the AAB-type heteroclinic cycles when parameters are chosen so that each AAB cycle is fragmentarily asymptotically stable within the positive orthant. Ascertain whether additional attractors are present or whether the AAB cycles exhaust all long-term attracting dynamics for this parameter regime.

Background

In Section 3.3.2, the authors consider parameters for which each of the five AAB subcycles (each equivalent to a Guckenheimer–Holmes-type cycle on coordinate planes) is fragmentarily asymptotically stable. Numerics show trajectories asymptote onto one such AAB cycle depending on initial conditions.

The authors explicitly state that it remains unproven whether other attracting dynamics exist for these parameter values, motivating a definitive analysis of possible additional attractors in this regime.