Other attracting dynamics in the RPSSL A-cycle regime

Determine whether the Rock–Paper–Scissors–Spock–Lizard (RPSSL) dynamical system defined in Section 3.3 (with A- and B-type heteroclinic connections) admits any attracting invariant sets besides the A-type heteroclinic cycle when parameters are chosen so that the A-cycle is fragmentarily asymptotically stable within the positive orthant. Establish conclusively whether additional attractors exist or not for this parameter regime.

Background

The RPSSL network in five dimensions consists of equilibria on coordinate axes linked by A-type and B-type heteroclinic connections. In the parameter regime of Section 3.3.1, the authors present numerical evidence that trajectories asymptote to the A-type subcycle, and no other long-term dynamics were observed.

Despite extensive simulations, the authors explicitly note a lack of proof excluding other possible attractors in this regime, motivating a precise determination of whether any additional attracting dynamics exist beyond the visible A-cycle.