Rock–Paper–Scissors is a well-known game, which also offers a simple model of competition between three species or, equivalently, strategies. When modelled in continuous time with ordinary differential equations, this system contains a heteroclinic cycle between the three distinct equilibrium solutions, each representing a steady state with just a single species. A slightly more general model results from the addition of two further species or strategies, giving a system of five strategies, now with each one dominant over two of the other four strategies. Such a game, called Rock–Paper–Scissors–Lizard–Spock, was popularised by the TV show ‘The Big Bang Theory’ in 2012, but equivalent networks have been around for much longer, even back to the first century BCE.
In a recent paper, LML Fellow Claire Postlethwaite and Alistair Rucklidge of the University of Leeds present the first comprehensive description of the dynamics of this system when described with ordinary differential equations (ODEs). Building on prior work on the stability of the entire network, they also discuss the dynamics that occur as trajectories approach the heteroclinic cycle. In the analysis, the authors first give an explicit method for computing the stability, not just of sub-cycles of a heteroclinic network, but of arbitrarily complex, repeating sequences of visits to the equilibria of the network. This yields information on whether trajectories which visit neighbourhoods of these equilibria in a particular order are attracted to, or repelled from, the network. The authors then also adapt conditions for heteroclinic cycle stability given in prior work to compute boundaries of stability.
Finally, Postlethwaite and Rucklidge also find a region of parameter space in which there is a complicated set of stability regions for increasingly complex sequences in which trajectories visit equilibria; these regions are reminiscent of the sausage-shaped resonance tongues seen in piecewise continuous systems. As they note, many other papers have proven the existence of complicated cycling behaviour of trajectories close to heteroclinic networks, and switching between different sub-cycles of the network. The authors believe this is the first work to give regions of parameter space where different cycling behaviours are stable and observable in numerical simulations, and to also explain how such stability is lost.
A pre-print of the paper is available at https://drive.google.com/file/d/1MkOinB4PcHKr7R12Js1Zp9h2lpXelnuj/view