The game of Rock-Paper-Scissors is an instructive example of cyclic competition between competing populations or strategies in evolutionary biology and game theory, where no single species is an overall winner. Mathematically, this cyclic behaviour can be modelled by ordinary differential equations containing heteroclinic cycles: sequences of equilibria along with trajectories that connect them in a cyclic manner. In simple examples, the equilibria are all similar to each other, in that they all have the same number of non-zero components. Here, we investigate a class of robust heteroclinic cycles that does not satisfy the usual assumption that all equilibria have the same number of non-zero components and that connections between equilibria lie in subspaces of equal dimension. We refer to these as robust heteroclinic cycles in pluridimensions. Potential applications include modelling the dynamics of evolving populations when there are transitions between equilibria corresponding to mixed populations with different numbers of species. With a few reasonable assumptions, we show that there are four distinct examples of robust heteroclinic cycles in pluridimensions between four equilibria in four dimensions, and study their stability. This involves generalizing the usual Poincare return map approach by allowing non-square transition matrices. We provide numerical illustrations of each of the four examples. Although our examples are in four dimensions, we present them in a manner that can be readily adapted to other problems in higher dimensions.

Joint work with Sofia Castro