The stability of heteroclinic cycles may be obtained from the value of the local stability index along each connection of the cycle. We establish a way of calculating the local stability index for quasi-simple cycles: cycles whose connections are one-dimensional and contained in flow-invariant spaces of equal dimension. These heteroclinic cycles exist both in symmetric and non-symmetric contexts. We make one assumption on the dynamics along the connections to ensure that the transition matrices have a convenient form. Our method applies to some classes of simple heteroclinic cycles and to various heteroclinic cycles arising in population dynamics, namely non-simple cycles, as well as to heteroclinic cycles that are part of a network. We illustrate our results with a quasi-simple (non-simple) cycle in a heteroclinic network for the dynamics of the Rock-Scissors-Paper game. Using applications to price setting models, we further illustrate the contribution of the Rock-Scissors-Paper game to the understanding of cyclic dominance in two-player games.

References:

1. [1]  L. Garrido-da-Silva and S.B.S.D. Castro (2018a) Stability of quasi-simple heteroclinic cycles. Dynamical Systems: an International Journal, https://doi.org/10.1080/14689367.2018.1445701. 
2. [2]  L. Garrido-da-Silva and S.B.S.D. Castro (2018b) Cyclic dominance in a two-person Rock-Scissors-Paper game. arXiv:1607.08748.