Stability of quasi-simple heteroclinic cycles

TL;DR

This paper introduces a method to calculate the local stability index for quasi-simple heteroclinic cycles, which are important in understanding the stability of complex dynamical systems in symmetric and non-symmetric contexts.

Contribution

It provides a new approach to compute the stability index for quasi-simple cycles, extending applicability to various heteroclinic cycles including those in population dynamics and heteroclinic networks.

Findings

Method applies to all simple heteroclinic cycles of type Z

Extends to non-simple heteroclinic cycles in population models

Illustrated with a cycle in a Rock-Scissors-Paper network

Abstract

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 1-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 all simple heteroclinic cycles of type Z and to various heteroclinic cycles arising in population dynamics, namely non-simple heteroclinic cycles, as well as to cycles that are part of a heteroclinic network. We illustrate our results with a non-simple cycle present in a heteroclinic network of the Rock-Scissors-Paper game.