Stability of heteroclinic cycles: a new approach

TL;DR

This paper introduces a new asymptotic method to analyze the stability of heteroclinic cycles in a three-dimensional network, with applications to game theory and computational dynamics.

Contribution

It presents a novel approach using a projective map to determine cycle stability, applicable to general networks and aiding numerical analysis.

Findings

Network stability depends on parameter ranges and interior equilibrium.

The projective map reduces complex dynamics to a one-dimensional analysis.

The method is potentially useful for broader classes of networks.

Abstract

This paper analyses the stability of cycles within a heteroclinic network lying in a three-dimensional manifold formed by six cycles, for a one-parameter model developed in the context of game theory. We show the asymptotic stability of the network for a range of parameter values compatible with the existence of an interior equilibrium and we describe an asymptotic technique to decide which cycle (within the network) is visible in numerics. The technique consists of reducing the relevant dynamics to a suitable one-dimensional map, the so called \emph{projective map}. Stability of the fixed points of the projective map determines the stability of the associated cycles. The description of this new asymptotic approach is applicable to more general types of networks and is potentially useful in computational dynamics.