Asymptotic stability of heteroclinic cycles of type Y

TL;DR

This paper introduces and analyzes the asymptotic stability of a new class of heteroclinic cycles called type Y, which generalize previous types by allowing flow-invariant subspaces of varying dimensions.

Contribution

It extends the theory of heteroclinic cycle stability to type Y cycles, relaxing eigenvalue distinctness and robustness assumptions, and characterizes stability via transition matrices.

Findings

Stability determined by eigenvalues and eigenvectors of transition matrices.

Type Y cycles can be stable or fragmentarily stable depending on eigenvalue conditions.

The approach generalizes previous stability criteria for heteroclinic cycles.

Abstract

We investigate stability of a new class of heteroclinic cycles that we call heteroclinic cycles of type Y. The cycles can be regarded as a generalisation of heteroclinic cycles of type Z introduced in [Podvigina, Nonlinearity 25, 2012]. The type Y cycles differ from the cycles of type Z in the following: The trajectories comprising a cycle of type Y belong to flow-invariant subspaces that can be of different dimensions. Unlike in the most studies of the stability of heteroclinic cycles, we do not require that the eigenvalues of the linearisations of the dynamical system near the equilibria are distinct. Instead of the common assumption that the cycles are robust, we prescribe flow-invariance of certain subspaces. Similarly to type Z cycles, asymptotic stability and fragmentary asymptotic stability of type Y cycles is determined by the eigenvalues and eigenvectors of transition matrices. The matrices are products of basic transition matrices that depend on the eigenvalues of linearisations and the dimensions of the contracting subspaces.