Asymptotic stability of heteroclinic cycles of type Y

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.