Behaviour of trajectories near a two-cycle heteroclinic network

Abstract

Heteroclinic networks and cycles are invariant sets comprised of interacting nodes connected by heteroclinic trajectories. Often the sets are not asymptotically stable but attract a positive measure set from its small neighbourhood. This property is called fragmentary asymptotic stability (f.a.s.). The definition implies that if a stable cycle is a subset of a heteroclinic network, then the entire network is stable. In general, the converse is wrong. In the examples given in the literature, the presence of spiralling due to complex eigenvalues in the linearization around an equilibrium implies switching between subcycles of the f.a.s. network, thus preventing individual cycles from being stable. We study the behaviour of trajectories near a heteroclinic network comprised of two cycles where the eigenvalues of the linearizations are real. The trajectories can be attracted to one of the cycles, or they can switch regularly or irregularly between them. To describe regular switching, we introduce the notions of an omnicycle and its trail-stability, and prove conditions for trail-stability of an omnicycle in the considered network.