Regular and irregular cycling near a heteroclinic network

Abstract

Heteroclinic networks are invariant sets containing more than one heteroclinic cycle. Such networks can appear robustly in equivariant vector fields. Previous authors have demonstrated that trajectories near heteroclinic networks can be attracted to one of a number of simultaneously ‘stable’ invariant subsets of the network. None of these invariant sets are asymptotically stable, but satisfy weaker definitions of stability. In this paper we discuss the behaviour of trajectories for one specific symmetric vector field. This vector field contains a robust heteroclinic network, and nearby trajectories display a variety of interesting dynamics. In particular, trajectories are observed to settle into a pattern of excursions around different parts of the network that we call ‘cycling sub-cycles’. Cycling patterns displaying different numbers of loops around the individual component cycles can be stable for the same parameter values, as can combinations of regular and irregular cycling. Analytic results for the regular cycling behaviour agree well with numerical simulations. We show that there exist parameter values where some trajectories display irregular cycling behaviour, in the sense that the numbers of loops around individual sub-cycles form a bounded aperiodic infinite sequence.