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Networks of interacting nodes connected by edges arise in almost every branchof scientific enquiry. The connectivity structure of the network can force theexistence of invariant subspaces, which would not arise in generic dynamicalsystems. These invariant subspaces can result in the appearance of robustheteroclinic cycles, which would otherwise be structurally unstable. Typically,the dynamics near a stable heteroclinic cycle is non-ergodic: mean residencetimes near the fixed points in the cycle are undefined, and there is apersistent slowing down. In this paper, we examine ring networks withnearest-neighbour or nearest-m-neighbour coupling, and show that there existclasses of heteroclinic cycles in the phase space of the dynamics. We show thatthere is always at least one heteroclinic cycle which can be asymptoticallystable, and thus the attracting dynamics of the network are expected to benon-ergodic. We conjecture that much of this behaviour persists in lessstructured networks and as such, non-ergodic behaviour is somehow typical.

Stability of heteroclinic cycles in rings of coupled oscillators