We examine the relation between a heteroclinic network as a flow-invariant set and directed graphs of possible connections between nodes. In particular, we show that there are robust realizations of a large class of transitive directed graphs that are not complete (i.e. not all unstable manifolds of nodes are included) but almost complete (i.e. complete up to a set of zero measure in the unstable manifold) and equable (i.e. all sets of connections from a node have the same dimension). Moreover, some of these almost complete and equable realizations have "completions", namely by adding extra nodes we can produce a larger network that is complete and may even be asymptotically stable. This is joint work with Sofia Castro (Porto) and Peter Ashwin (Exeter).

# On realizing graphs as complete heteroclinic networks

Alexander Lohse

Hamburg University