For a vector field, heteroclinic cycle is a cyclic sequence of equilibria and connecting trajectories. A heteroclinic network is a connected union of finitely many heteroclinic cycles. It is possible to assign a graph to a heteroclinic network by identifying equilibria and connections in the network with nodes and edges in the graph.

Associated to heteroclinic networks there are interesting, and often complex, dynamics which are realised when the network is asymptotically stable. A necessary condition for asymptotic stability is that the unstable manifold of all equilibria be contained in the heteroclinic network and such a network is called complete. Completeness ensures the network is visible as an attractor, but may be too strict a condition. An almost complete network is such that only a set of measure zero in the unstable manifold is not contained in the network, thus ensuring strong attraction properties.

It is possible to construct a vector field that supports the existence of a heteroclinic network, N, corresponding to a particular graph. We show how to achieve this construction in such a way that the network N is (almost) complete. The completion process is not unique and neither is it indifferent from the point of view of the dynamics near the (almost) complete network.

This is joint work with P. Ashwin (Exeter) and A. Lohse (Hamburg).