Given a heteroclinic network, there is an associated graph such that the vertices of the graph correspond to the equilibria of the network and an edge corresponds to a connection between equilibria. Classification of ac-networks is carried out by describing all possible types of asso- ciated graphs.

We prove sufficient conditions for asymptotic stability of certain robust heteroclinic networks. The proof is given as a series of theorems and lemmas that are applicable to the ac-networks and to more general types of networks. Finally, we apply these results to derive conditions for asymptotic stability of several heteroclinic networks.

Joint work with Isabel Labouriau and Sofia Castro.