On designing heteroclinic networks from graphs

Robust heteroclinic networks are invariant sets that can appear as attractors in symmetrically coupled or otherwise constrained dynamical systems. These networks may have a complicated structure determined to a large extent by the constraints and dimension of the system. As these networks are of great interest as dynamical models of biological and cognitive processes, it is useful to understand how particular directed graphs can be realised as attracting robust heteroclinic networks between states in phase space. This paper presents two methods of realising arbitrarily complex directed graphs as robust heteroclinic networks for flows generated by ODEs—we say the ODEs realise the graphs as heteroclinic networks between equilibria that represent the vertices. Suppose we have a directed graph on n v vertices with n e edges. The “simplex realisation” embeds the graph as an invariant set of a flow on an ( n v − 1 ) -simplex. This method realises the graph as long as it is one- and two-cycle free. The “cylinder realisation” embeds a graph as an invariant set of a flow on a ( n e + 1 ) -dimensional space. This method realises the graph as long as it is one-cycle free. In both cases we realise the graph as an invariant set within an attractor, and discuss some illustrative examples, including the influence of noise and parameters on the dynamics. In particular we show that the resulting heteroclinic network may or may not display “memory” of the vertices visited.