On designing heteroclinic networks from graphs

TL;DR

This paper introduces two methods to realize complex directed graphs as robust heteroclinic networks in dynamical systems, enabling better understanding and modeling of biological and cognitive processes.

Contribution

It presents the simplex and cylinder realizations, allowing arbitrary graphs to be embedded as heteroclinic networks in ODE flows under certain conditions.

Findings

Simplex realization embeds graphs on an (n_v-1)-simplex, requiring no one- or two-cycles.

Cylinder realization embeds graphs in (n_e+1)-dimensional space, requiring no one-cycles.

The paper discusses noise effects and vertex memory in the heteroclinic networks.

Abstract

Robust heteroclinic networks are invariant sets that can appear as attractors in symmetrically coupled or otherwise constrained dynamical systems. These networks may have a very complicated structure that is poorly understood and 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 graphs can be realised as robust heteroclinic networks that are attracting. This paper presents two methods of realizing arbitrarily complex directed graphs as robust heteroclinic networks for flows generated by ODEs---we say the ODEs {\em 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 find 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.