Arbitrarily large heteroclinic networks in fixed low-dimensional state space

Abstract

We consider heteroclinic networks between $n\in \mathbb{N}$ nodes where the only connections are those linking each node to its two subsequent neighbouring ones. Using a construction method where all nodes are placed in a single one-dimensional space and the connections lie in coordinate planes, we show that it is possible to robustly realise these networks in ${\mathbb{R}}^6$ for any number of nodes $n$ using a polynomial vector field. This bound on the space dimension (while the number of nodes in the network goes to $\infty$) is a novel phenomenon and a step towards more efficient realisation methods for given connection structures in terms of the required number of space dimensions. We briefly discuss some stability properties of the generated heteroclinic objects.