Arbitrarily large heteroclinic networks in fixed low-dimensional state space

Abstract

We consider heteroclinic networks between n∈N nodes where the only connections are those linking each node to its two subsequent neighboring 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 realize these networks in R6 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 ∞) is a novel phenomenon and a step toward more efficient realization 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.