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arXiv · 2303.17922

Arbitrarily large heteroclinic networks in fixed low-dimensional state space

Sofia B. S. D. Castro·Alexander Lohse
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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.

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Sofia B. S. D. Castro, Alexander Lohse. 2023-03-31. Arbitrarily large heteroclinic networks in fixed low-dimensional state space. https://arxiv.org/abs/2303.17922

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