Heteroclinic Dynamics in Network Dynamical Systems with Higher-Order Interactions

TL;DR

This paper investigates the emergence of heteroclinic structures in network dynamical systems with higher-order interactions, highlighting the role of directed hypergraphs in supporting such complex dynamics.

Contribution

It demonstrates that directed hypergraphs enable heteroclinic structures in higher-order network dynamics, unlike undirected hypergraphs which face obstructions.

Findings

Directed hypergraphs support heteroclinic structures.

Undirected hypergraphs face obstructions to heteroclinic dynamics.

Heteroclinic structures organize global features of network dynamics.

Abstract

Heteroclinic structures organize global features of dynamical systems. We analyze whether heteroclinic structures can arise in network dynamics with higher-order interactions which describe the nonlinear interactions between three or more units. We find that while commonly analyzed model equations such as network dynamics on undirected hypergraphs may be useful to describe local dynamics such as cluster synchronization, they give rise to obstructions that allow to design heteroclinic structures in phase space. By contrast, directed hypergraphs break the homogeneity and lead to vector fields that support heteroclinic structures.