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Heteroclinic Dynamics in Network Dynamical Systems with Higher-Order Interactions

Christian Bick, Sören von der Gracht

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.

Heteroclinic Dynamics in Network Dynamical Systems with Higher-Order Interactions

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.
Paper Structure (33 sections, 17 theorems, 80 equations, 4 figures)

This paper contains 33 sections, 17 theorems, 80 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Heteroclinic Cycles and Higher-Order Interactions
  3. Robust Heteroclinic Cycles
  4. Guckenheimer--Holmes Cycle
  5. Constructing Robust Heteroclinic Cycles
  6. Hypergraphs and Network Dynamics
  7. Network Dynamics on Hypergraphs
  8. Interactions for One-Dimensional Node Dynamics
  9. Obstruction to Heteroclinic Cycles for Network Dynamics on Undirected Hypergraphs
  10. The Guckenheimer--Holmes Cycle
  11. The Field Cycle
  12. The Guckenheimer--Holmes Cycle in Directed Hypergraphs
  13. Realisation in a Directed Classical Network
  14. Only True 2-to-1 Connections
  15. Directed Edges and True 2-to-1 Connections
  16. ...and 18 more sections

Key Result

Theorem 3.1

The Guckenheimer--Holmes system eq:gh_cubic cannot be realized in a network dynamical system on an undirected hypergraph $\mathcal{H}$ on three vertices eq:hypernetdyn-undirected.

Figures (4)

  • Figure 1: Sketch of the Guckenheimer--Holmes (left) and Field cycle (right).
  • Figure 2:
  • Figure 3:
  • Figure 4:

Theorems & Definitions (45)

  • Theorem 3.1
  • proof
  • Remark 1
  • Theorem 3.2
  • proof
  • Remark 2
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • ...and 35 more