Coupled Hypergraph Maps and Chaotic Cluster Synchronization

TL;DR

This paper introduces coupled hypergraph maps (CHMs), a new class of higher-order dynamical systems on hypergraphs, and analyzes their complex chaotic and periodic synchronization behaviors using spectral theory and numerical methods.

Contribution

It proposes the novel CHMs framework, combining hypergraph spectral analysis with stability and numerical methods to explore complex synchronization phenomena.

Findings

Robust regions of chaotic cluster synchronization identified

Key differences between Laplacian and hypergraph Laplacian coupling

Detection of periodic and quasi-periodic patterns

Abstract

Coupled map lattices (CMLs) are prototypical dynamical systems on networks/graphs. They exhibit complex patterns generated via the interplay of diffusive/Laplacian coupling and nonlinear reactions modelled by a single iterated map at each node; the maps are often taken as unimodal, e.g., logistic or tent maps. In this letter, we propose a class of higher-order coupled dynamical systems involving the hypergraph Laplacian, which we call coupled hypergraph maps (CHMs). By combining linearized (in-)stability analysis of synchronized states, hypergraph spectral theory, and numerical methods, we detect robust regions of chaotic cluster synchronization occurring in parameter space upon varying coupling strength and the main bifurcation parameter of the unimodal map. Furthermore, we find key differences between Laplacian and hypergraph Laplacian coupling and detect various other classes of periodic and quasi-periodic patterns. The results show the high complexity of coupled graph maps and indicate that they might be an excellent universal model class to understand the similarities and differences between dynamics on classical graphs and dynamics on hypergraphs.