Coupled hypergraph maps and chaotic cluster synchronization

¹ Faculty of Mathematics, Technical University of Munich - Boltzmannstr. 3, 85748 Garching b. München, Germany
² Complexity Science Hub Vienna - Josefstädter Str. 39, 1080 Vienna, Austria
³ Max Planck Institute for Mathematics in the Sciences - Inselstr. 22, 04103 Leipzig, Germany
⁴ The Alan Turing Institute, The British Library - London NW1 2DB, UK
⁵ University of Southampton - University Rd, Southampton SO17 1BJ, UK
⁶ Santa Fe Institute for the Sciences of Complexity - 1399 Hyde Park Road Santa Fe, NM 87501, USA

^(a) ckuehn@ma.tum.de (corresponding author)

Received: 18 May 2021
Accepted: 3 August 2021

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.