Coupled hypergraph maps and chaotic cluster synchronization

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.

Dynamics on higher-order networks: a review

“…MSF formalism has also been generalized for hypernetworks in [ 76 ] to treat the linear stability of the phenomenon of synchronization, where the special class of Laplace-type interactions has also been examined. The dynamical systems known as coupled map lattices are extended to the scenario of higher-order interactions, namely to the concept of coupled hypergraph maps [ 77 ]. The process of cluster synchronization is investigated in such a system through the analysis of a hypernetwork Laplacian for different chaotic discrete dynamical systems.…”

Random walks and Laplacians on hypergraphs: When do they match?

“…Moreover, Section 5 is devoted to the comparison between the hypergraph random walk Laplacians and the normalized Laplacians for oriented hypergraphs that were motivated by chemical interactions and introduced in [21]; for investigations into properties of these hypergraph Laplacians see e.g. [6,22,29,32,33]. We observe that, while these latter operators do not encode properties of random walks in general, their spectrum is nevertheless capable of encoding qualitative properties of the hypergraph that are not encoded by the spectra of the random walk Laplacians.…”

“…Moreover, in Section 6 we study mathematical foundations of Coupled Hypergraph Maps (CHMs): dynamical systems on hypergraphs that were introduced in [6] as a generalization of Coupled Map Lattices (CMLs) [23]. CHMs are based on Laplacian-type coupling, therefore the generated dynamics changes when a different type of Laplace operator is considered.…”

Intralayer and interlayer synchronization in multiplex network with higher-order interactions

“…In this regard, most of the studies on synchronization involving higher-order structures consider simplicial complexes [31][32][33][34] to represent non-pairwise interactions due to their simple geometrical representation. Very few have taken hypergraphs [35][36][37][38][39] into consideration. The extension of complete synchronization to puniform hypergraphs have been studied in Ref.…”