Dynamical systems on hypergraphs

TL;DR

This paper introduces a framework for modeling high-order interactions in dynamical systems using hypergraphs, analyzing pattern formation under different coupling conditions with applications to ecological and chemical models.

Contribution

It presents a novel hypergraph-based approach to study pattern emergence in complex dynamical systems with high-order interactions.

Findings

Conditions for spontaneous pattern formation identified.

Application to generalized Volterra system demonstrated.

Application to Brusselator model shown.

Abstract

We present a general framework that enables one to model high-order interaction among entangled dynamical systems, via hypergraphs. Several relevant processes can be ideally traced back to the proposed scheme. We shall here solely elaborate on the conditions that seed the spontaneous emergence of patterns, spatially heterogeneous solutions resulting from the many-body interaction between fundamental units. In particular we will focus, on two relevant settings. First, we will assume long-ranged mean field interactions between populations, and then turn to considering diffusive-like couplings. Two applications are presented, respectively to a generalised Volterra system and the Brusselator model.