Mean-field theory for double-well systems on degree-heterogeneous networks

TL;DR

This paper develops a degree-based mean-field theory to analyze the dynamics of coupled double-well systems on networks, providing insights into tipping points and multistage transitions in complex systems.

Contribution

It introduces a low-dimensional mean-field approach to understand coupled tipping dynamics in degree-heterogeneous networks, capturing critical transitions accurately.

Findings

The theory accurately predicts tipping points and equilibria.

Numerical simulations support multistage tipping transitions.

The approach simplifies high-dimensional network dynamics.

Abstract

Many complex dynamical systems in the real world, including ecological, climate, financial, and power-grid systems, often show critical transitions, or tipping points, in which the system's dynamics suddenly transit into a qualitatively different state. In mathematical models, tipping points happen as a control parameter gradually changes and crosses a certain threshold. Tipping elements in such systems may interact with each other as a network, and understanding the behavior of interacting tipping elements is a challenge because of the high dimensionality originating from the network. Here we develop a degree-based mean-field theory for a prototypical double-well system coupled on a network with the aim of understanding coupled tipping dynamics with a low-dimensional description. The method approximates both the onset of the tipping point and the position of equilibria with a reasonable accuracy. Based on the developed theory and numerical simulations, we also provide evidence for multistage tipping point transitions in networks of double-well systems.