A closed form for Jacobian reconstruction from timeseries and its application as an early warning signal in network dynamics

TL;DR

This paper introduces a closed-form analytical method to reconstruct the Jacobian matrix from time series data, enabling early warning signals for critical transitions in complex networked systems like ecological models.

Contribution

The authors derive a novel closed-form solution for Jacobian estimation from time series, facilitating analysis of nonlinear dynamical systems and early warning detection.

Findings

Accurate Jacobian reconstruction from time series is feasible.

Leading eigenvalue serves as an effective early warning indicator.

Method performs well on complex ecological network models.

Abstract

The Jacobian matrix of a dynamical system describes its response to perturbations. Conversely one can estimate the Jacobian matrix by carefully monitoring how the system responds to environmental noise. Here we present a closed form analytical solution for the calculation of a system's Jacobian from a timeseries. Being able to access a system's Jacobian enables us to perform a broad range of mathematical analyses by which deeper insights into the system can be gained. Here we consider in particular the computation of the leading Jacobian eigenvalue as an early warning signal for critical transition. To illustrate this approach we apply it to ecological meta-foodweb models, which are strongly nonlinear dynamical multi-layer networks. Our analysis shows that accurate results can be obtained, although the data demand of the method is still high.