General multi-scale estimates for Lyapunov data of Perron-Frobenius matrices. The case of diluted autocatalytic chemical reaction networks

TL;DR

This paper introduces a recursive, multi-scale method inspired by quantum field theory to estimate Lyapunov data of Perron-Frobenius matrices in autocatalytic chemical networks, even with limited kinetic rate information.

Contribution

It presents a novel recursive multi-scale algorithm for precise Lyapunov data estimation in chemical reaction networks, inspired by renormalization group techniques.

Findings

Provides explicit rational function estimates of Lyapunov eigenvalues and eigenvectors.

Compatible with scarce kinetic rate data in chemical systems.

Applicable to diluted autocatalytic reaction networks.

Abstract

Autocatalytic chemical reaction networks are dynamical systems whose linearization around zero, dX/dt = AX, is represented by a Perron-Frobenius matrix A with positive Lyapunov exponent; this exponent gives the growth rate of the species concentration vector X in the diluted regime, i.e. in a vicinity of zero. We introduce here a new, general recursive procedure providing precise quantitative information about Lyapunov data, namely, the Lyapunov eigenvalue, and left and right eigenvectors. Our estimates are based on a multi-scale algorithm inspired from Wilson's renormalization group method in quantum field theory, and Markov chain arguments introduced in (Nghe & Unterberger). They are compatible with the very scarce knowledge of kinetic rates (coefficients of A) generally available in chemistry, and take on the form of simple rational functions of the latter.