Lyapunov Function Partial Differential Equations for Chemical Reaction Networks: Some Special Cases

TL;DR

This paper introduces a PDE-based method to generate Lyapunov functions for stability analysis in chemical reaction networks, applicable to various network types including complex balanced and certain higher-dimensional cases.

Contribution

It develops a novel PDE approach for Lyapunov function construction, extending stability analysis tools for complex and higher-dimensional chemical reaction networks.

Findings

Solution of PDEs is dissipative for any network.

Method effectively analyzes asymptotic stability of complex balanced networks.

Examples demonstrate the efficiency of the proposed approach.

Abstract

In this paper, we develop a method to generate the Lyapunov function for stability analysis for chemical reaction networks. Based on the Chemical Master Equation, we derive the Lyapunov Function partial differential equations (PDEs), whose solution approximates the scaling non-equilibrium potential and serves as the candidate Lyapunov function for the given network. We further prove that for any chemical reaction network the solution (if exists) of the PDEs is dissipative. Moreover, the proposed method of Lyapunov Function PDEs is qualified for analyzing the asymptotic stability of complex balanced networks, all networks with $1$-dimensional stoichiometric subspace and some special networks with more than $2$-dimensional stoichiometric subspace if some moderate conditions are added. Several examples are presented to illustrate the efficiency of the method.