A robust Lyapunov criterion for non-oscillatory behaviors in biological interaction networks

TL;DR

This paper develops a robust Lyapunov-based method to verify non-oscillatory behavior in biological interaction networks, enabling analysis of complex biochemical systems where traditional methods fail.

Contribution

It introduces a constructive approach using piecewise linear Lyapunov functions to certify global non-oscillatory dynamics in nonlinear biological networks.

Findings

Successfully applied to enzymatic cycles with unknown global behavior

Rules out oscillations, limit cycles, and quasi-periodic solutions

Provides a practical algorithm for biological network analysis

Abstract

We introduce the notion of non-oscillation, propose a constructive method for its robust verification, and study its application to biological interaction networks (also known as, chemical reaction networks). We begin by revisiting Muldowney's result on non-existence of periodic solutions based on the study of the variational system of the second additive compound of the Jacobian of a nonlinear system. We show that exponential stability of the latter rules out limit cycles, quasi-periodic solutions, and broad classes of oscillatory behavior. We focus then on nonlinear equations arising in biological interaction networks with general kinetics, and we show that the dynamics of the aforementioned variational system can be embedded in a linear differential inclusion. We then propose algorithms for constructing piecewise linear Lyapunov functions to certify global robust non-oscillatory behavior. Finally, we apply our techniques to study several regulated enzymatic cycles where available methods are not able to provide any information about their qualitative global behavior.