An algorithm for finding weakly reversible deficiency zero realizations of polynomial dynamical systems

TL;DR

This paper presents an algorithm to identify weakly reversible deficiency zero realizations of polynomial dynamical systems, which simplifies their analysis by ensuring stable, predictable behavior.

Contribution

The paper introduces a novel algorithm for finding weakly reversible deficiency zero realizations of polynomial dynamical systems, aiding in their mathematical analysis.

Findings

Algorithm successfully finds $WR_0$ realizations when they exist.

Systems with $WR_0$ realizations have simple, stable dynamics.

The method facilitates the analysis of complex polynomial systems.

Abstract

Systems of differential equations with polynomial right-hand sides are very common in applications. On the other hand, their mathematical analysis is very challenging in general, due to the possibility of complex dynamics: multiple basins of attraction, oscillations, and even chaotic dynamics. Even if we restrict our attention to mass-action systems, all of these complex dynamical behaviours are still possible. On the other hand, if a polynomial dynamical system has a weakly reversible deficiency zero ($WR_0$) realization, then its dynamics is known to be remarkably simple: oscillations and chaotic dynamics are ruled out and, up to linear conservation laws, there exists a single positive steady state, which is asymptotically stable. Here we describe an algorithm for finding $WR_0$ realizations of polynomial dynamical systems, whenever such realizations exist.