# Polynomial Dynamical Systems, Reaction Networks, and Toric Differential Inclusions

## TL;DR

This paper links polynomial dynamical systems to reaction network models using Euclidean embedded graphs, introduces toric differential inclusions, and explores their implications for longstanding conjectures in dynamical systems.

## Contribution

It establishes a geometric framework connecting polynomial systems with reaction networks and introduces toric differential inclusions as a tool for analyzing global stability conjectures.

## Key findings

- Polynomial dynamical systems can be represented by Euclidean embedded graphs.
- Toric differential inclusions have a distinctive geometric structure.
- Embedding polynomial systems into toric inclusions offers new approaches to global stability conjectures.

## Abstract

Some of the most common mathematical models in biology, chemistry, physics, and engineering, are polynomial dynamical systems, i.e., systems of differential equations with polynomial right-hand sides. Inspired by notions and results that have been developed for the analysis of reaction networks in biochemistry and chemical engineering, we show that any polynomial dynamical system on the positive orthant $\mathbb R^n_{> 0}$ can be regarded as being generated by an oriented graph embedded in $\mathbb R^n$, called $\mathit{Euclidean \ embedded \ graph}$. This allows us to recast key conjectures about reaction network models (such as the Global Attractor Conjecture, or the Persistence Conjecture) into more general versions about some important classes of polynomial dynamical systems. Then, we introduce $\mathit{toric \ differential \ inclusions}$, which are piecewise constant autonomous dynamical systems with a remarkable geometric structure. We show that if a Euclidean embedded graph $G$ has some reversibility properties, then any polynomial dynamical system generated by $G$ can be embedded into a toric differential inclusion. We discuss how this embedding suggests an approach for the proof of the Global Attractor Conjecture and Persistence Conjecture.