Simple chemical systems with chaos

TL;DR

This paper identifies and characterizes simple three-dimensional chemical dynamical systems capable of chaos, revealing their structural properties and demonstrating their widespread existence with minimal complexity.

Contribution

It proves fundamental properties of chaotic chemical dynamical systems and computationally finds minimal examples with few monomials and reactions, filling a gap in existing literature.

Findings

Chaotic CDSs in 3D have at least 6 monomials, including a negative quadratic one.

20 chaotic 3D CDSs with minimal monomials and quadratic reactions were identified.

Chaotic CDSs are more common and structurally simple than previously documented.

Abstract

A number of simple chaotic three-dimensional dynamical systems (DSs) with quadratic polynomials on the right-hand sides are reported in the literature, containing exactly 5 or 6 monomials of which only 1 or 2 are quadratic. However, none of these simple systems are chemical dynamical systems (CDSs) - a special subset of polynomial DSs that model the dynamics of mass-action chemical reaction networks (CRNs). In particular, only a small number of three-dimensional quadratic CDSs with chaos are reported, all of which have at least 9 monomials and at least 3 quadratics, with CRNs containing at least 7 reactions and at least 3 quadratic ones. To bridge this gap, in this paper we prove some basic properties of chaotic CDSs, including that those in three dimensions have at least 6 monomials, at least one of which is negative and quadratic. We then use these results to computationally find 20 chaotic three-dimensional CDSs with 6 monomials and as few as 4 quadratics, or 7 monomials and as few as 2 quadratics. At the CRN level, some of these systems have 4 reactions of which only 3 are quadratic, or 5 reactions with only 2 being quadratic. These results quantify structural complexity of chaotic CDSs, and indicate that they are ubiquitous.