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Simple chemical systems with chaos

Tomislav Plesa, Julien Clinton Sprott

TL;DR

The paper addresses whether chaotic dynamics can occur in the simplest possible chemical-dynamical systems (CDSs) in three dimensions. It proves two foundational results—the Sign theorem and the Nonlinearity theorem—that constrain the minimal monomial structure required for chaos, and then uses these insights to perform a large-scale computational search. The authors identify $20$ chaotic CDSs with only $6$ or $7$ monomials (and as few as $4$ quadratic reactions), demonstrating chaos in notably simple CRNs. This work shows that chaotic CDSs are relatively ubiquitous and provides a concrete catalog of minimal examples, while posing several open problems about the precise structural limits of chaos in CDSs.

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.

Simple chemical systems with chaos

TL;DR

The paper addresses whether chaotic dynamics can occur in the simplest possible chemical-dynamical systems (CDSs) in three dimensions. It proves two foundational results—the Sign theorem and the Nonlinearity theorem—that constrain the minimal monomial structure required for chaos, and then uses these insights to perform a large-scale computational search. The authors identify chaotic CDSs with only or monomials (and as few as quadratic reactions), demonstrating chaos in notably simple CRNs. This work shows that chaotic CDSs are relatively ubiquitous and provides a concrete catalog of minimal examples, while posing several open problems about the precise structural limits of chaos in CDSs.

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.
Paper Structure (6 sections, 3 theorems, 12 equations, 1 figure, 2 tables)

This paper contains 6 sections, 3 theorems, 12 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Theory
  4. Examples
  5. Discussion
  6. Appendix: Chemical reaction networks

Key Result

Theorem 3.1

$($Sign theorem$)$ Let $\mathbb{V} \subset \mathbb{R}_{\ge}^N$ be a compact invariant set for DS$(eq:DS)$. Assume that there exists $\mathbf{x}_0 \in \mathbb{V}$ such that the solution $\mathbf{x}(t;\mathbf{x}_0)$ of $(eq:DS)$ spends infinite amount of time in some non-empty compact set $\mathbb{K}

Figures (1)

  • Figure 1: Chemical systems with chaos. Numerically detected chaotic attractors in the $(x,y)$-space for the $20$ chemical systems presented in Tables \ref{['tab:one']}--\ref{['tab:two']}.

Theorems & Definitions (14)

  • Definition 2.1
  • Definition 2.2
  • Example 2.1
  • Example 2.2
  • Definition 2.3
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • ...and 4 more