Chemical systems with chaos

TL;DR

This paper demonstrates that chemical dynamical systems can exhibit complex chaotic behaviors previously observed in abstract polynomial systems, by developing a theory to map polynomial dynamics into chemically realizable models.

Contribution

The authors develop a theory to map polynomial dynamical systems into chemical systems with fewer nonlinear terms, enabling the realization of chaos in chemical reaction networks.

Findings

Quadratic and cubic chemical systems can display various types of chaos.

Explicit examples of chemical systems with one-wing, two-wing, and hidden chaos are constructed.

Chaos can occur in chemical systems with a small number of nonlinear reactions.

Abstract

Three-dimensional polynomial dynamical systems (DSs) can display chaos with various properties already in the quadratic case with only one or two quadratic monomials. In particular, one-wing chaos is reported in quadratic DSs with only one quadratic monomial, while two-wing and hidden chaos in quadratic DSs with only two quadratic monomials. However, none of the reported DSs can be realized with chemical reactions. To bridge this gap, in this paper, we investigate chaos in chemical dynamical systems (CDSs) - a subset of polynomial DSs that can model the dynamics of mass-action chemical reaction networks. To this end, we develop a fundamental theory for mapping polynomial DSs into CDSs of the same dimension and with a reduced number of non-linear terms. Applying this theory, we show that, under suitable robustness assumptions, quadratic CDSs, and cubic CDSs with only one cubic, can display a rich set of chaotic solutions already in three dimensions. Furthermore, we construct some relatively simple three-dimensional examples, including a quadratic CDS with one-wing chaos and three quadratics, a cubic CDS with two-wing chaos and one cubic, and a quadratic CDS with hidden chaos and five quadratics.