Chemical systems with chaos

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.