Simple chemical systems with chaos

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.