Simple chemical systems with chaos

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 five or six monomials of which only one or two 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 nine monomials and at least three quadratics, with CRNs containing at least seven reactions and at least three 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 six monomials, at least one of which is negative and quadratic. We then use these results to numerically identify 20 chaotic three-dimensional CDSs with positive Lyapunov exponents, each having either six monomials with as few as four quadratic terms, or seven monomials with as few as two quadratic terms. At the CRN level, some of these systems have four reactions, of which only three are quadratic, or five reactions, of which only two are quadratic. These results quantify structural complexity of chaotic CDSs, and indicate that they are ubiquitous.