Chemical systems with limit cycles

Abstract

The dynamics of a chemical reaction network (CRN) is often modelled under the assumption of mass action kinetics by a system of ordinary differential equations (ODEs) with polynomial right-hand sides that describe the time evolution of concentrations of chemical species involved. Given an arbitrarily large integer K∈N, we show that there exists a CRN such that its ODE model has at least K stable limit cycles. In particular, we show that N(K)≤K+2, where N(K) is the minimal number of chemical species that a CRN with K limit cycles can have. Bounds on the minimal number of chemical reactions and on the minimal size of CRNs with at most second-order kinetics are also provided for CRNs with K limit cycles.