Chemical systems with limit cycles

Radek Erban, Hye-Won Kang

Published 2022-11-10Version 1

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 $K \in {\mathbb N}$, we show that there exists a CRN such that its ODE model has at least $K$ stable limit cycles. In particular, we show that $N(K) \le K+2$, 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.