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Chemical systems with limit cycles

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. Such a CRN can be constructed with reactions of at most second order provided that the number of chemical species grows linearly with $K$. Bounds on the minimal number of chemical species and the minimal number of chemical reactions are presented for CRNs with $K$ stable limit cycles and at most second order or seventh order kinetics. We also show that CRNs with only two chemical species can have $K$ stable limit cycles, when the order of chemical reactions grows linearly with $K$.