Chemical systems with limit cycles

TL;DR

This paper demonstrates that chemical reaction networks can be constructed to exhibit an arbitrarily large number of stable limit cycles, with bounds on species and reactions, including cases with only two species.

Contribution

It introduces methods to construct CRNs with many stable limit cycles, providing bounds on species and reactions, and explores minimal conditions for multiple cycles.

Findings

CRNs can have at least K stable limit cycles for any large K.

Construction of CRNs with reactions of at most second order and linearly growing species.

CRNs with only two species can also have K stable limit cycles with higher reaction order.

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 $K \in {\mathbb N}$, 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 $K$.