# Planar chemical reaction systems with algebraic and non-algebraic limit cycles

## TL;DR

This paper explores how chemical reaction systems can produce multiple stable cycles, using mathematical models to determine their maximum number.

## Contribution

The paper introduces new lower bounds for limit cycles in chemical systems and constructs systems with multiple stable algebraic limit cycles.

## Key findings

- A chemical system with four stable algebraic limit cycles is constructed using a degree-4 algebraic curve.
- Lower bounds on modified Hilbert numbers are established for algebraic and non-algebraic limit cycles.
- The maximum number of ovals in a degree-4 algebraic curve is used to determine the number of stable limit cycles.

## Abstract

The Hilbert number H(n) is defined as the maximum number of limit cycles of a planar autonomous system of ordinary differential equations (ODEs) with right-hand sides containing polynomials of degree at most \documentclass[12pt]{minimal}
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\begin{document}$$n \in {{\mathbb {N}}}$$\end{document}n∈N. The dynamics of chemical reaction systems with two chemical species can be (under mass-action kinetics) described by such planar autonomous ODEs, where n is equal to the maximum order of the chemical reactions in the system. Analogues of the Hilbert number H(n) for three different classes of chemical reaction systems are investigated: (i) chemical systems with reactions up to the n-th order; (ii) systems with up to n-molecular chemical reactions; and (iii) weakly reversible chemical reaction networks. In each case (i), (ii) and (iii), the question on the number of limit cycles is considered. Lower bounds on the modified Hilbert numbers are provided for both algebraic and non-algebraic limit cycles. Furthermore, given a general algebraic curve \documentclass[12pt]{minimal}
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\begin{document}$$h(x,y)=0$$\end{document}h(x,y)=0 of degree \documentclass[12pt]{minimal}
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\begin{document}$$n_h \in {{\mathbb {N}}}$$\end{document}nh∈N and containing one or more ovals in the positive quadrant, a chemical system is constructed which has the oval(s) as its stable algebraic limit cycle(s). The ODEs describing the dynamics of the constructed chemical system contain polynomials of degree at most \documentclass[12pt]{minimal}
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\begin{document}$$n=2\,n_h+1.$$\end{document}n=2nh+1. Considering \documentclass[12pt]{minimal}
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\begin{document}$$n_h \ge 4,$$\end{document}nh≥4, the algebraic curve \documentclass[12pt]{minimal}
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\begin{document}$$h(x,y)=0$$\end{document}h(x,y)=0 can contain multiple closed components with the maximum number of ovals given by Harnack’s curve theorem as \documentclass[12pt]{minimal}
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\begin{document}$$1+(n_h-1)(n_h-2)/2$$\end{document}1+(nh-1)(nh-2)/2, which is equal to 4 for \documentclass[12pt]{minimal}
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\begin{document}$$n_h=4.$$\end{document}nh=4. Algebraic curve \documentclass[12pt]{minimal}
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\begin{document}$$h(x,y)=0$$\end{document}h(x,y)=0 with \documentclass[12pt]{minimal}
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\begin{document}$$n_h=4$$\end{document}nh=4 and the maximum number of four ovals is used to construct a chemical system which has four stable algebraic limit cycles.