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

Gheorghe Craciun, Radek Erban
{"title":"Planar chemical reaction systems with algebraic and non-algebraic limit cycles","authors":"Gheorghe Craciun, Radek Erban","doi":"arxiv-2406.05057","DOIUrl":null,"url":null,"abstract":"The Hilbert number $H(n)$ is defined as the maximum number of limit cycles of\na planar autonomous system of ordinary differential equations (ODEs) with\nright-hand sides containing polynomials of degree at most $n \\in {\\mathbb N}$.\nThe dynamics of chemical reaction systems with two chemical species can be\n(under mass-action kinetics) described by such planar autonomous ODEs, where\n$n$ is equal to the maximum order of the chemical reactions in the system.\nGeneralizations of the Hilbert number $H(n)$ to three different classes of\nchemical reaction networks are investigated: (i) chemical systems with\nreactions up to the $n$-th order; (ii) systems with up to $n$-molecular\nchemical reactions; and (iii) weakly reversible chemical reaction networks. In\neach case (i), (ii) and (iii), the question on the number of limit cycles is\nconsidered. Lower bounds on the generalized Hilbert numbers are provided for\nboth algebraic and non-algebraic limit cycles. Furthermore, given a general\nalgebraic curve $h(x,y)=0$ of degree $n_h \\in {\\mathbb N}$ and containing one\nor more ovals in the positive quadrant, a chemical system is constructed which\nhas the oval(s) as its stable algebraic limit cycle(s). The ODEs describing the\ndynamics of the constructed chemical system contain polynomials of degree at\nmost $n=2\\,n_h+1.$ Considering $n_h \\ge 4,$ the algebraic curve $h(x,y)=0$ can\ncontain multiple closed components with the maximum number of ovals given by\nHarnack's curve theorem as $1+(n_h-1)(n_h-2)/2$, which is equal to 4 for\n$n_h=4.$ Algebraic curve $h(x,y)=0$ with $n_h=4$ and the maximum number of four\novals is used to construct a chemical system which has four stable algebraic\nlimit cycles.","PeriodicalId":501325,"journal":{"name":"arXiv - QuanBio - Molecular Networks","volume":"25 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - QuanBio - Molecular Networks","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2406.05057","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

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 $n \in {\mathbb N}$. 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. Generalizations of the Hilbert number $H(n)$ to three different classes of chemical reaction networks 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 generalized Hilbert numbers are provided for both algebraic and non-algebraic limit cycles. Furthermore, given a general algebraic curve $h(x,y)=0$ of degree $n_h \in {\mathbb N}$ 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 $n=2\,n_h+1.$ Considering $n_h \ge 4,$ the algebraic curve $h(x,y)=0$ can contain multiple closed components with the maximum number of ovals given by Harnack's curve theorem as $1+(n_h-1)(n_h-2)/2$, which is equal to 4 for $n_h=4.$ Algebraic curve $h(x,y)=0$ with $n_h=4$ and the maximum number of four ovals is used to construct a chemical system which has four stable algebraic limit cycles.
具有代数和非代数极限循环的平面化学反应系统
希尔伯特数$H(n)$被定义为一个平面自主常微分方程(ODEs)系统的最大极限循环数,该系统的右边包含{\mathbb N}$中最多$n /度的多项式。本文研究了希尔伯特数 $H(n)$对三类不同化学反应网络的泛化:(i) 具有高达 $n$ 三阶反应的化学系统;(ii) 具有高达 $n$ 分子化学反应的系统;以及 (iii) 弱可逆化学反应网络。在(i)、(ii)和(iii)的每种情况下,都考虑了极限循环次数的问题。为代数和非代数极限循环提供了广义希尔伯特数的下界。此外,给定一条在{\mathbb N}$中阶数为$n_h \的广义代数曲线$h(x,y)=0$,并且在正象限中包含一个或多个椭圆形,就可以构造出一个以椭圆形为其稳定代数极限循环的化学系统。考虑到 $n_h \ge 4,$代数曲线$h(x,y)=0$可以包含多个闭合分量,根据哈纳克曲线定理,椭圆的最大数目为$1+(n_h-1)(n_h-2)/2$,当$n_h=4 时等于 4。利用 n_h=4$ 的代数曲线 $h(x,y)=0$和最大椭圆数来构造一个化学系统,该系统有四个稳定的代数极限循环。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
自引率
0.00%
发文量
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信