Characterization of the Riccati and Abel Polynomial Differential Systems Having Invariant Algebraic Curves

IF 1.9 4区 数学 Q2 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS
Jaume Giné, Jaume Llibre
{"title":"Characterization of the Riccati and Abel Polynomial Differential Systems Having Invariant Algebraic Curves","authors":"Jaume Giné, Jaume Llibre","doi":"10.1142/s0218127424500664","DOIUrl":null,"url":null,"abstract":"<p>The Riccati polynomial differential systems are differential systems of the form <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>x</mi></mrow><mrow><mi>′</mi></mrow></msup><mo>=</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></math></span><span></span>, <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>y</mi></mrow><mrow><mi>′</mi></mrow></msup><mo>=</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">+</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>1</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mi>y</mi><mo stretchy=\"false\">+</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span><span></span>, where <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>c</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span><span></span> and <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span><span></span> for <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>i</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></math></span><span></span> are polynomial functions. We characterize all the Riccati polynomial differential systems having an invariant algebraic curve. We show that the coefficients of the first four highest degree terms of the polynomial in the variable <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>y</mi></math></span><span></span> defining the invariant algebraic curve determine completely the Riccati differential system. A similar result is obtained for any Abel polynomial differential system.</p>","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":null,"pages":null},"PeriodicalIF":1.9000,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Journal of Bifurcation and Chaos","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1142/s0218127424500664","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0

Abstract

The Riccati polynomial differential systems are differential systems of the form x=c0(x), y=b0(x)+b1(x)y+b2(x)y2, where c0 and bi for i=0,1,2 are polynomial functions. We characterize all the Riccati polynomial differential systems having an invariant algebraic curve. We show that the coefficients of the first four highest degree terms of the polynomial in the variable y defining the invariant algebraic curve determine completely the Riccati differential system. A similar result is obtained for any Abel polynomial differential system.

具有不变代数曲线的里卡蒂和阿贝尔多项式微分系统的特征
Riccati 多项式微分方程系是形式为 x′=c0(x),y′=b0(x)+b1(x)y+b2(x)y2 的微分方程系,其中 i=0,1,2 的 c0 和 bi 是多项式函数。我们描述了所有具有不变代数曲线的 Riccati 多项式微分方程系统。我们证明,定义不变代数曲线的变量 y 的多项式的前四个最高阶项的系数完全决定了 Riccati 微分系统。对于任何阿贝尔多项式微分方程系,都可以得到类似的结果。
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
International Journal of Bifurcation and Chaos
International Journal of Bifurcation and Chaos 数学-数学跨学科应用
CiteScore
4.10
自引率
13.60%
发文量
237
审稿时长
2-4 weeks
期刊介绍: The International Journal of Bifurcation and Chaos is widely regarded as a leading journal in the exciting fields of chaos theory and nonlinear science. Represented by an international editorial board comprising top researchers from a wide variety of disciplines, it is setting high standards in scientific and production quality. The journal has been reputedly acclaimed by the scientific community around the world, and has featured many important papers by leading researchers from various areas of applied sciences and engineering. The discipline of chaos theory has created a universal paradigm, a scientific parlance, and a mathematical tool for grappling with complex dynamical phenomena. In every field of applied sciences (astronomy, atmospheric sciences, biology, chemistry, economics, geophysics, life and medical sciences, physics, social sciences, ecology, etc.) and engineering (aerospace, chemical, electronic, civil, computer, information, mechanical, software, telecommunication, etc.), the local and global manifestations of chaos and bifurcation have burst forth in an unprecedented universality, linking scientists heretofore unfamiliar with one another''s fields, and offering an opportunity to reshape our grasp of reality.
×
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