Global Analysis of Riccati Quadratic Differential Systems

Joan C. Artés, J. Llibre, D. Schlomiuk, N. Vulpe
{"title":"Global Analysis of Riccati Quadratic Differential Systems","authors":"Joan C. Artés, J. Llibre, D. Schlomiuk, N. Vulpe","doi":"10.1142/s0218127424500044","DOIUrl":null,"url":null,"abstract":"In this paper, we study the family of quadratic Riccati differential systems. Our goal is to obtain the complete topological classification of this family on the Poincaré disk compactification of the plane. The family was partially studied before but never from a truly global viewpoint. Our approach is global and we use geometry to achieve our goal. The geometric analysis we perform is via the presence of two invariant parallel straight lines in any generic Riccati system. We obtain a total of 119 topologically distinct phase portraits for this family. Furthermore, we give the complete bifurcation diagram in the 12-dimensional space of parameters of this family in terms of invariant polynomials, meaning that it is independent of the normal forms in which the systems may be presented. This bifurcation diagram provides an algorithm to decide for any given quadratic system in any form it may be presented, whether it is a Riccati system or not, and in case it is to provide its phase portrait.","PeriodicalId":13688,"journal":{"name":"Int. J. Bifurc. Chaos","volume":"4 4","pages":"2450004:1-2450004:46"},"PeriodicalIF":0.0000,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Int. J. Bifurc. Chaos","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1142/s0218127424500044","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

In this paper, we study the family of quadratic Riccati differential systems. Our goal is to obtain the complete topological classification of this family on the Poincaré disk compactification of the plane. The family was partially studied before but never from a truly global viewpoint. Our approach is global and we use geometry to achieve our goal. The geometric analysis we perform is via the presence of two invariant parallel straight lines in any generic Riccati system. We obtain a total of 119 topologically distinct phase portraits for this family. Furthermore, we give the complete bifurcation diagram in the 12-dimensional space of parameters of this family in terms of invariant polynomials, meaning that it is independent of the normal forms in which the systems may be presented. This bifurcation diagram provides an algorithm to decide for any given quadratic system in any form it may be presented, whether it is a Riccati system or not, and in case it is to provide its phase portrait.
里卡提二次微分系统的全局分析
本文研究二次里卡提微分方程系。我们的目标是在平面的波恩卡莱圆盘压缩上获得该族的完整拓扑分类。以前曾对该族进行过部分研究,但从未从真正的全局角度进行过研究。我们的方法是全局性的,我们利用几何来实现我们的目标。我们进行的几何分析是通过在任何通用里卡提系统中存在两条不变的平行直线来实现的。我们总共得到了 119 个拓扑学上不同的相位肖像。此外,我们用不变多项式给出了该族 12 维参数空间中的完整分岔图,这意味着它与系统可能呈现的正常形式无关。该分岔图提供了一种算法,可用于判定任何给定二次方程系统的任何呈现形式是否为里卡提系统,如果是,则提供其相位图。
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