{"title":"论基本方程组解的旋转和极限循环","authors":"Grigor Barsegian","doi":"10.1515/gmj-2024-2042","DOIUrl":null,"url":null,"abstract":"This article discusses the rotations (windings) of solutions to the basic system of equations <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi>y</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>=</m:mo> <m:mrow> <m:msub> <m:mi>F</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2042_eq_0102.png\"/> <jats:tex-math>{y^{\\prime}=F_{1}(x,y)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi>x</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>=</m:mo> <m:mrow> <m:msub> <m:mi>F</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2042_eq_0099.png\"/> <jats:tex-math>{x^{\\prime}=F_{2}(x,y)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. This allows us to return to the topic of known limit cycles from a much broader point of view, in particular, it makes it possible to describe the conditions for the existence of limit cycles.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"71 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the rotations and limit cycles of solutions to the basic system of equations\",\"authors\":\"Grigor Barsegian\",\"doi\":\"10.1515/gmj-2024-2042\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This article discusses the rotations (windings) of solutions to the basic system of equations <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:msup> <m:mi>y</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>=</m:mo> <m:mrow> <m:msub> <m:mi>F</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_gmj-2024-2042_eq_0102.png\\\"/> <jats:tex-math>{y^{\\\\prime}=F_{1}(x,y)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:msup> <m:mi>x</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>=</m:mo> <m:mrow> <m:msub> <m:mi>F</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_gmj-2024-2042_eq_0099.png\\\"/> <jats:tex-math>{x^{\\\\prime}=F_{2}(x,y)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. This allows us to return to the topic of known limit cycles from a much broader point of view, in particular, it makes it possible to describe the conditions for the existence of limit cycles.\",\"PeriodicalId\":55101,\"journal\":{\"name\":\"Georgian Mathematical Journal\",\"volume\":\"71 1\",\"pages\":\"\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-08-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Georgian Mathematical Journal\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/gmj-2024-2042\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Georgian Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/gmj-2024-2042","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
本文讨论基本方程组 y ′ = F 1 ( x , y ) {y^{\prime}=F_{1}(x,y)} 和 x ′ = F 2 ( x , y ) {x^{\prime}=F_{2}(x,y)} 的旋转（绕组）解。这样，我们就可以从更广阔的视角回到已知极限循环的话题上来，特别是，它使我们有可能描述极限循环存在的条件。
On the rotations and limit cycles of solutions to the basic system of equations
This article discusses the rotations (windings) of solutions to the basic system of equations y′=F1(x,y){y^{\prime}=F_{1}(x,y)} and x′=F2(x,y){x^{\prime}=F_{2}(x,y)}. This allows us to return to the topic of known limit cycles from a much broader point of view, in particular, it makes it possible to describe the conditions for the existence of limit cycles.
期刊介绍：
The Georgian Mathematical Journal was founded by the Georgian National Academy of Sciences and A. Razmadze Mathematical Institute, and is jointly produced with De Gruyter. The concern of this international journal is the publication of research articles of best scientific standard in pure and applied mathematics. Special emphasis is put on the presentation of results obtained by Georgian mathematicians.
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