{"title":"具有多项式换向器的六度均匀等时中心系统的全局相位特征","authors":"Li-na Guo, Ai-yong Chen, Shuai-feng Zhao","doi":"10.1007/s10255-024-1081-z","DOIUrl":null,"url":null,"abstract":"<p>This paper studies the global phase portraits of uniform isochronous centers system of degree six with polynomial commutator. Such systems have the form <span>\\(\\dot x = - y + xf(x,\\,y),\\,\\,\\dot y = x + yf(x,\\,y)\\)</span>, where <i>f</i>(<i>x, y</i>) = <i>a</i><sub>1</sub><i>x</i> + <i>a</i><sub>2</sub><i>xy</i> + <i>a</i><sub>3</sub><i>xy</i><sup>2</sup> + <i>a</i><sub>4</sub><i>xy</i><sup>3</sup> + <i>a</i><sub>5</sub><i>xy</i><sup>4</sup> = <i>xσ</i>(<i>y</i>), and any zero of 1 + <i>a</i><sub>1</sub><i>y</i> + <i>a</i><sub>2</sub><i>y</i><sup>2</sup> + <i>a</i><sub>3</sub><i>y</i><sup>3</sup> + <i>a</i><sub>4</sub><i>y</i><sup>4</sup> + <i>a</i><sub>5</sub><i>y</i><sup>5</sup>, <span>\\(y = \\bar y\\)</span> is an invariant straight line. At last, all global phase portraits are drawn on the Poincaré disk.</p>","PeriodicalId":6951,"journal":{"name":"Acta Mathematicae Applicatae Sinica, English Series","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Global Phase Portraits of Uniform Isochronous Centers System of Degree Six with Polynomial Commutator\",\"authors\":\"Li-na Guo, Ai-yong Chen, Shuai-feng Zhao\",\"doi\":\"10.1007/s10255-024-1081-z\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>This paper studies the global phase portraits of uniform isochronous centers system of degree six with polynomial commutator. Such systems have the form <span>\\\\(\\\\dot x = - y + xf(x,\\\\,y),\\\\,\\\\,\\\\dot y = x + yf(x,\\\\,y)\\\\)</span>, where <i>f</i>(<i>x, y</i>) = <i>a</i><sub>1</sub><i>x</i> + <i>a</i><sub>2</sub><i>xy</i> + <i>a</i><sub>3</sub><i>xy</i><sup>2</sup> + <i>a</i><sub>4</sub><i>xy</i><sup>3</sup> + <i>a</i><sub>5</sub><i>xy</i><sup>4</sup> = <i>xσ</i>(<i>y</i>), and any zero of 1 + <i>a</i><sub>1</sub><i>y</i> + <i>a</i><sub>2</sub><i>y</i><sup>2</sup> + <i>a</i><sub>3</sub><i>y</i><sup>3</sup> + <i>a</i><sub>4</sub><i>y</i><sup>4</sup> + <i>a</i><sub>5</sub><i>y</i><sup>5</sup>, <span>\\\\(y = \\\\bar y\\\\)</span> is an invariant straight line. At last, all global phase portraits are drawn on the Poincaré disk.</p>\",\"PeriodicalId\":6951,\"journal\":{\"name\":\"Acta Mathematicae Applicatae Sinica, English Series\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-06-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Acta Mathematicae Applicatae Sinica, English Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s10255-024-1081-z\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematicae Applicatae Sinica, English Series","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10255-024-1081-z","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Global Phase Portraits of Uniform Isochronous Centers System of Degree Six with Polynomial Commutator
This paper studies the global phase portraits of uniform isochronous centers system of degree six with polynomial commutator. Such systems have the form \(\dot x = - y + xf(x,\,y),\,\,\dot y = x + yf(x,\,y)\), where f(x, y) = a1x + a2xy + a3xy2 + a4xy3 + a5xy4 = xσ(y), and any zero of 1 + a1y + a2y2 + a3y3 + a4y4 + a5y5, \(y = \bar y\) is an invariant straight line. At last, all global phase portraits are drawn on the Poincaré disk.
期刊介绍：
Acta Mathematicae Applicatae Sinica (English Series) is a quarterly journal established by the Chinese Mathematical Society. The journal publishes high quality research papers from all branches of applied mathematics, and particularly welcomes those from partial differential equations, computational mathematics, applied probability, mathematical finance, statistics, dynamical systems, optimization and management science.
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