{"title":"用平均理论求平面微分系统极限环","authors":"Houdeifa Melki, A. Makhlouf","doi":"10.30538/psrp-oma2021.0095","DOIUrl":null,"url":null,"abstract":"In this article, we consider the limit cycles of a class of planar polynomial differential systems of the form \\[\\dot{x}=-y+\\varepsilon (1+\\sin ^{n}\\theta )xP(x,y)\\] \\[ \\dot{y}=x+\\varepsilon (1+\\cos ^{m}\\theta )yQ(x,y), \\] where \\(P(x,y)\\) and \\(Q(x,y)\\) are polynomials of degree \\(n_{1}\\) and \\(n_{2}\\) respectively and \\(\\varepsilon\\) is a small parameter. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of a linear center \\( \\dot{x}=-y, \\dot{y}=x,\\) by using the averaging theory of first order.","PeriodicalId":52741,"journal":{"name":"Open Journal of Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2021-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Limit cycles of a planar differential system via averaging theory\",\"authors\":\"Houdeifa Melki, A. Makhlouf\",\"doi\":\"10.30538/psrp-oma2021.0095\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this article, we consider the limit cycles of a class of planar polynomial differential systems of the form \\\\[\\\\dot{x}=-y+\\\\varepsilon (1+\\\\sin ^{n}\\\\theta )xP(x,y)\\\\] \\\\[ \\\\dot{y}=x+\\\\varepsilon (1+\\\\cos ^{m}\\\\theta )yQ(x,y), \\\\] where \\\\(P(x,y)\\\\) and \\\\(Q(x,y)\\\\) are polynomials of degree \\\\(n_{1}\\\\) and \\\\(n_{2}\\\\) respectively and \\\\(\\\\varepsilon\\\\) is a small parameter. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of a linear center \\\\( \\\\dot{x}=-y, \\\\dot{y}=x,\\\\) by using the averaging theory of first order.\",\"PeriodicalId\":52741,\"journal\":{\"name\":\"Open Journal of Mathematical Analysis\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2021-12-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Open Journal of Mathematical Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.30538/psrp-oma2021.0095\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Open Journal of Mathematical Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.30538/psrp-oma2021.0095","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Limit cycles of a planar differential system via averaging theory
In this article, we consider the limit cycles of a class of planar polynomial differential systems of the form \[\dot{x}=-y+\varepsilon (1+\sin ^{n}\theta )xP(x,y)\] \[ \dot{y}=x+\varepsilon (1+\cos ^{m}\theta )yQ(x,y), \] where \(P(x,y)\) and \(Q(x,y)\) are polynomials of degree \(n_{1}\) and \(n_{2}\) respectively and \(\varepsilon\) is a small parameter. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of a linear center \( \dot{x}=-y, \dot{y}=x,\) by using the averaging theory of first order.