{"title":"A Refined Global Poincaré–Bendixson Annulus with the Limit Cycle of the Rayleigh System","authors":"Y. Li, A. A. Grin, A. V. Kuzmich","doi":"10.1134/s0012266124060028","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> New methods for constructing two Dulac–Cherkas functions are developed using which a\nbetter, depending on the parameter <span>\\(\\lambda >0\\)</span>, inner\nboundary of the Poincaré–Bendixson annulus <span>\\(A(\\lambda ) \\)</span> is found for the Rayleigh system. A procedure is\nproposed for directly finding a polynomial whose zero level set contains a transversal oval used as\nthe outer boundary of <span>\\(A(\\lambda )\\)</span>. An\ninterval for <span>\\(\\lambda \\)</span> is specified with which the best outer boundary of\nthe annulus <span>\\( A(\\lambda )\\)</span> is a closed contour composed of two\narcs of the constructed oval and two arcs of unclosed curves of the zero level set of one of the\nDulac–Cherkas functions. Thus, a refined global Poincaré–Bendixson annulus for the\nlimit cycle of the Rayleigh system is presented.\n</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"103 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0012266124060028","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
Abstract
New methods for constructing two Dulac–Cherkas functions are developed using which a
better, depending on the parameter \(\lambda >0\), inner
boundary of the Poincaré–Bendixson annulus \(A(\lambda ) \) is found for the Rayleigh system. A procedure is
proposed for directly finding a polynomial whose zero level set contains a transversal oval used as
the outer boundary of \(A(\lambda )\). An
interval for \(\lambda \) is specified with which the best outer boundary of
the annulus \( A(\lambda )\) is a closed contour composed of two
arcs of the constructed oval and two arcs of unclosed curves of the zero level set of one of the
Dulac–Cherkas functions. Thus, a refined global Poincaré–Bendixson annulus for the
limit cycle of the Rayleigh system is presented.
期刊介绍：
Differential Equations is a journal devoted to differential equations and the associated integral equations. The journal publishes original articles by authors from all countries and accepts manuscripts in English and Russian. The topics of the journal cover ordinary differential equations, partial differential equations, spectral theory of differential operators, integral and integral–differential equations, difference equations and their applications in control theory, mathematical modeling, shell theory, informatics, and oscillation theory. The journal is published in collaboration with the Department of Mathematics and the Division of Nanotechnologies and Information Technologies of the Russian Academy of Sciences and the Institute of Mathematics of the National Academy of Sciences of Belarus.
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