{"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":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0012266124060028","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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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.
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