{"title":"Limit Cycles in a Class of Planar Discontinuous Piecewise Quadratic Differential Systems with a Non-regular Line of Discontinuity (II)","authors":"Dongping He, Jaume Llibre","doi":"10.1007/s00009-024-02714-0","DOIUrl":null,"url":null,"abstract":"<p>In our previous work, we have studied the limit cycles for a class of discontinuous piecewise quadratic polynomial differential systems with a non-regular line of discontinuity, which is formed by two rays starting from the origin and forming an angle <span>\\(\\alpha = \\pi /2\\)</span>. The unperturbed system is the quadratic uniform isochronous center <span>\\(\\dot{x} = -y + x y\\)</span>, <span>\\(\\dot{y} = x + y^2\\)</span> with a family of periodic orbits surrounding the origin. In this paper, we continue to investigate this kind of piecewise differential systems, but now the angle between the two rays is <span>\\(\\alpha \\in (0,\\pi /2)\\cup [3\\pi /2,2\\pi )\\)</span>. Using the Chebyshev theory, we prove that the maximum number of hyperbolic limit cycles that can bifurcate from these periodic orbits using the averaging theory of first order is exactly 8 for <span>\\(\\alpha \\in (0,\\pi /2)\\cup [3\\pi /2,2\\pi )\\)</span>. Together with our previous work, which concerns on the case of <span>\\(\\alpha =\\pi /2\\)</span>, we can conclude that using the averaging theory of first order the maximum number of hyperbolic limit cycles is exactly 8, when this quadratic center is perturbed inside the above-mentioned classes separated by a non-regular line of discontinuity with <span>\\(\\alpha \\in (0,\\pi /2]\\cup [3\\pi /2,2\\pi )\\)</span>.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00009-024-02714-0","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
In our previous work, we have studied the limit cycles for a class of discontinuous piecewise quadratic polynomial differential systems with a non-regular line of discontinuity, which is formed by two rays starting from the origin and forming an angle \(\alpha = \pi /2\). The unperturbed system is the quadratic uniform isochronous center \(\dot{x} = -y + x y\), \(\dot{y} = x + y^2\) with a family of periodic orbits surrounding the origin. In this paper, we continue to investigate this kind of piecewise differential systems, but now the angle between the two rays is \(\alpha \in (0,\pi /2)\cup [3\pi /2,2\pi )\). Using the Chebyshev theory, we prove that the maximum number of hyperbolic limit cycles that can bifurcate from these periodic orbits using the averaging theory of first order is exactly 8 for \(\alpha \in (0,\pi /2)\cup [3\pi /2,2\pi )\). Together with our previous work, which concerns on the case of \(\alpha =\pi /2\), we can conclude that using the averaging theory of first order the maximum number of hyperbolic limit cycles is exactly 8, when this quadratic center is perturbed inside the above-mentioned classes separated by a non-regular line of discontinuity with \(\alpha \in (0,\pi /2]\cup [3\pi /2,2\pi )\).
期刊介绍:
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.