Francisco Braun , Leonardo Pereira Costa da Cruz , Joan Torregrosa
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We apply this technique to non-smooth perturbations of the four families of isochronous centers of the Loud family, <span><math><msub><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, <span><math><msub><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, <span><math><msub><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>, and <span><math><msub><mrow><mi>S</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>, as well as to non-smooth perturbations of non-smooth centers given by putting different <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>’s in each zone. To show the coverage of our approach, we apply its first order, which is equivalent to averaging theory of the first order, in perturbations of the already mentioned centers considering all the straight lines through the origin. Then we apply the second order of our approach to perturbations of the above centers for a specific oblique straight line. Here in order to argue we introduce certain blow-ups in the perturbative parameters. As a consequence of our study, we obtain examples of piecewise quadratic systems with at least 12 limit cycles. By analyzing two previous works of the literature claiming much more limit cycles we found some mistakes in the calculations. Therefore, the best lower bound for the number of limit cycles of a piecewise quadratic system is up to now the 12 limit cycles found in the present paper.</p></div>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the number of limit cycles in piecewise planar quadratic differential systems\",\"authors\":\"Francisco Braun , Leonardo Pereira Costa da Cruz , Joan Torregrosa\",\"doi\":\"10.1016/j.nonrwa.2024.104124\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We consider piecewise quadratic perturbations of centers of piecewise quadratic systems in two zones determined by a straight line through the origin. By means of expansions of the displacement map, we are able to find isolated zeros of it, without dealing with the unsurprising difficult integrals inherent in the usual averaging approach. We apply this technique to non-smooth perturbations of the four families of isochronous centers of the Loud family, <span><math><msub><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, <span><math><msub><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, <span><math><msub><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>, and <span><math><msub><mrow><mi>S</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>, as well as to non-smooth perturbations of non-smooth centers given by putting different <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>’s in each zone. 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引用次数: 0
摘要
我们考虑的是由通过原点的直线确定的两个区域内的逐次二次系统中心的逐次二次扰动。通过对位移图的展开,我们可以找到其孤立的零点,而无需处理通常的平均方法中固有的、不足为奇的困难积分。我们将这一技术应用于劳德家族四个等时中心系列(S1、S2、S3 和 S4)的非光滑扰动,以及通过在每个区域放置不同 Si 而得到的非光滑中心的非光滑扰动。为了说明我们的方法的覆盖范围,我们将其一阶(相当于一阶平均理论)应用于上述中心的扰动,考虑所有通过原点的直线。然后,我们将二阶方法应用于特定斜直线的上述中心扰动。为了论证这一点,我们在扰动参数中引入了某些炸毁。通过研究,我们得到了至少有 12 个极限循环的片断二次系统的例子。通过分析之前两篇声称极限周期更多的文献,我们发现了计算中的一些错误。因此,片断二次系统极限循环次数的最佳下限就是本文发现的 12 次极限循环。
On the number of limit cycles in piecewise planar quadratic differential systems
We consider piecewise quadratic perturbations of centers of piecewise quadratic systems in two zones determined by a straight line through the origin. By means of expansions of the displacement map, we are able to find isolated zeros of it, without dealing with the unsurprising difficult integrals inherent in the usual averaging approach. We apply this technique to non-smooth perturbations of the four families of isochronous centers of the Loud family, , , , and , as well as to non-smooth perturbations of non-smooth centers given by putting different ’s in each zone. To show the coverage of our approach, we apply its first order, which is equivalent to averaging theory of the first order, in perturbations of the already mentioned centers considering all the straight lines through the origin. Then we apply the second order of our approach to perturbations of the above centers for a specific oblique straight line. Here in order to argue we introduce certain blow-ups in the perturbative parameters. As a consequence of our study, we obtain examples of piecewise quadratic systems with at least 12 limit cycles. By analyzing two previous works of the literature claiming much more limit cycles we found some mistakes in the calculations. Therefore, the best lower bound for the number of limit cycles of a piecewise quadratic system is up to now the 12 limit cycles found in the present paper.
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.