{"title":"Studying the Upper Bounds of the Numbers of Zeros of Abelian Integrals by the Law of Polynomials","authors":"Lijun Hong, Jinling Liu, Xiaochun Hong","doi":"10.1142/s0218127424500810","DOIUrl":null,"url":null,"abstract":"<p>For the quadratic reversible systems of genus one, all of their periodic orbits are higher-order algebraic curves. When they are perturbed by polynomials of degree <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi></math></span><span></span>, the numbers of zeros of their Abelian integrals will change and we study the upper bounds of these numbers by using the methods of Riccati equation and Picard–Fuchs equation. We consider both the highest and lowest degrees of polynomials, and more importantly, we consider the law of polynomials and the range of values for their variables. Consequently, some laws of the polynomials are discovered and many upper bounds are obtained, and these upper bounds are sharper than the results obtained by other techniques.</p>","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":"58 1","pages":""},"PeriodicalIF":1.9000,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Journal of Bifurcation and Chaos","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1142/s0218127424500810","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
Abstract
For the quadratic reversible systems of genus one, all of their periodic orbits are higher-order algebraic curves. When they are perturbed by polynomials of degree , the numbers of zeros of their Abelian integrals will change and we study the upper bounds of these numbers by using the methods of Riccati equation and Picard–Fuchs equation. We consider both the highest and lowest degrees of polynomials, and more importantly, we consider the law of polynomials and the range of values for their variables. Consequently, some laws of the polynomials are discovered and many upper bounds are obtained, and these upper bounds are sharper than the results obtained by other techniques.
对于属一的二次可逆系统，它们的周期轨道都是高阶代数曲线。当它们受到 n 阶多项式的扰动时，它们的阿贝尔积分的零点个数将发生变化，我们利用里卡蒂方程和皮卡尔-富克斯方程的方法研究了这些零点个数的上限。我们既考虑多项式的最高度，也考虑多项式的最低度，更重要的是，我们考虑多项式的规律及其变量的取值范围。因此，我们发现了多项式的一些规律，得到了许多上界，而且这些上界比其他技术得到的结果更加尖锐。
期刊介绍：
The International Journal of Bifurcation and Chaos is widely regarded as a leading journal in the exciting fields of chaos theory and nonlinear science. Represented by an international editorial board comprising top researchers from a wide variety of disciplines, it is setting high standards in scientific and production quality. The journal has been reputedly acclaimed by the scientific community around the world, and has featured many important papers by leading researchers from various areas of applied sciences and engineering.
The discipline of chaos theory has created a universal paradigm, a scientific parlance, and a mathematical tool for grappling with complex dynamical phenomena. In every field of applied sciences (astronomy, atmospheric sciences, biology, chemistry, economics, geophysics, life and medical sciences, physics, social sciences, ecology, etc.) and engineering (aerospace, chemical, electronic, civil, computer, information, mechanical, software, telecommunication, etc.), the local and global manifestations of chaos and bifurcation have burst forth in an unprecedented universality, linking scientists heretofore unfamiliar with one another''s fields, and offering an opportunity to reshape our grasp of reality.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
