Limit cycles appearing from the perturbation of a cubic isochronous center

Limit cycles appearing from the perturbation of a cubic isochronous center | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Limit cycles appearing from the perturbation of a cubic isochronous center Jihua Yang, Qipeng Zhang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6356777/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 12 May, 2025 Read the published version in Qualitative Theory of Dynamical Systems → Version 1 posted 12 You are reading this latest preprint version Abstract For the polynomial differential system$$\dot{x}=-y+\sum\limits_{i+j=3}\alpha_{i,j}x^iy^j,\ \dot{y}=x+\sum\limits_{i+j=3}\beta_{i,j}x^iy^j,\ \alpha_{i,j},\beta_{i,j}\in\mathbb{R},$$ Pleshkan (Differ. Equations, 1969) established that the origin is an isochronous center of this system if and only if it can be transformed into one of the canonical forms $S^ _1$, $S^ _2$, $S^ _3$ or $S^ _4$. Except for case $S^ _1$, the bifurcations of limit cycles in these four types of isochronous differential systems remain unexplored. In this paper, we focus on the bifurcation of limit cycles for the system $S^ _2$ under perturbations by an arbitrary polynomial vector field. By employing the Abelian integral, we derive an upper bound for the number of limit cycles that can emerge from such perturbations. The lower bounds are also provided for $n=1, 2, 3, 4$, and numerical simulations are conducted. Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 12 May, 2025 Read the published version in Qualitative Theory of Dynamical Systems → Version 1 posted Editorial decision: Revision requested 14 Apr, 2025 Reviews received at journal 14 Apr, 2025 Reviews received at journal 13 Apr, 2025 Reviews received at journal 10 Apr, 2025 Reviewers agreed at journal 07 Apr, 2025 Reviewers agreed at journal 05 Apr, 2025 Reviewers agreed at journal 03 Apr, 2025 Reviewers agreed at journal 03 Apr, 2025 Reviewers invited by journal 03 Apr, 2025 Editor assigned by journal 03 Apr, 2025 Submission checks completed at journal 03 Apr, 2025 First submitted to journal 01 Apr, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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