{"title":"Periodic Perturbations of Oscillators on a Plane","authors":"Yu. N. Bibikov, E. V. Vasil’eva","doi":"10.1134/s1063454124010059","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The results of research carried out in the 21st century at the Department of Differential Equations of St. Petersburg State University are reviewed. The subject of study is the problem of stability of the zero solution to the second-order equation describing periodic perturbations of the oscillator with a nonlinear restoring force under reversible and conservative perturbations. Such perturbations are classified as transcendental perturbations, for which the solution of the problem of stability requires taking into account all terms in the expansion in a series of the right-hand side of the equation. The problem of stability under transcendental perturbations was formulated in 1893 by A.M. Lyapunov. The results regarding stability presented in this review were obtained using the methods of KAM theory: perturbations of the oscillator with infinitely small and infinitely large oscillation frequencies were considered; conditions for the existence of quasi-periodic solutions in any vicinity of the time axis are determined, from which the stability (not asymptotic) of the zero solution of the perturbed equation follows; and the conditions are found for the stability of the zero solution for a Hamiltonian system with two degrees of freedom, the unperturbed part of which is described by a pair of oscillators (in this case conservative perturbations are considered).</p>","PeriodicalId":43418,"journal":{"name":"Vestnik St Petersburg University-Mathematics","volume":"36 1","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Vestnik St Petersburg University-Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1134/s1063454124010059","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The results of research carried out in the 21st century at the Department of Differential Equations of St. Petersburg State University are reviewed. The subject of study is the problem of stability of the zero solution to the second-order equation describing periodic perturbations of the oscillator with a nonlinear restoring force under reversible and conservative perturbations. Such perturbations are classified as transcendental perturbations, for which the solution of the problem of stability requires taking into account all terms in the expansion in a series of the right-hand side of the equation. The problem of stability under transcendental perturbations was formulated in 1893 by A.M. Lyapunov. The results regarding stability presented in this review were obtained using the methods of KAM theory: perturbations of the oscillator with infinitely small and infinitely large oscillation frequencies were considered; conditions for the existence of quasi-periodic solutions in any vicinity of the time axis are determined, from which the stability (not asymptotic) of the zero solution of the perturbed equation follows; and the conditions are found for the stability of the zero solution for a Hamiltonian system with two degrees of freedom, the unperturbed part of which is described by a pair of oscillators (in this case conservative perturbations are considered).
期刊介绍：
Vestnik St. Petersburg University, Mathematics is a journal that publishes original contributions in all areas of fundamental and applied mathematics. It is the prime outlet for the findings of scientists from the Faculty of Mathematics and Mechanics of St. Petersburg State University. Articles of the journal cover the major areas of fundamental and applied mathematics. The following are the main subject headings: Mathematical Analysis; Higher Algebra and Numbers Theory; Higher Geometry; Differential Equations; Mathematical Physics; Computational Mathematics and Numerical Analysis; Statistical Simulation; Theoretical Cybernetics; Game Theory; Operations Research; Theory of Probability and Mathematical Statistics, and Mathematical Problems of Mechanics and Astronomy.
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
