{"title":"On the Existence of Periodic Solutions of a System of Second-Order Ordinary Differential Equations with a Quasihomogeneous Nonlinearity","authors":"A. N. Naimov, M. V. Bystretsky","doi":"10.1134/s0012266124050112","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> In the present paper, we study an a priori estimate and the existence of periodic solutions\nof a given period for a system of second-order ordinary differential equations with the main\nquasihomogeneous nonlinearity. It is proved that an a priori estimate of periodic solutions takes\nplace if the corresponding unperturbed system does not have nonzero bounded solutions. Under\nthe conditions of the a priori estimate, using methods for calculating the mapping degree of vector\nfields, a criterion for the existence of periodic solutions is stated and proved for any perturbation\nin a given class. The results obtained differ from earlier results in that the set of zeros of the main\nnonlinearity is not taken into account.\n</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"383 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0012266124050112","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In the present paper, we study an a priori estimate and the existence of periodic solutions
of a given period for a system of second-order ordinary differential equations with the main
quasihomogeneous nonlinearity. It is proved that an a priori estimate of periodic solutions takes
place if the corresponding unperturbed system does not have nonzero bounded solutions. Under
the conditions of the a priori estimate, using methods for calculating the mapping degree of vector
fields, a criterion for the existence of periodic solutions is stated and proved for any perturbation
in a given class. The results obtained differ from earlier results in that the set of zeros of the main
nonlinearity is not taken into account.
期刊介绍：
Differential Equations is a journal devoted to differential equations and the associated integral equations. The journal publishes original articles by authors from all countries and accepts manuscripts in English and Russian. The topics of the journal cover ordinary differential equations, partial differential equations, spectral theory of differential operators, integral and integral–differential equations, difference equations and their applications in control theory, mathematical modeling, shell theory, informatics, and oscillation theory. The journal is published in collaboration with the Department of Mathematics and the Division of Nanotechnologies and Information Technologies of the Russian Academy of Sciences and the Institute of Mathematics of the National Academy of Sciences of Belarus.
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