{"title":"Logistic Equation with Long Delay Feedback","authors":"S. A. Kashchenko","doi":"10.1134/s0012266124020010","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study the local dynamics of the delay logistic equation with an additional feedback\ncontaining a large delay. Critical cases in the problem of stability of the zero equilibrium state are\nidentified, and it is shown that they are infinite-dimensional. The well-known methods for\nstudying local dynamics based on the theory of invariant integral manifolds and normal forms do\nnot apply here. The methods of infinite-dimensional normalization proposed by the author are\nused and developed. As the main results, special nonlinear boundary value problems of parabolic\ntype are constructed, which play the role of normal forms. They determine the leading terms of\nthe asymptotic expansions of solutions of the original equation and are called quasinormal forms.\n</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"24 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-06-05","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/s0012266124020010","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
We study the local dynamics of the delay logistic equation with an additional feedback
containing a large delay. Critical cases in the problem of stability of the zero equilibrium state are
identified, and it is shown that they are infinite-dimensional. The well-known methods for
studying local dynamics based on the theory of invariant integral manifolds and normal forms do
not apply here. The methods of infinite-dimensional normalization proposed by the author are
used and developed. As the main results, special nonlinear boundary value problems of parabolic
type are constructed, which play the role of normal forms. They determine the leading terms of
the asymptotic expansions of solutions of the original equation and are called quasinormal forms.
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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