{"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":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0012266124020010","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","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.
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