{"title":"Delay in Solving Autonomous Singularly Perturbed Equations Near an Unstable Equilibrium Position","authors":"K. S. Alybaev, A. M. Juraev, M. N. Nurmatova","doi":"10.1134/s1995080224600791","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>This paper considers an autonomous system of singularly perturbed equations of fast variables, consisting of <span>\\(2n\\)</span> first-order equations and one equation of a slow variable. The first approximation matrix of singularly perturbed equations has pairwise complex conjugate eigenvalues. The system has an equilibrium position, and the stability of the equilibrium position is lost by all eigenvalues at some value of the slow variable. It is proven that the solution of a singularly perturbed equation remains near an unstable equilibrium position during a finite time. Thus, the solution is delayed near the unstable equilibrium position. Early works considered cases when the stability of the equilibrium position is lost by one pair of complex conjugate eigenvalues.</p>","PeriodicalId":46135,"journal":{"name":"Lobachevskii Journal of Mathematics","volume":"13 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Lobachevskii Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1134/s1995080224600791","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
This paper considers an autonomous system of singularly perturbed equations of fast variables, consisting of \(2n\) first-order equations and one equation of a slow variable. The first approximation matrix of singularly perturbed equations has pairwise complex conjugate eigenvalues. The system has an equilibrium position, and the stability of the equilibrium position is lost by all eigenvalues at some value of the slow variable. It is proven that the solution of a singularly perturbed equation remains near an unstable equilibrium position during a finite time. Thus, the solution is delayed near the unstable equilibrium position. Early works considered cases when the stability of the equilibrium position is lost by one pair of complex conjugate eigenvalues.
期刊介绍：
Lobachevskii Journal of Mathematics is an international peer reviewed journal published in collaboration with the Russian Academy of Sciences and Kazan Federal University. The journal covers mathematical topics associated with the name of famous Russian mathematician Nikolai Lobachevsky (Lobachevskii). The journal publishes research articles on geometry and topology, algebra, complex analysis, functional analysis, differential equations and mathematical physics, probability theory and stochastic processes, computational mathematics, mathematical modeling, numerical methods and program complexes, computer science, optimal control, and theory of algorithms as well as applied mathematics. The journal welcomes manuscripts from all countries in the English language.
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