{"title":"Two Contrasting Examples of Multidimensional Differential Systems with Lyapunov Extreme Instability","authors":"A. A. Bondarev","doi":"10.1134/s0001434624010036","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> Using specific examples, we constructively show that, in dimensions greater than <span>\\(1\\)</span>, the Lyapunov extreme instability of a differential system, i.e., the property that the phase curves of all nonzero solutions starting sufficiently close to zero leave any prescribed compact set, does not imply that these solutions go arbitrarily far away from zero in the sense of Perron or in the upper-limit sense as <span>\\(t\\to\\infty\\)</span>. Namely, we construct two Lyapunov extremely unstable systems such that all solutions of the first system tend to zero, while the solutions of the second system are divided into two types: all nonzero solutions starting in the closed unit ball tend to infinity in norm, and all the other solutions tend to zero. Further, both systems constructed in the paper have zero first approximation along the zero solution. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Notes","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0001434624010036","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
Using specific examples, we constructively show that, in dimensions greater than \(1\), the Lyapunov extreme instability of a differential system, i.e., the property that the phase curves of all nonzero solutions starting sufficiently close to zero leave any prescribed compact set, does not imply that these solutions go arbitrarily far away from zero in the sense of Perron or in the upper-limit sense as \(t\to\infty\). Namely, we construct two Lyapunov extremely unstable systems such that all solutions of the first system tend to zero, while the solutions of the second system are divided into two types: all nonzero solutions starting in the closed unit ball tend to infinity in norm, and all the other solutions tend to zero. Further, both systems constructed in the paper have zero first approximation along the zero solution.
期刊介绍：
Mathematical Notes is a journal that publishes research papers and review articles in modern algebra, geometry and number theory, functional analysis, logic, set and measure theory, topology, probability and stochastics, differential and noncommutative geometry, operator and group theory, asymptotic and approximation methods, mathematical finance, linear and nonlinear equations, ergodic and spectral theory, operator algebras, and other related theoretical fields. It also presents rigorous results in mathematical physics.
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