{"title":"Stability indices of non-hyperbolic equilibria in two-dimensional systems of ODEs","authors":"Alexander Lohse","doi":"10.1080/14689367.2022.2119941","DOIUrl":null,"url":null,"abstract":"We consider families of systems of two-dimensional ordinary differential equations with the origin 0 as a non-hyperbolic equilibrium. For any number , we show that it is possible to choose a parameter in these equations such that the stability index is precisely . In contrast to that, for a hyperbolic equilibrium x it is known that either or . Furthermore, we discuss a system with an equilibrium that is locally unstable but globally attracting, highlighting some subtle differences between the local and non-local stability indices.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1080/14689367.2022.2119941","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 1
Abstract
We consider families of systems of two-dimensional ordinary differential equations with the origin 0 as a non-hyperbolic equilibrium. For any number , we show that it is possible to choose a parameter in these equations such that the stability index is precisely . In contrast to that, for a hyperbolic equilibrium x it is known that either or . Furthermore, we discuss a system with an equilibrium that is locally unstable but globally attracting, highlighting some subtle differences between the local and non-local stability indices.
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