{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,1]],"date-time":"2022-04-01T11:51:37Z","timestamp":1648813897700},"reference-count":8,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2015,4,1]],"date-time":"2015-04-01T00:00:00Z","timestamp":1427846400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Proc. Camb. Phil. Soc."],"published-print":{"date-parts":[[2015,7]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The notion of an exponential contraction is only one among many possible rates of contraction of a nonautonomous system, while for an autonomous system all contractions are exponential. We consider the notion of an <jats:italic>L<\/jats:italic><jats:sup>1<\/jats:sup> contraction that includes exponential contractions as a very particular case and that is naturally adapted to the variation-of-parameters formula. Both for discrete and continuous time, we show that under very general assumptions the notion of an <jats:italic>L<\/jats:italic><jats:sup>1<\/jats:sup> contraction persists under sufficiently small linear and nonlinear perturbations, also maintaining the type of stability. As a natural development, we establish a version of the Grobman\u2013Hartman theorem for nonlinear perturbations of an <jats:italic>L<\/jats:italic><jats:sup>1<\/jats:sup> contraction.<\/jats:p>","DOI":"10.1017\/s0305004115000158","type":"journal-article","created":{"date-parts":[[2015,4,1]],"date-time":"2015-04-01T14:20:21Z","timestamp":1427898021000},"page":"23-46","source":"Crossref","is-referenced-by-count":0,"title":["Stability of <i>L<\/i><sup>1<\/sup> contractions"],"prefix":"10.1017","volume":"159","author":[{"given":"LUIS","family":"BARREIRA","sequence":"first","affiliation":[]},{"given":"CLAUDIA","family":"VALLS","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2015,4,1]]},"reference":[{"key":"S0305004115000158_ref7","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(73)90245-X"},{"key":"S0305004115000158_ref4","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1963-0152718-3"},{"key":"S0305004115000158_ref1","first-page":"880","article-title":"Homeomorphism of systems of differential equations","volume":"128","author":"Grobman","year":"1959","journal-title":"Dokl. Akad. Nauk SSSR"},{"key":"S0305004115000158_ref2","first-page":"77","article-title":"Topological classification of neighbourhoods of a singularity in n-space","volume":"56","author":"Grobman","year":"1962","journal-title":"Mat. Sb. (N.S.)"},{"key":"S0305004115000158_ref3","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1960-0121542-7"},{"key":"S0305004115000158_ref5","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0396(69)90083-7"},{"key":"S0305004115000158_ref8","doi-asserted-by":"publisher","DOI":"10.2307\/2373513"},{"key":"S0305004115000158_ref6","first-page":"263","article-title":"On the local structure of hyperbolic points in Banach spaces","volume":"40","author":"Palis","year":"1968","journal-title":"An. Acad. Brasil. Ci."}],"container-title":["Mathematical Proceedings of the Cambridge Philosophical Society"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0305004115000158","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,20]],"date-time":"2019-04-20T18:57:49Z","timestamp":1555786669000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0305004115000158\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,4,1]]},"references-count":8,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2015,7]]}},"alternative-id":["S0305004115000158"],"URL":"https:\/\/doi.org\/10.1017\/s0305004115000158","relation":{},"ISSN":["0305-0041","1469-8064"],"issn-type":[{"value":"0305-0041","type":"print"},{"value":"1469-8064","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,4,1]]}}}