{"title":"V-Shaped Action Functional with Delay","authors":"Urs Frauenfelder","doi":"10.1134/S1560354723510020","DOIUrl":null,"url":null,"abstract":"<div><p>In this note we introduce the V-shaped action functional with delay in a symplectization,\nwhich is an intermediate action functional between the Rabinowitz action functional\nand the V-shaped action functional. It lives on the same space as the\nV-shaped action functional, but its gradient flow equation is a delay equation\nas in the case of the Rabinowitz action functional. We show that there is a smooth interpolation\nbetween the V-shaped action functional and the V-shaped action functional with delay\nduring which the critical points and its actions are fixed. Moreover, we prove that there\nis a bijection between gradient flow lines of the V-shaped action functional with delay\nand the ones of the Rabinowitz action functional.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"28 4","pages":"364 - 373"},"PeriodicalIF":0.8000,"publicationDate":"2023-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Regular and Chaotic Dynamics","FirstCategoryId":"4","ListUrlMain":"https://link.springer.com/article/10.1134/S1560354723510020","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
In this note we introduce the V-shaped action functional with delay in a symplectization,
which is an intermediate action functional between the Rabinowitz action functional
and the V-shaped action functional. It lives on the same space as the
V-shaped action functional, but its gradient flow equation is a delay equation
as in the case of the Rabinowitz action functional. We show that there is a smooth interpolation
between the V-shaped action functional and the V-shaped action functional with delay
during which the critical points and its actions are fixed. Moreover, we prove that there
is a bijection between gradient flow lines of the V-shaped action functional with delay
and the ones of the Rabinowitz action functional.
期刊介绍：
Regular and Chaotic Dynamics (RCD) is an international journal publishing original research papers in dynamical systems theory and its applications. Rooted in the Moscow school of mathematics and mechanics, the journal successfully combines classical problems, modern mathematical techniques and breakthroughs in the field. Regular and Chaotic Dynamics welcomes papers that establish original results, characterized by rigorous mathematical settings and proofs, and that also address practical problems. In addition to research papers, the journal publishes review articles, historical and polemical essays, and translations of works by influential scientists of past centuries, previously unavailable in English. Along with regular issues, RCD also publishes special issues devoted to particular topics and events in the world of dynamical systems.
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