{"title":"Bifurcation Analysis of Bogdanov–Takens Bifurcations in Delay Differential Equations","authors":"M. M. Bosschaert, Yu. A. Kuznetsov","doi":"10.1137/22m1527532","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 553-591, March 2024. <br/> Abstract. In this paper, we will perform the parameter-dependent center manifold reduction near the generic and transcritical codimension two Bogdanov–Takens bifurcation in classical delay differential equations. Using an approximation to the homoclinic solutions derived with a generalized Lindstedt–Poincaré method, we develop a method to initialize the continuation of the homoclinic bifurcation curves emanating from these points. The normal form transformation is derived in the functional analytic perturbation framework for dual semigroups (sun-star calculus) using a normalization technique based on the Fredholm alternative. The obtained expressions give explicit formulas, which have been implemented in the freely available bifurcation software package DDE-BifTool. The effectiveness is demonstrated on various models","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/22m1527532","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 553-591, March 2024. Abstract. In this paper, we will perform the parameter-dependent center manifold reduction near the generic and transcritical codimension two Bogdanov–Takens bifurcation in classical delay differential equations. Using an approximation to the homoclinic solutions derived with a generalized Lindstedt–Poincaré method, we develop a method to initialize the continuation of the homoclinic bifurcation curves emanating from these points. The normal form transformation is derived in the functional analytic perturbation framework for dual semigroups (sun-star calculus) using a normalization technique based on the Fredholm alternative. The obtained expressions give explicit formulas, which have been implemented in the freely available bifurcation software package DDE-BifTool. The effectiveness is demonstrated on various models
期刊介绍：
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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