{"title":"A Bifurcation Lemma for Invariant Subspaces","authors":"John M. Neuberger, Nándor Sieben, James W. Swift","doi":"10.1137/23m1595540","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 2, Page 1610-1635, June 2024. <br/> Abstract.The bifurcation from a simple eigenvalue (BSE) theorem is the foundation of steady-state bifurcation theory for one-parameter families of functions. When eigenvalues of multiplicity greater than one are caused by symmetry, the equivariant branching lemma (EBL) can often be applied to predict the branching of solutions. The EBL can be interpreted as the application of the BSE theorem to a fixed point subspace. There are functions which have invariant linear subspaces that are not caused by symmetry. For example, networks of identical coupled cells often have such invariant subspaces. We present a generalization of the EBL, where the BSE theorem is applied to nested invariant subspaces. We call this the bifurcation lemma for invariant subspaces (BLIS). We give several examples of bifurcations and determine if BSE, EBL, or BLIS applies. We extend our previous automated bifurcation analysis algorithms to use the BLIS to simplify and improve the detection of branches created at bifurcations.","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-06-21","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/10.1137/23m1595540","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 2, Page 1610-1635, June 2024. Abstract.The bifurcation from a simple eigenvalue (BSE) theorem is the foundation of steady-state bifurcation theory for one-parameter families of functions. When eigenvalues of multiplicity greater than one are caused by symmetry, the equivariant branching lemma (EBL) can often be applied to predict the branching of solutions. The EBL can be interpreted as the application of the BSE theorem to a fixed point subspace. There are functions which have invariant linear subspaces that are not caused by symmetry. For example, networks of identical coupled cells often have such invariant subspaces. We present a generalization of the EBL, where the BSE theorem is applied to nested invariant subspaces. We call this the bifurcation lemma for invariant subspaces (BLIS). We give several examples of bifurcations and determine if BSE, EBL, or BLIS applies. We extend our previous automated bifurcation analysis algorithms to use the BLIS to simplify and improve the detection of branches created at bifurcations.
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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