{"title":"On the Isolation/Nonisolation of a Cosymmetric Equilibrium and Bifurcations in its Neighborhood","authors":"Leonid G. Kurakin, Aik V. Kurdoglyan","doi":"10.1134/S1560354721030047","DOIUrl":null,"url":null,"abstract":"<div><p>A dynamical system with a cosymmetry is considered.\nV. I. Yudovich showed that a noncosymmetric\nequilibrium of such a system under the conditions of\nthe general position is a member of a one-parameter\nfamily.\nIn this paper, it is assumed that the equilibrium is\ncosymmetric, and the linearization matrix of the\ncosymmetry is nondegenerate.\nIt is shown that, in the case of an odd-dimensional\ndynamical system, the equilibrium is also nonisolated\nand belongs to a one-parameter family of equilibria.\nIn the even-dimensional case, the cosymmetric equilibrium is,\ngenerally speaking, isolated.\nThe Lyapunov – Schmidt method is used to study bifurcations\nin the neighborhood of the cosymmetric equilibrium when\nthe linearization matrix has a double kernel.\nThe dynamical system and its cosymmetry depend on a real\nparameter.\nWe describe scenarios of branching for families of\nnoncosymmetric equilibria.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"26 3","pages":"258 - 270"},"PeriodicalIF":0.8000,"publicationDate":"2021-06-03","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/S1560354721030047","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
A dynamical system with a cosymmetry is considered.
V. I. Yudovich showed that a noncosymmetric
equilibrium of such a system under the conditions of
the general position is a member of a one-parameter
family.
In this paper, it is assumed that the equilibrium is
cosymmetric, and the linearization matrix of the
cosymmetry is nondegenerate.
It is shown that, in the case of an odd-dimensional
dynamical system, the equilibrium is also nonisolated
and belongs to a one-parameter family of equilibria.
In the even-dimensional case, the cosymmetric equilibrium is,
generally speaking, isolated.
The Lyapunov – Schmidt method is used to study bifurcations
in the neighborhood of the cosymmetric equilibrium when
the linearization matrix has a double kernel.
The dynamical system and its cosymmetry depend on a real
parameter.
We describe scenarios of branching for families of
noncosymmetric equilibria.
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.