{"title":"Dynamical phenomena connected with stability loss of equilibria and periodic trajectories","authors":"A. Neishtadt, D. Treschev","doi":"10.1070/RM10023","DOIUrl":null,"url":null,"abstract":"This is a study of a dynamical system depending on a parameter . Under the assumption that the system has a family of equilibrium positions or periodic trajectories smoothly depending on , the focus is on details of stability loss through various bifurcations (Poincaré–Andronov– Hopf, period-doubling, and so on). Two basic formulations of the problem are considered. In the first, is constant and the subject of the analysis is the phenomenon of a soft or hard loss of stability. In the second, varies slowly with time (the case of a dynamic bifurcation). In the simplest situation , where is a small parameter. More generally, may be a solution of a slow differential equation. In the case of a dynamic bifurcation the analysis is mainly focused around the phenomenon of stability loss delay. Bibliography: 88 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"76 1","pages":"883 - 926"},"PeriodicalIF":1.4000,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Russian Mathematical Surveys","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1070/RM10023","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 1
Abstract
This is a study of a dynamical system depending on a parameter . Under the assumption that the system has a family of equilibrium positions or periodic trajectories smoothly depending on , the focus is on details of stability loss through various bifurcations (Poincaré–Andronov– Hopf, period-doubling, and so on). Two basic formulations of the problem are considered. In the first, is constant and the subject of the analysis is the phenomenon of a soft or hard loss of stability. In the second, varies slowly with time (the case of a dynamic bifurcation). In the simplest situation , where is a small parameter. More generally, may be a solution of a slow differential equation. In the case of a dynamic bifurcation the analysis is mainly focused around the phenomenon of stability loss delay. Bibliography: 88 titles.
期刊介绍:
Russian Mathematical Surveys is a high-prestige journal covering a wide area of mathematics. The Russian original is rigorously refereed in Russia and the translations are carefully scrutinised and edited by the London Mathematical Society. The survey articles on current trends in mathematics are generally written by leading experts in the field at the request of the Editorial Board.