The dynamics of vector fields with singularities
type: Item
instance of: scholarly article
title: The dynamics of vector fields with singularities
zbMATH Open document ID: 1494.37016
publication date: 2022-02-14T00:00:00Z
review text: The article is a survey of recent results on dynamics of vector fields exhibiting singularities that are accumulated by recurrent regular points robustly. The author considers the notion of singular hyperbolicity as introduced in [\textit{C. A. Morales} et al., Ann. Math. (2) 160, No. 2, 375--432 (2004; Zbl 1071.37022)]. This notion is useful to characterize a rich family of vector fields with singularities, including the Lorenz attractor. The more general notion of multi-singular hyperbolicity introduced in [\textit{C. Bonatti} and \textit{A. Da Luz}, J. Eur. Math. Soc. (JEMS 23, No. 8, 2649--2705 (2021; Zbl 1487.37037)] is also considered. These notions are relevant to describe dense sets of vector fields. Several results and conjectures along this line of research are reviewed by the present author. This article also covers several results on ergodic theory of singular hyperbolic dynamics or multi-singular hyperbolic dynamics. The last section of the article contains an analysis of the extended linear Poincaré flow and the rescaled linear Poincaré flow showing their use for the study of vector fields with singularities.
reviewed by: Miguel Paternain
Mathematics Subject Classification ID: 37D05
Mathematics Subject Classification ID: 37D10
Mathematics Subject Classification ID: 37D15
Mathematics Subject Classification ID: 37D20
Mathematics Subject Classification ID: 37D25
Mathematics Subject Classification ID: 37D30
Mathematics Subject Classification ID: 37C10
Mathematics Subject Classification ID: 37C83
zbMATH DE Number: 7473247
zbMATH Keywords: vector field
zbMATH Keywords: singularity
zbMATH Keywords: Morse-Smale system
zbMATH Keywords: horseshoe
zbMATH Keywords: singular hyperbolicty
author: Da-Wei Yang
MaRDI profile type: Publication
full work available at URL: s10114-022-0471-5
OpenAlex ID: W4210339993
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The dynamics of vector fields with singularities (DOI)
DOI: 10.1007/S10114-022-0471-5

The dynamics of vector fields with singularities (published in)
published in: Acta Mathematica Sinica, English Series