Qualitative theory of flows on surfaces (a review)
type: Item
instance of: scholarly article
title: Qualitative theory of flows on surfaces (a review)
zbMATH Open document ID: 0939.37024
DOI: 10.1007/BF02433487
published in: Journal of Mathematical Sciences (New York)
publication date: 1998-10-22T00:00:00Z
review text: Within the last fifty years the geometric theory of dynamical systems was widely appreciated and nowadays this field is accessible for a broad scientific audience. In fact, the nonlinear dynamics has become a subject with many applications in general sciences. In this paper many concepts and results on dynamical systems defined in 2-dimensional manifolds obtained in the last decades are gathered and discussed. The summary of this work says that: ``The geometric study of flows on manifolds has a long history in mathematics, even though it was not considered as a special field until it appeared in the works of Poincaré, Lyapunov, and Bendixson. After that, the subject developed rapidly, and, at the present time, it is the focus of extensive research. It is impossible to touch upon all sides of the geometric theory of flows in one article. The purpose of this review is to present some aspects of the geometric theory of flows on 2-manifolds (of course, within the frame of our knowledge). Our main interest is the topological structure and the topological equivalence of surface flows, asymptotic properties of semitrajectories and the closing lemma, etc. (see the contents). However, our article was not written for specialists only; we have also attempted to provide an introduction to the subject. We would like to thank the many mathematicians who took part in valuable discussions of problems of the theory of dynamical systems, and especially D. Anosov, I. Bronshtein, V. Grines, G. Levitt, M. Malkin, Shigenori Matsumoto, I. Nikolaev, and A. Zorich.'' The article is organized as follows: Section 1: Introduction of basic concepts, some useful constructions, terminology and notations. Section 2: The structure of limit sets. Section 3: The topological structure of a flow. Section 4: Space of flows (structural stability, Morse-Smale flows and bifurcation). Section 5: Topological classification of flows (more on equivalence of flows). Section 6: On the equivalence between the top…
reviewed by: Marco A. Teixeira
Mathematics Subject Classification ID: 37E35
Mathematics Subject Classification ID: 37C99
Mathematics Subject Classification ID: 37-02
Mathematics Subject Classification ID: 37C15
zbMATH DE Number: 1213213
zbMATH Keywords: flow
zbMATH Keywords: vector field
zbMATH Keywords: structural stability
zbMATH Keywords: survey
zbMATH Keywords: Morse-Smale flows and bifurcation
author: S. Kh. Aranson
author: Evgenii V. Zhuzhoma
MaRDI profile type: Publication
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Qualitative theory of flows on surfaces (a review) (full work available at URL)
full work available at URL: https://doi.org/10.1007/bf02433487

Qualitative theory of flows on surfaces (a review) (OpenAlex ID)
OpenAlex ID: W2021792413