Spaces of dynamical systems
type: Item
instance of: scholarly article
title: Spaces of dynamical systems
zbMATH Open document ID: 1294.37002
published in: De Gruyter Studies in Mathematical Physics
publication date: 2012-04-13T00:00:00Z
full work available at URL: http://www.degruyter.com/viewbooktoc/product/174334
reviewed by: Kealey Dias
Mathematics Subject Classification ID: 37-02
Mathematics Subject Classification ID: 37C20
Mathematics Subject Classification ID: 37D05
Mathematics Subject Classification ID: 37C25
Mathematics Subject Classification ID: 37C05
Mathematics Subject Classification ID: 37C50
Mathematics Subject Classification ID: 37-03
Mathematics Subject Classification ID: 01A60
zbMATH DE Number: 6023866
zbMATH Keywords: dynamical systems
zbMATH Keywords: hyperbolic orbits and sets
zbMATH Keywords: structural stability
zbMATH Keywords: generic properties
zbMATH Keywords: fixed points
zbMATH Keywords: periodic points
zbMATH Keywords: smooth mappings and diffeomorphisms
zbMATH Keywords: pseudotrajectories
zbMATH Keywords: shadowing
author: Sergei Yu. Pilyugin
MaRDI profile type: Publication
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review text: \textit{S. Yu. Pilyugin}'s ``Spaces of Dynamical Systems'' provides a formal introduction to the main objects and results of the modern theory of dynamical systems, concentrating on hyperbolicity and structural stability. This text is intended to complement the author's earlier text ``Introduction to structurally stable systems of differential equations'' [Translated from the Russian by the author. Basel etc: Birkhäuser Verlag. (1992; Zbl 0747.34031)]. These two books have perceptible overlap in content, though neither text requires the other as a prerequisite. The chief dissimilarity between the two texts is that the earlier text focused on the dynamics of smooth flows, while much of the current work emphasizes discrete time dynamical systems, only presenting the results for flows which are analogues of the discrete time case. In addition to classical statements and examples, the book introduces the reader to a few specialized topics, including shadowing of pseudotrajectories. The style is rigorous theorem/proof mathematical format, with a detailed and clear exposition. The main strength of this book is that the author includes complete proofs of results that other publications frequently leave to the reader. The book foregoes lengthy introductions in favor of contextualizing results by concise historical comments. NEWLINENEWLINENEWLINE``Spaces of Dynamical Systems'' assumes a solid command of topics from an undergraduate education in math: functional analysis, topology, linear algebra, calculus, differential equations, and analysis. This writing would be best suited for early graduate students or professional mathematicians new to the field of dynamics. There are a few explicit exercises embedded in the text, however, supplementation might be advantageous if the book is to be used in a course. This book contains twelve chapters and two appendices (together a total of 221 pages), and 7 grayscale figures.NEWLINENEWLINEThe first three chapters set the groundwork for …
description: scientific article; zbMATH DE number 6023866