Dynamical systems and linear algebra
type: Item
instance of: scholarly article
zbMATH Open document ID: 1306.37001
author: Fritz Colonius
author: Wolfgang Kliemann
publication date: 2014-11-05T00:00:00Z
Mathematics Subject Classification ID: 37-01
Mathematics Subject Classification ID: 34-01
Mathematics Subject Classification ID: 93C05
Mathematics Subject Classification ID: 15A99
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 39A06
zbMATH DE Number: 6365923
zbMATH Keywords: dynamical systems
zbMATH Keywords: linear differential and difference equations
zbMATH Keywords: Lyapunov exponents, projective spaces, Grassmann linear manifolds
zbMATH Keywords: skew product systems
zbMATH Keywords: Floquet theory
zbMATH Keywords: \(\alpha\)-limit sets
zbMATH Keywords: \(\omega\)-limit sets
zbMATH Keywords: chain transitivity
zbMATH Keywords: ergodic theory
zbMATH Keywords: random dynamical systems
MaRDI profile type: Publication
title: Dynamical systems and linear algebra
published in: Graduate Studies in Mathematics
review text: This book is to devoted to dynamical systems whose behavior is determined by real quadratic matrices \(A \in {\mathbb R}^{d \times d}\). The corresponding systems are described by linear differential equations of types \(\dot{x} = Ax\), \(A \in {\mathbb R}^{d \times d}\), or \(\dot{x} = A(t)x\) \(A(t) \in {\mathbb R}^{d \times d}\), \(t \in {\mathbb R}\) in the case of continuous time \(t\) and with linear difference equation \(x_{n+1} = Ax_n\), \(A \in {\mathbb R}^{d \times d}\) or \(x_{n+1} = A_nx_n\), \(A_n \in {\mathbb R}^{d \times d}\), \(n \in {\mathbb Z}\) in the case of discrete time \(n\). It is natural that linear algebra plays a key role in the analysis of such systems; moreover this analysis leads to the study, characterization and modification of many objects in linear algebra, such as similarity matrices, spectral analysis of matrices and their special classes, and so on. As a result in many linear algebra textbooks (e.g., by F. R. Gantmacher or R. A. Horn and C. R. Johnson) one can find basic results about linear differential and difference equations, however, the serious and fundamental account of the theory of these equations from the view-point of dynamical systems does not exist. And this book is among the first ones dealing with the curious interplay between linear algebra and the theory of dynamical systems. Of course, it is impossible to cover the immensity of this play, but this book is a nice step in the corresponding direction. The book contains 11 chapters divided into two parts; each chapter begins from a small introduction, which describes what one can find in the chapter, further, it runs some sections with basic text, section with exercises, and a section ``Orientation, notes and references'' (explanation of stated results and references). The book is completed with a bibliography (144 items) and an index.NEWLINENEWLINEThe first part (5 chapters) of the book, ``Matrices and linear dynamical systems'', deals with the autonomous case. Cha…
reviewed by: Peter P. Zabreĭko
description: scientific article; zbMATH DE number 6365923