Ordinary differential equations. An introduction to nonlinear analysis. Transl. from the German by Gerhard Metzen
type: Item
instance of: scholarly article
title: Ordinary differential equations. An introduction to nonlinear analysis. Transl. from the German by Gerhard Metzen
zbMATH Open document ID: 0708.34002
author: Herbert Amann
published in: De Gruyter Studies in Mathematics
publication date: 1993-06-05T00:00:00Z
review text: The book under review consists of 6 chapters with 27 sections of nearly equal size. The first chapter gives an extended introduction to various fields where ordinary differential equations occur, and also sketches some classes of equations which admit solutions in closed form. As a matter of fact, most equations cannot be solved explicitly (or, stated more drastically, all equations which may be solved have been solved). This emphasizes the need of providing information on the qualitative behaviour of hypothetic solutions, without knowing them, and this is the main purpose of the Chapters 2-6 of this book. The contents of these chapters go as follows. Chapter 2 is concerned with the usual existence and continuity theorems for initial value problems in Euclidean space. The last section of this chapter contains elementary results on semiflows and flows. The theory of linear equations is treated in detail in Chapter 3. As usual, particular emphasis is put on properties of the Cauchy function (evolution operator) generated by a strongly continuous linear operator function \(A=A(t)\). As is well known, the stationary case A(t)\(\equiv A\) admits a complete and precise description, e.g. in the classification of equilibrium points, for necessary and sufficient stability conditions, or in characterizing the \(\epsilon\)-dichotomy (hyperbolicity) of critical points. Qualitative properties are dealt with in Chapter 4, pointing out the importance of Lyapunov stability and Lyapunov functions, limit sets and attractors, stable and unstable manifolds. Moreover, the Grobman-Hartman linearization principle (at hyperbolic critical points) is discussed, and some consequences are derived. Chapter 5 is entirely devoted to periodic solutions. The simple observation that the fixed points of the Poincaré operator (translation operator along the trajectories) are precisely the initial values which give rise to periodic solutions, motivates the need of treating Brouwer's fixed point princip…
Mathematics Subject Classification ID: 34-02
Mathematics Subject Classification ID: 34A12
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34D20
zbMATH DE Number: 194474
zbMATH Keywords: epsilon dichotomy
zbMATH Keywords: flows
zbMATH Keywords: Cauchy function
zbMATH Keywords: evolution operator
zbMATH Keywords: equilibrium points
zbMATH Keywords: Lyapunov functions
zbMATH Keywords: limit sets
zbMATH Keywords: attractors
zbMATH Keywords: unstable manifolds
zbMATH Keywords: Grobman-Hartman linearization principle
zbMATH Keywords: Poincaré operator
zbMATH Keywords: topological degree theory
zbMATH Keywords: Floquet theory
zbMATH Keywords: bifurcation theory
zbMATH Keywords: Hopf bifurcation
reviewed by: Jürgen Appell
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 194474