Analysis and simulation of chaotic systems
type: Item
instance of: scholarly article
title: Analysis and simulation of chaotic systems
zbMATH Open document ID: 0777.34001
author: Frank Hoppensteadt
published in: Applied Mathematical Sciences
publication date: 1993-06-17T00:00:00Z
review text: This textbook is based on lectures which the author has given to graduate students in mathematics and engineering at the University of Michigan. The whole book covers a course lasting a full year. The book is divided into two roughly equal parts, each with four chapters. Part I is an introduction to nonlinear systems, and Part II is about perturbation methods. It is assumed that students are familiar with the methods and theory of differential equations up to a level equivalent to a course based on, for example, Theory of Ordinary Differential Equations by \textit{E. A. Coddington} and \textit{N. Levinson} (McGraw-Hill, New York, 1955; Zbl 0064.330). Chapter 1 contains a very brief review (25 pages) of oscillations in linear systems including linear systems with periodic coefficients, Fourier methods and equations with variable coefficients. This brevity is inevitably at the expense of explanation. For example, Mathieu's equation gets a one page treatment including the stability diagram, with very little explanation of its derivation. Autonomous equations are treated at some length in Chapter 2. Some of the section headings, theorem titles and key words give the flavour and breath of its content. They include Poincaré-Bendixson theory, van der Pol's equation, potential problems, Denjoy's theorem, dissipative systems, the cusp catastrophe, stroboscopic methods, Sarkovski's sequence, mappings of the plane and equations with time delays. This is a very accessible, 60 page review of some basic concepts and techniques for autonomous systems. Lyapunov and orbital theories are discussed in Chapter 3, whilst Chapter 4 looks at some algebraic and topological aspects of nonlinear oscillations including fixed point theorems. Part II starts with a chapter on regular perturbation methods. The regular perturbation theorem for initial value problems is proved. Forced oscillations are introduced in Chapter 6. Resonance and nonresonant forcing are discussed with the usual example of…
Mathematics Subject Classification ID: 34-01
Mathematics Subject Classification ID: 34C15
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 70K50
Mathematics Subject Classification ID: 58-01
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C29
Mathematics Subject Classification ID: 37D45
zbMATH DE Number: 204988
zbMATH Keywords: averaging
zbMATH Keywords: textbook
zbMATH Keywords: nonlinear oscillations
zbMATH Keywords: perturbation methods
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 204988