Analysis and simulation of chaotic systems
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 Duffing's equation. The Poincaré-Lindsteadt method is also applied to Duffing's equation, and subharmonic responses and the frequency-response surface are derived. The chapter ends with a section on basins of attraction and fractal measures. Averaging is the subject of Chapter 8. The nonlinear averaging theorem is proved, and there is some discussion of highly oscillatory differential and difference equations. The book concludes with a chapter on quasistatic-state methods. Each chapter concludes with a small set of exercises. In a book of 305 pages, the author has managed to include a large number of topics. They are generally covered in short readable sections. Even the book moves at quite a pace, proofs of key theorems have not been sacrificed. The author has achieved his aim of producing a concise textbook on nonlinear systems suitable for graduate students and nonspecialist workers in dynamical systems.
scientific article; zbMATH DE number 204988