UK Nonlinear News, November 2001
Springer-Verlag, Applied Mathematical Science Series Vol 94.
The author must be commended for his organisation of the subject matter.
The book begins with brief overviews of Linear Systems (Chapter 1) and
Dynamical Systems (Chapter 2). The author is quick to point out that there
''[...] are many excellent texts and research monographs dealing with these
topics in great detail''. However, the main aim of the first two
chapters is to provide the readers
with an adequate background for the later more advanced topics.
Chapter 3 is devoted to techniques used to study the stability of nonlinear systems. Particular emphasis is placed on the stability of systems under persistent disturbances (sometimes referred to as total stability). Future chapters require an understanding of this concept.
Bifurcation and topological methods form the basis of Chapter 4. The author begins by deriving some implicit function theorems, followed by a description of common techniques used in the bifurcation analyses of nonlinear systems. This chapter concludes by considering some useful methods for finding solutions via geometrical means, in particular the use of fixed-point theorems.
Regular perturbation methods are dealt with in Chapter 5, whereas the main focus of Chapter 6 is perturbation methods for forced oscillators (singular perturbation methods). Several interesting aspects of resonant forcing are illustrated via Duffing's equation. This chapter concludes with a discussion of the boundaries of basins of attraction and fractal measure.
In my opinion the final two chapters are the highlight of this book. Chapter 7 describes the method of averaging whereas Chapter 8 presents the techniques of quasistatic-state approximations. These chapters provide one of the best explanations of these topics that I have read.
In the introduction the author states that he used material from this book to teach a variety of students (both in Engineering and Mathematics departments). He claims that students with a strong background in Differential Equations can proceed directly to Chapters 4 -- 8, whereas a course on computer simulations could deal with sections from Chapters 2, 4 7 and 8. As an instructor I found these suggestions as well as the author's use of a variety of examples from different areas to be very useful. Overall Frank Hoppensteadt is to be commended for delivering a clear and concise text on these topics.
UK Nonlinear News thanks Springer-Verlag for providing a review copy of this book.
A listing of books reviewed in UK Nonlinear News is available.