`UK Nonlinear News`,
`August 2001`

Birkhauser Press

ISBN: 0-8176-4119-X

Number of pages: 661 (CD-ROM included)

£54.00

This book (1st edition published in 1997 and this 2nd edition published in 2000) is another addition to a growing list of textbooks which attempt to incorporate mathematical software (in this case Maple) as a tool in the study and investigation of nonlinear phenomena. The authors have done this extremely well and made the whole learning process very much 'hands on' -- one of the best attempts at using a mathematical package as an auxiliary tool that I have come across. When I ordered this book for the library a few months ago, I wasn't sure that I made the correct decision to have my Mathematics department pay for a book which had more than 130 pages devoted to experimental activities (a total of 30 experiments to be exact). However after using this book to teach nonlinear dynamics to an Honours student, I'm very certain that I made the correct decision. In fact I soon added a copy to my personal collection of nonlinear books since the library's copy is always out on loan!

I commend Richard Enns and George McGuire for their organisation of the subject matter, choice of examples (which were pleasantly varied) and clarity in explanantion of some quite complex concepts. My student (who had very little prior experience with Maple, and limited computational techniques) found the integration of analytic and computational methods presented in the text to be excellent. She now feels that she has acquired sufficient Maple programming skills to apply it to her other courses. One of the criticisms of the first edition (which was unjust in my opinion) was the authors' rather brief introduction to Maple. In the present edition, the authors have clearly attempted to rectify this by giving "...a more gentle approach to the Maple system". This is the main focus of Chapter 1.

Chapters 2 and 3 introduce the readers to nonlinear phenomena by giving a brief discussion of a variety of problems drawn mainly from mechanics, ecology, biology, chemistry, electrical phenomena, pattern formation, solitons, chaos and maps. The topological, analytical and numerical methods that are normally used in the study of nonlinear problems are presented in chapters 4, 5 and 6 respectively. Although some readers may find the treatment of these topics to be somewhat superficial, I feel that such a criticism is unwarranted. There are numerous books on any one of these topics alone and the authors must be given credit for explaining the main points in each chapter. The exercises found in these chapters enable students not only to grasp the concepts and ideas described by the authors but also to experiment and ``play'' with the Maple worksheets. My student certainly gained a great deal from completing the exercises in these chapters.

Limit cycles, forced oscillators, jump phenomena and hysteresis, entrainment, quasiperiodicity, and chaotic oscillators are discussed in chapters 7 and 8 (unfortunately Hamiltonian chaos is only briefly addressed). Chapter 9 is devoted to nonlinear maps. My student focussed mainly on this chapter. Her favourite sections in this chapter dealt with controlling chaos (based on proportional feedback) and Saturn's Rings (an example of three-dimensional maps).

Chapter 10 deals with nonlinear partial differential equation phenomena with coverage of topics such as Bäcklund transformations, KdV wave solution, sine-Gordon equation and the three-wave problem. Chapter 11 discusses the standard numerical methods which are normally used to solve partial differential equations. Finally chapter 12 introduces the inverse scattering method (a topic I was relatively unfamiliar with before reading this book). I confess that I found the last three chapters to be somewhat 'lean', anyone who wishes to use this book to teach these topics may wish to supplement their lectures with some additional material. Once again the exercises found in these chapters are crucial in order to 'drive home' the concepts. The authors admits that the material found in the latter three chapters are "...mathematically more sophisticated" and they assume that the students do have some prior knowledge and familiarity of standard partial differential equations and their solutions.

As mentioned earlier, I was initially uninterested in the experimental activities found in Part II of the book. Most of these experiments require some mechanical equipment or electrical circuits. However, I have been assured by a colleague in the Physics department that almost all of the equipment mentioned in the text is easily obtained, and I am now convinced that some of these experiments can be used as lecture demonstrations. The three experiments pertaining to nonlinear resonance caught my eye. Although it is useful to read about a phenomena or investigate it via Maple, observing the phenomena via a demonstration or performing an experiment definitely 'brings the point home'. Perhaps we in the Mathematics department (unlike my Physics or Engineering colleagues) don't do enough of such practical demonstrations.

In summary, the authors have gone to a great deal of effort to bring alive the ideas and concepts of nonlinear phenomena. This book will definitely be a favourite amongst instructors, particularly those who want to move away from the traditional 'chalk-and-talk' lecturing system to a more hands-on learning environment (where students explore and investigate via the Maple worksheets or even undertake some experiments), interspersed by short lectures by the instructor to introduce topics and summarize results. I will certainly utilize this text for a variety of courses that I teach. I have shown it to some of my colleagues who are also beginning to admire the approach taken by Enns and McGuire.

P.S. Apparently a hardcopy solutions manual along with a disk containing all Maple solutions will be made available to instructors who adapt this book as an official textbook at their institution. I am unable to comment on the merit of this supplementary material since they were not yet available when I enquired with the publishers. Perhaps I may be able to comment on their usefulness in a future issue of UK Nonlinear News.

A listing of books reviewed in `UK Nonlinear News`
is available.

UK Nonlinear News thanks Birkhauser Press for providing a review copy of this book.

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Last Updated: 6th August 2001.