626 BOOK REVIEWS

[2] Y. MEYER, Wavelets and Operators, CambridgeUniversity Press, Cambridge, 1993.

[3] M. V. WICKERHAUSER, Adapted Wavelet Analy-sis from Theory to Software, A. K. Peters,Wellesley, MA, 1994.

MICHAEL FRAZIERMichigan State University

Periodic Motions. By Mikl6s Farkas. Springer-Verlag, New York, 1995. $54.50, 577 pp., cloth.ISBN 0-387-94204-1.

For those of us who carry out research involvingperiodic solutions of ordinary differential equa-tions, this is the book we have been waiting for.It contains as complete a story as possible (ina book of this size) of the existence and sta-

bility of periodic solutions of autonomous andnonautonomous equations, with techniques rang-ing from perturbations to bifurcations, all writtenin an easy-to-read, easy-to-follow style.

The introductory chapter reviews conceptsneeded in the discussions to follow. Topics suchas dynamical systems and stability are briefly re-viewed. Basically, it is assumed that the readeris already familiar with such things as Liapunovfunctions. The next chapter contains a thoroughdiscussion ofperiodic solutions of linear systems.It begins with unforced systems, moves on toforced systems, and finishes with examples fromequations of Hill and Mathieu.

In Chapter 3, there is an extensive discussion ofautonomous systems in the plane. Separate sec-tions are devoted to Lienard’s equation, Duffing’sequation, the Lotka-Volterra predator-prey sys-tem, and Hilbert’s sixteenth problem. Of course,each of these topics has volumes of literature de-voted to them, which are not reproduced in thistext, nor should they be. But the main essencesof the problems are presented, and the referencesshow where to find the more extensive results.

Chapter 4 deals with periodic time-dependentsystems, including the forced Lienard and Duff-ing equations. There is also an interesting ex-

ample of competition between populations in a

periodically changing environment.In Chapter 5 there is a return to autonomous

systems, but of arbitrary dimension. Here sub-jects such as orbital stability, Poincar6 maps, andinvariant manifolds are treated. Examples in-clude higher order competition and cooperationamong biological populations.

The text concludes with the two most impor-tant and interesting chapters which are on pertur-bations and bifurcations. Chapter 6 deals withthe question of when periodicity and stability ofsolutions is preserved under perturbation. In par-ticular the method of averaging, so familiar to theanalysts of the former Soviet Union, is discussedalong with singular perturbations.

In the final chapter, the most important topiccurrently in the treatment of periodic solutionsis discussed, namely, bifurcation theory. Again,there are entire texts devoted to this topic contain-ing many results that are not found here, but thereferences tell the reader where to look. How-ever, two aspects of this chapter are noteworthy.There is a discussion of the Hopf bifurcation the-orem, one of the most useful theorems of modernbifurcation theory. Also there is an introductionto zip bifurcation (for competitive systems), de-veloped by Professor Farkas, which is a beautifultheory in which periodic solutions open up froma curve of equilibria as a parameter changes, justlike a zipper.

Finally, a substantial bibliography is includedin this book, which is a must for all researchersin this field.

H. I. FREEDMANUniversity ofAlberta

Parallel Computing Using the Prefix Prob-lem. By S. Lakshmivarahan and Sudarshan K.Dhall. Oxford University Press, New York, 1994.$59.95,294 pp., hardcover. ISBN 0-19-508849.

The usual definition of the prefix computation isthe following:

Given elements al aN from a setA that is closed under an associativebinary operation., compute the termsdi ai . . ai for N.

On a single processor, this computationis straightforward and generates little interest.What better proof than the lack of any referenceto the problem in [8]? Enter parallel computa-tion, and the landscape suddenly changes. Oneof the most popular recent books on algorithmsand data structures devotes the better part of itschapter on parallel algorithms to prefix compu-tations [4, Chap. 30]. Karp and Ramachandran

Other terms that have been used for this or closelyrelated computations are suf x [1] and scan [3].

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