Perturbation methods, bifurcation theory and computer algebra / Richard H. Rand, Dieter Armbruster

As stated clearly in the preface the purpose of the authors was “to provide computer algebra programs which implement a number of popular perturbation methods”. They do this with great success, and the result is a fine textbook, that must be worked and not only read. The background necessary to read the book is quite elementary, one needs only some background in differential equations and familiarity with computers. The computer algebra system used in the book is MACSYMA, and an appendix is provided to guide the reader, but one needs more. The material is presented in the form of worked examples and the reader is supposed to do a lot of work to understand really what is going on. Computer programs are provided to guide the numerical experiments. Besides the perturbation methods presented there is a lot of information about elementary bifurcations: folds, pitchforks, Hoff, both ordinary and partial differential equations are discussed. Each chapter of the book describes a differential perturbation method: Lindstedt, Center manifold, Normal forms, Two variable expansion method, Averaging, Lie transforms, Lyapunov-Schmidt reduction. Each method is nicely discussed, through very good examples. The methods are compared. Limitation and advantages of each are clearly pointed out in form of examples. Finally, it is a pleasure to “work” the book. (Zentralblatt)