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Book review

Author(s) Kirchgraber, U. (ed.)
Walther, H.O. (ed.)
Title Dynamics Reported, vol. I
Publisher John Wiley & Sons,
B.G. Teuboer
Year of publication 1988
Reviewed by Paul A. Blaga

The theory of dynamical systems has a fairly rapid evolution and it is very difficult for someone to keep pace with this evolution. Therefore, it is useful, from time to time, to have a collection of review papers containing the new results and providing, at the same time, a guide through the journal literature. It is the aim of this series of books, entitled "Dynamics Reported", whose first volume is now under review, to carry out this task.

The volume includes five review papers, written by authorities in their fields, treating five different subjects from dynamical systems theory. The first paper ("Mather Sets for Twist Maps and Geodesics on Tori", by V. Bangert) presents an introduction to Mather-Aubry theory on the dynamics of monotone twist-maps. The second paper ("Connecting Orbits in Scalar Reaction Diffusion Equation", by P. Brunovsky and B. Fiedler) establishes connections between a hyperbolic stationary solution and other solutlons of the one-dimensional reaction-diffusion equation with Dirichlet boundary conditions. The following article ("Qualitative Theory of Nonlinear Resonance by Averaging and Dynamical Systems Methods", by J. Murdock), is a self-contained presentation of the theory of averaging for periodic and quasiperiodic systems, especially as related to nonlinear resonance. The goal of the fourth work ("An Algorithmic Approach for Solving Singularly Perturbed Initial Value Problems", by K. Nipp) is to find a chain of approximations to the solution of a singularly perturbed initial value problem for an ordinary differential equation in a systematic way, while the last paper ("Exponential Dichotomies, the Shadowing Lemma and Transversal Homoclinic Points", by K.J. Palmer), offers a new approach to some important results in dynamical systems theory, including the Smale theorem on chaotic behaviour of trajectories of diffeomorphismes near transversel homoclinic points.

In all the cases, the expositions are fairly pedagogic and there is a plenty of worked examples. The authors pay attention mainly to typical cases rather than to most general ones, although the essential theorems are provided with complete and carefully written proofs. The book is, without any doubt, of a great value for any mathematician or other scientist interested in dynamical systems. Especially the graduate students will benefit very much by reading this book, and getting acquainted with the newest achievements in the field.

Every paper has a long list of references, but I ought to say that there is no index, although it would have been very useful.