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Book review

Author(s) Kirchgraber, U. (ed.)
Walther, H.O. (ed.)
Title Dynamics Reported, vol. II
Publisher John Wiley and Sons,
B.G. Teubner
Year of publication 1989
Reviewed by Paul A. Blaga

This is-the second volume of the series of books entitled "Dynamics Reported" and includlng review papers written by experts in dynamical systems on some key topics of this theory.

The book comprises a number of five contributions and I'll try in the following to provide a short description of them. The first paper ("Invariant Manifolds for Semilinear Partial Differential Equations", by P.W. Bates end C.K.R.T. Jones) is devoted to the construction of of stable, unstable and central manifolds for systems with linearized somigroups, which are not necessarily analytic, but are strongly continuous. The second contribution ("Formally Symmetric Normal Forms and Genericity", by H.W. Broer and F. Takens), examines the global dynamics of some glasses of dynamical systems and shows that in several cases these dynamics are generically very complicated, because of the formal symmetries in normal forms. The third work ("Some Quadratic Systems with at most One Limit Cycle", by W.A. Coppel) gives some sufficient conditions for a quadratic systems to have at most one limit cycle. The following paper ("Centre Manifolds, Normal Forms and Elementary Bifurcations", by A. Vanderbauwhede) contains an exposltion of centre manifold theory, an introduction to normal forms and applications of these two theories to bifurcation problems, while the last contribution ("Models for Dynamics", by J.C. Willems), presents a framework for modelling dynamical systems which interact with their environment.

Every work contains a great number of examples and applications to concrete dynamical systems, some of them of special interest for physicists. The expositions are pedagogic and, to a large extent, self-contained. The subjects of the papers are of a great theoretical and practical importance. These are only few reasons to warmly recommend this book, which will be of a great help to the scientists and graduate students working in dynamical systems. As I wrote in the review of the first volume, such a kind of books are indispensable for someone wishing to keep the pace with the developments from his own area of research.

Each paper is accompanied by a large list of references. There is no index. The book is very accurate printed.