Global Analysis of Dynamical Systems Festschrift dedicated to Floris Takens for his 60th birthday Henk Broer, Bernd Krauskopf & Gert Vegter (eds.) Institute of Physics Publ., Bristol-Philadelphia (2001), 464 pp. price £ 60.-;
Reviewer: Ferdinand Verhulst, Mathematisch Instituut, University of Utrecht, The Netherlands.

Level: advanced.
This book covers a great variety of topics in dynamical systems reflecting the wide interest and many scientific contacts of Floris Takens. He wrote seminal papers on singularities of vector fields, bifurcation theory, reconstruction in time-series analysis and many other subjects. It was a good idea to start with the reprint of an early and influential publication by Takens: Forced oscillations and bifurcations. This was published in 1974 by the Mathematisch Instuut of Utrecht University as Communication vol. 3 and such was the low quality of typing and reproduction in those times that people always referred to this as the `Utrecht preprint'. They could not believe this was the real publication. It did not stop people to absorb the interesting results in this `Utrecht preprint' on bifurcations, the role of symmetries and normal forms.
In 20 papers the text ranges from the pure mathematical side of dynamical systems to applications, including a note by Ruelle, a paper by long-term collaborators Palis and Yoccoz, interesting accounts of work by former students (Floris supervised around 20 PhD students) and friends.
Seven papers deal with topics in bifurcation theory, two are concerned with homoclinic bifurcations and two with hamiltonian dynamics; there is one paper on time-series analysis and four papers deal with ergodic theory and one-dimensional maps. To give an impression of the contents we discuss briefly some of the papers.

Strong resonances and Takens's Utrecht preprint by Bernd Krauskopf discusses the p/q-resonance for periodic solutions of a vector field. In particular the 1/4-resonance is a notoriously hard case where this author gave major contributions. At the same time the paper is a nice survey of the literature (58 references) and new results.

Exponential confinement of chaos in the bifurcation sets of real analytic diffeomorphisms by Henk W. Broer and Robert Roussarie deals with bifurcations which are connected with planar vector fields which depend periodically on time. Normalization (averaging) produces vector fields which are symmetric and integrable. A well-known implication is that in the original system, chaos is confined to -- as the authors prove -- exponential or super-exponential thin sets. This is in its generality an important result.

An unfolding theory approach to bursting in fast-slow systems by Martin Golubitsky, Kreimir Josi and Tasso J. Kaper.
Bursting phenomena have drawn a lot of attention which is motivated both by modeling in neurophysiology and the interesting mathematics associated with it. A natural mathematical context is the field of fast-slow systems (where the slow system traces periodic behaviour and alternates with spike-like oscillations) with higher codimension bifurcations. For instance among periodic bursters in a nerve cell model, bursters are characterized by codimension two. Other bursters even have codimension three. In this paper a detailed classification is given with explicit normal forms and suggestions for numerical implementation.

There are many more interesting papers in this Festschrift. It is not only a nice tribute to Floris Takens but at the same time a valuable and stimulating addition to the dynamical systems literature.