Catastrophe theory is an important and very difficult branch of Mathematics. The inventor of catastrophe theory is R. Thom in the 1960’s (see his monograph “Stabilité structurelle et morphogénèse”. Reading, Mass.: Benjamin (1973; Zbl 0294.92001)). This theory has a wide range of applications in dynamical systems, in particular bifurcation and chaos, as well as in biology, economics and chemical kinetics.
The book under review is the second edition, the first one being published in (1993; Zbl 0782.58001). In a foreword, René Thom himself considers this book unique because its approach is pragmatic, the material self-contained and the style as elementary as possible. It assumes only knowledge of calculus and elementary algebra at an advanced undergraduate level, being accessible to a wide audience. This edition contains two new chapters (11 and 12) on the genericity and stability of unfoldings.
There is no other book where complete proofs of the celebrated Thom’s classification theorems for elementary catastrophes and for versal unfoldings are given using only advanced undergraduate methods. The proof of Thom’s classification theory for degenerate critical points is based on Morse’s lemma and reduction lemma, which are proved in chapters 1 and 3, respectively. In chapter 2 the notions of catastrophe and stability are explained and the first two elementary catastrophes, the fold and the cusp, are introduced.
In chapter 4 a linearization lemma is stated in order to prove Mather’s necessary condition for determinacy. An interesting consequence of this lemma is a general lifting property for curves in orbits of transformation groups.
Chapter 5 deals with the codimension of a smooth function \(f\), which gives the minimal number of parameters needed for a versal unfolding of \(f\). Finite determinacy is equivalent to having a finite codimension.
The proof of Thom’s classification of the seven elementary catastrophes is given in chapter 6. This theorem classifies degenerate critical points of smooth functions up to codimension at most 4. It can be extended to codimensions at most 6. The classification for codimensions 5 and 6 is detailed in the exercises to this chapter. This concludes the first part of the investigation.
Chapter 7 is an introduction to the main goal of catastrophe theory: to classify smooth functions according to their behaviour under perturbation or, equivalently, to find their versal unfolding. The fundamental theorem on universal unfoldings is stated. The next 3 chapters contain the theorems needed to prove this theorem. Transversality is the major tool for determining whether a smooth function admits a versal unfolding or not. The most difficult part of the proof of Thom’s fundamental theorem on universal unfolding is based on the Malgrange-Mather theorem, whose proof derives basically from [M. Golubitsky and V. Guillemin, Stable mappings and their singulatities (Springer, New York) (1973; Zbl 0294.58004)].
In chapter 11 it is shown that the versal unfoldings of germs of codimension at most 4 form an open and dense subset of the set of all unfoldings with at most 4 parameters. Thus Thom’s classification theorem is generic. The main result in chapter 12 is that stability and versality are equivalent to each other.
The original chapters have been revised and new material was included. The most important is the theorem of uniqueness of the residual singularity. The present book is still self-contained. Many interesting examples and exercises as well as many illustrations enhance the understanding and readability of the book under review.