Normal Forms and Unfoldings for Local Dynamical Systems J. Murdock, Springer Monographs in Mathematics, Springer Verlag 2003, 494 p., price € 79,95, ISBN 0-387-95464-3.
Reviewer: Henk Broer, University of Groningen, The Netherlands.

The theory of local dyamical systems studies neighborhoods of a given equilibrium point, in particular the dynamical behavior that is generically possible. In the book under review the dynamical systems are autonomous (or occasionally time-periodic) systems of ordinary differential equations. At the equilibrium point the first concern is with the linear part, but next also the rest of the Taylor series comes into view as well as unfoldings or deformations of these by parameters. The interest then is with the dynamical behavior as a function of these parameters, in particular with transitions or bifurcations between different types of behavior. The number of parameters needed to see a given type of equilibrium in a persistent way, is called the codimension of the `singularity'. Usually models are considered by truncating the Taylor series at a suitable order. The question then is what properties of the truncated family of systems are persistent. Such properties concern the bifurcation of other equilibria and periodic solutions, their stable and unstable manifold, homo- or heteroclinic connections, etc. The persistence problem for families of systems can be restricted to a given universe, like to the world of Hamiltonian, of volume preserving, of equivariant or of reversible systems.

One major tool of this research is the theory of normal forms, which at least goes back to Poincaré. It deals with the adjustment of the Taylor series at the equilibrium by changes of coordinates. The adjustment aims to be a simplification, which often amounts to the elimination of as many as possible terms in the series. Normal form theory in itself already covers a large area of research, with historical contributors like Birkhoff, Sternberg, Bruno, Gustafson, Takens and many others. The techniques at equilibria largely translate to periodic and quasi-periodic orbits and also it turns out that the preservation of a symplectic or volume form, invariance (or equivariance) under a symmetry group can be preserved by using a Lie-formalism. In the latter case also reversing of symmetries can be taken into account. Moreover, in this same setting the local normal form at a fixed point of a diffeomorphism (Takens approximation by a flow) can be dealt with. This scope is what I expected when seeing the general title of this almost 500 page work.

The book under review however does not address these general issues, but quickly narrows down to four major normal form `styles', among which the semisimple style, the `Elphick-Iooss' inner product style and the `Cushman-Sanders' sl(2) style. All styles refer to different solutions of the problem to characterize `complementary' subspaces of homogeneously polynomial vector fields that can not be eliminated by changes of coordinates. The treatment has a definite algorithmic and algebraic nature, aiming at the computation of normal forms in concrete examples with help of computer-algebra packages. Background for this is the theory of representations and invariants. An appendix of more than 60 pages provides an algebraic background for this.

To my knowledge the monograph under review is the first succesful attempt to deal with the `Elphick-Iooss' inner product style and the `Cushman-Sanders' sl(2) style at a larger scale. The emphasis on the computability of normal forms by computer-algebraic tools is useful for applications in concrete examples. As the author has it in the preface, the book ``... does not require any specific knowledge other than the basic theories of advanced calculus, differential equations and linear algebra.'' Moreover, the book is meant to be self-contained. From this on the one hand a certain emphasis on the algebraic aspects can be understood. On the other hand, however, my general feeling is that the whole style is dominated too much by algorithms and algebra.

To explain this further, let me elaborate briefly on the most common normal form style, the semisimple style. Here the linear part S of the vector field at the equilibrium is semisimple and the corresponding ad-operator £ = ad_S = [S, - ], defined on the spaces of homogeneous polynomial vector fields, is semisimple as well. In this case the image im £ consists of terms that can be eliminated by changes of coordinates. A natural choice for the `complementary' space then is the kernel ker £. The elements of this space are vector fields that commute with the linear part, which gives rise to a toroidal symmetry (in the center manifold). This includes the case of the 1-torus or circle. As a consequence, any truncated normal form polynomial can be reduced in dimension by dividing out this symmetry. This kind of reduction is important for many applications, in particular for bifurcation theory, as can be seen from a number of textbooks. In the Hamiltonian setting this method is closely related to classical reductions in integrable systems.

Moreover, when the linear part of the vector field at the equilibrium point has a Jordan splitting A = S + N, with S semisimple and N nilpotent, one still can take care that the remaining terms all commute with S. This again makes a toroidal reduction possible. However, now the nilpotent term N gives the possibility of even further restricting the complementary space. The option of toroidal reduction is one of the reasons why many users of normal form theory restrict to the semisimple style. In the nilpotent case the semisimple part vanishes and the above considerations break down. This is one good reason to resort to one of the other normal form styles.

In the present book, these more general issues of normal form theory are not dealt with in any substantial way: there is no mentioning at all of this toroidal symmetry. Indeed, this would have required a more generally mathematical entrance to the theory, with a good eye for applications. As a further illustration of this let me mention that, when introducing universal unfoldings of matrices the author uses a computation with series and not the more geometric theory of `matrices depending on parameters' by Arnol'd. Terms as `versal' and `transversal' do not occur in the book. A similar remark holds for differential geometric aspects of the Lie theory approach and general remarks on normal form approaches that automatically preserve given structures, again see above. This narrowness of scope also is reflected in the bibliography, where I found many items missing. Among those I like to mention F. Takens, Singularities of vector fields, Publ Math IHES 43 1974, which is fundamental both for the semisimple and nilpotent case.

In summary, I like to raise the question for what readership the book was written. The author has two disclaimers in the preface saying that the text is not an `introduction' to the subject and that ``... it is is unlikely that a student will find this book to be understandable without some previous exposure to the subject of dynamical systems, ....'' I would agree to both statements. In addition to this, I have doubts that any reader, not already acquainted with an appropriate amount of algebra, would find the text accessible, this notwithstanding the appendices. In that respect the text really seems to address experts in the field. As said before, the text succesfully addresses computer-algebraic aspects of certain normal form computations that are useful for applications in concrete examples.

Nevertheless, assuming a wider background in both algebra and (differential) geometry, could have led to a different book with a broader scope as sketched before. Such a strategy would have done more justice to the present title and size of the book, which then also might have been more interesting for an even larger group of experts.