`UK Nonlinear News`,
`February 2004`

Springer Monographs in Mathematics, Springer-Verlag 2003.

`Normal-form theory` has become a celebrated topic which is widely
used in nonlinear science. Its basic application is the classification of
dynamical systems into categories by transforming nonlinear vector fields into
a standard form which is determined by their linear part; one expects that the
solution sets of those dynamical systems whose vector fields lie in the same
category should exhibit similar features. In essence, the vector field is
expanded in a power series and a sequence of near-identity transformations is
constructed which systematically remove `non-resonant' homogeneous quadratic,
cubic, quartic, ... terms without affecting those of lower order; the
definition of the term `non-resonant' is part of the theory. The resulting
vector field is said to be in normal form up to a certain order, and up to
this order contains only `resonant' terms. The book under review is chiefly
concerned with the mechanics of carrying out this procedure.

The author uses terminology which I have not seen elsewhere (and avoids the
words `resonant' and `non-resonant' as much as possible). The procedure of
constructing and handling the near-identity transformations is called a
`normal-form format.' For example, one can use a single transformation with
coefficients determined recursively or a sequence of transformations applied
iteratively, and the transformations can be constructed directly or via
auxiliary generating functions; the author discusses five such normal-form
formats in detail. Regardless of how the actual transformation theory
is organised, each step of the normal-form theory requires the solution of an
inhomogeneous linear equation for the
`homological operator` £ (which is fixed by the chosen
normal-form format) in the space of homogeneous polynomials of the relevant
order. To solve the equation it is necessary to write this space as the
direct sum of the range of £ and a complement, the precise choice of
which is called the `normal-form style.' The author treats five normal-form
styles, which he terms semisimple (applicable only when
the linear part of the vector field is associated with a semisimple matrix),
extended semisimple, inner product (previously championed by G. Elphick and G.
Iooss),
simplified and sl(`2`) (previously championed by R. Cushman and J.
Sanders). The choice of a normal-form format and style completely determines
the normal form, and there remain the tasks of characterising the terms in the
transformed vector field (the `description problem') and calculating their
coefficients (the `computation problem').

After some introductory examples, the author begins his treatment in Chapter 2
with a detailed discussion of the `splitting problem', that is the task of
finding a direct-sum decomposition of a vector space `V` into the
range of a linear operator `L` on `V` and a suitable
complement. Several choices of complement are discussed which later evolve
into the `normal-form styles' (splittings associated with the homological
operator £) discussed above. The following Chapters 3 and 4 on
respectively linear and nonlinear normal forms make up the heart of the book.
Chapter 3 is dedicated to normal forms related to unfoldings of linear vector
fields. At issue is a linear vector field depending upon a parameter
ε the vector field is expanded in a power series in ε and
the normal-form theory is used to remove non-resonant terms in the higher-order
ε-dependent terms. Many of the ideas used in subsequent chapters
are introduced and developed here (and in lengthy appendices),
in particular a thorough explanation of five
`normal-form formats' (transformation theories) and the use of `normal-form
styles.'
This material is expanded into full nonlinear normal-form theory in Chapter 4,
where
the description and computation problems are also comprehensively discussed.
Ring and module theory are brought to bear on the former, while algorithmic
methods
suitable for symbolic computation are developed for the latter.

The final two chapters are devoted to some central applications of nonlinear normal forms. One of the great advantages of `truncated normal forms' (obtained by placing a vector field into normal form up to a certain order and dropping higher-order terms) is that their geometrical structures such as stable, unstable and centre manifolds and fibrations of the centre-stable and centre-unstable manifolds over the centre manifold assume an obvious, almost explicit structure and are readily discussed. (Indeed, `truncated normal forms' are often completely integrable.) Chapter 5 presents a detailed discussion of this feature and addresses related approximation issues: how well does for example the centre manifold of a truncated normal form approximate the centre-manifold of the vector field obtained by adding higher-order terms? Finally, in Chapter 6, the author turns to local bifurcation theory in which local bifurcation problems are studied using normal forms. The emphasis here is upon the classical unfolding situation in which a parameter is varied so that the nature of the linear part of the vector field near the origin changes, and associated steady-state, periodic or homoclinic bifurcation is detected according to the signs of certain coefficients in an appropriate nonlinear normal form. The key fact here is that only a small number of coefficients of low-order terms is required to settle the issue (for the `full' vector field as well as the `truncated normal form'), and the author explains the use of `Newton diagrams' to indicate which coefficients are important in this respect.

This book certainly represents a very thorough treatment of the anatomy of
normal-form transformations; in addition to the detailed explanations of
normal-form `formats' and `styles' there is a comprehensive explanation of
canonical forms for matrices (for example the Jordan canonical forms), a
detailed algorithmic approach to the `description' and `computation' problems
and two lengthy appendices on ring and module theory. It may serve well as a
reference work on this aspect of normal-form theory, and indeed the author
achieves his stated aim of providing an encyclopedia of results and
explanations which are not easily found in the existing literature. Given this
thoroughness, it is perhaps surprising that, with the exception of the rather
cursory Section 4.9 on Hamiltonian systems, the author does not present a
systematic treatment of normal forms for vector fields with symmetries.
(Incidentally, I disagree with his statement that `an inner-product
normal-form style for the Hamiltonian case has not been worked out' - it is
presented in detail by Elphick (`Global aspects of Hamiltonian normal forms,'
`Phys. Lett. A` ** 127**, pp. 418--424,
1988) and Meyer & Hall (`Introduction to Hamiltonian dynamics and the
`N`-body problem,' Springer-Verlag, 1992, Section VII.C.2).)

I suspect that the book is, however, unlikely to be of interest to the core
readership of `UK Nonlinear News` whose primary interest is in the
use of normal forms as a tool for the analysis of phenomological problems.
These readers would be more interested a book which catalogues the solution to
the `description problem' for normal forms in a wide range of unfolding
scenarios (which goes far beyond the limited number of more or less standard
examples presented here) together with a list of `tips and tricks' for
computing important coefficients as efficiently as possible
(rather than an algorithmic procedure).
In this respect a book such as `Topics in Bifurcation Theory and
Applications` by G. Iooss &
M. Adelmeyer, World Scientific, in which a normal form with an `inner product
style' is developed and applied to a large number of unfolding scenarios,
would be much more helpful.

`UK Nonlinear News` thanks
Springer-Verlag
for providing a review copy of this book.

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