# Normal Forms and Unfoldings for Local Dynamical Systems

## By Mark Groves

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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