This is the web page for the book


 by James Murdock, published by Springer-Verlag in 2002.  This page will contain errata, comments, and references to new work related to what is contained in the book.

HERE IS A LINK TO THE Springer-Verlag web page for the book.

If you have found any errata not listed here, or published any papers that answer questions raised in the book or in any other way advance the program of research to which the book belongs, please let me know by email to, and I will post your contributions here.


p. 10, last line: g = - f_{theta}(as it is two lines above).  This does not affect the argument.

p. 24, last line: k times beta should be added to c, not to d.  Similarly, at the top of page 25, c+k(beta) is renamed as c.  This does not affect the argument.

p. 25 and 26, equations (1.2.16) and (1.2.17):  beta should multiply xy, not y^2.  (It is correct throughout the rest of the discussion.)

p. 201, two lines below (4.5.30): vector field should be vector space.

p. 286 equation (4) in theorem 4.10.5: in the last term the coefficient is alpha sub 2 nu.

p. 374 middle displayed equation: q = p + Ak.

p. 377, five lines from bottom:  script ell = 2, not 4.

Section 6.4 contains an error that is corrected in the third paper listed below (under "recent work").  This error only arises when A contains Jordan blocks of different size with the same eigenvalue.  The error did not appear in the original paper on which section 6.4 was based.


Hypernormal form theory: foundations and algorithms, J. Diff. Eq. 205 (2004), 434-465.  This paper may be viewed as a continuation of section 4.10 of the book.

A new transvectant algorithm for nilpotent normal forms.  Joint paper with J. A. Sanders. J. Diff. Eq., 238 (2007), 234-256.  This extends the material in section 4.7 of the book.

An improved theory of asymptotic unfoldings.  Joint paper with David Malonza.J. Diff. Eq. 247 (2009), 685-709.  At the time the book was written, unfolding computations (by my method, explained in section 6.4) were only available for the simplified normal form style.  This limitation is removed in this paper.  In addition, an error that slipped into section 6.4 is corrected.  (See errata above.)


One of my early goals in writing this book was to sort out for myself the issues surrounding what I eventually decided to call formats and styles.  During the course of the writing, my opinion about the ``best'' format and style changed several times.  My current views are as follows.

Of course for beginners the best format is 1a; it is the only format I use in my introductory graduate course in dynamical systems.  For more advanced purposes my current view is that format 2b is the best.  Here are the reasons:

1.  As a generated format, it allows for the preservation of sub-Lie-algebras and other representation spaces.
2.  Format 2a can be regarded as a subset of format 2b, in the sense that every format 2a generator is a legitimate format 2b generator as well.  So, for instance, if one has written software for format 2b, it can be used to carry out normalization by format 2a whenever that is desired; just iterate with several homogeneous generators.
3. Unlike format 2c, the generator of an inverse transformation is minus the generator of the transformation.
4. Format 2b is best for hypernormalization, as discussed in section 4.10.  (See the middle paragraph on page 293.)
5. Format 2b is best for quadratic convergence (although this topic is barely mentioned in the book).

A case can be made that the best normal form style is the sl(2) style, although it is not popular.  The arguments would be as follows:

1. The sl(2) style always coincides with the semisimple style when the linear part is semisimple.  This is not true of the inner product normal form, except when the (semisimple) linear part is normal (commutes with its conjugate transpose).
2,  The sl(2) style is always an extended semisimple normal form.  This extends the first point to the case when the linear part is not semisimple.  Again, the inner product normal form is only an extended semisimple normal form if the semisimple part of the linear part is normal.
3.  Because of points 1 and 2, the sl(2) normal form always has the symmetries (equivariance) that comes from the semisimple part of the linear part.  This usually provides a number of preserved foliations, as shown in chapter 5.
4.  When the nilpotent part is nontrivial, the sl(2) normal form also has a foliation that is not preserved but is nearly preserved, in the sense of Lemma 5.4.2.  It is true that we do not yet know how to make use of this lemma, but nothing like it is true in other normal form styles.
5.  The sl(2) normal form may admit the most efficient set of algorithms for computation of normal forms by computer algebra (when a nilpotent part is present).. These include the pushdown algorithm, which computes the projection without the necessity to solve any systems of linear equations.  The pushdown algorithm is not terribly efficient for problems of low dimension, but in higher dimensions is probably more efficient than methods that require solving equations.

The chief drawback to the sl(2) normal form, aside from its steep learning curve (requiring sl(2) representation theory), are the ``nuisance factors'' (the pressures) that inevitably enter in some way, no matter how the nilpotent part is embedded into an sl(2) triad.  If one's goal is merely to eliminate as many entries as possible in the vector field when written as a column vector, and if the linear part is in Jordan form, then the best style is the simplified normal form.  This is an extended semisimple normal form, so it does have some equivariance properties.

In the early stages of writing this book I changed notations several times.  Finally it became necessary to "lock in" the notation that I would use, and not allow myself to change it again.  I did this by formulating the three principles stated on page xv.  I have recently thought through this issue once again.  For the most part I like the notations in the book, except that (as much as possible) it would be preferable to avoid the use of  "anti-objects" (group anti-homomorphisms and group and Lie algebra anti-representations).  Most of the "anti-objects" in the book arose from an effort to match the notations in common use among applied mathematicians.  But in fact there is a conflict among notations preferred by pure mathematicians, and it is (almost) impossible to eliminate all anti-objects and still have a set of notations that covers the whole range of operations that arise in different parts of the subject.  The best that I can do is as follows:   Click here for an essay about notation for normal form theory.  (This is in pdf format and can be read if you have Adobe Acrobat.)


I learned about Belitskii's work two weeks before the deadline for the manuscript.  I was able to include one reference at that time, and another at the time the proofs were corrected.  However, even at that time I did not understand the full extent of Belitskii's contributions to the subject, and in fact recently rediscovered something that Belitiskii had already proved many years ago.

Belitskii discovered and developed what is called in my book the inner product normal form style, independently of Elphick et. al. and many years earlier.  Furthermore, his work is not limited to the normal form, but extends to what are called hypernormal forms in my book; his version of the hypernormal form may be called the inner product hypernormal form style, because it is based on the same inner product used for the normal form.  The hypernormal form spaces in this hypernormal form style are the kernels of the adjoints of Lie operators defined by nonlinear vector fields, and these adjoints are not themselvers vector fields, but higher-order linear differential operators.  None of the other work on hypernormal forms (reported in considerable detail in the Notes and References to Section 4.10 of the book) is at all concerned with defining a hypernormal form "style" (in the sense that word is used in the book, namely, a systematic rule for selection of complementary subspaces to the image of a Lie operator); rather, the selection of such complements is arbitrary.  So far, the inner product sytle is the only one that has been extended from the context of normal forms to that of  hypernoral forms.  (In addition to all of this, Belitskii went on to consider ``normalization beyond all orders,'' that is, normalization of the full smooth vector field rather than just its formal power series.  The results are technical and incomplete, and do not fit in well with the philosophy of my book.)  This is all described in  Reference [14] in the bibliography of my book.  However, Reference [14] is only a summary, and does not give a complete list of references for the original work.  No reference is given in [14] to indicate the location of the original paper or papers in which Belitskii worked out the inner product normal and hypernormal forms, and some of the references that are given seem to be incorrect.  For instance, in the middle of page 38 of Reference [14] Belitskii says "a complete proof is given in [6]'", but reference [6] (in Belitskii's bibliography) is only two pages long and contains no proofs (unless the English version is not a complete translation of the Russian version, which I have not seen).

This reference [6] is the same as Reference [15] in my book, which is slightly garbled: the page numbers should be 59-60 (not 46-67).  I am still trying to find all of Belitskii's work on this topic that has been translated into English.  I may post additional comments here later.