This is the web page for the book
NORMAL FORMS AND UNFOLDINGS FOR LOCAL DYNAMICAL SYSTEMS,
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.
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 jmurdock@iastate.edu, and I will post your contributions here.
ERRATA IN MY BOOK
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.
RECENT WORK
My paper "Hypernormal form theory: foundations and algorithms," J. Diff.
Eq. 205 (2004) 434-465, may be viewed as a continuation of section 4.10 of
the book.
WHAT IS THE BEST FORMAT? STYLE? NOTATION?
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.
FORMAT:
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).
STYLE:
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.
At the time the book was written, unfolding computations (by my method)
were only available for the simplified normal form. Although I have
not worked it out yet, I think that I understand how to fix this so that
unfoldings can be done in the inner product and sl(2) normal forms. The
unfolding computation is a simple special case of (partial) hypernormalization.
Essentially my error was that when I computed unfoldings, I only used
weak hypernormalization (as defined in section 4.10, but with parameters
added), and weak hypernormalization is not enough.
NOTATION:
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.)
ABOUT BELITSKII
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.