Author's comments (16 October 2006)
Reviewer 3 makes a claim that the purpose of computing to higher order is to get asymptotic validity over longer intervals of time. I would not agree that this is the purpose. It is one of many purposes, but it only works under certain circumstances (when the linear term is semisimple, and all if its eigenvalues are simple and lie on the imaginary axis.) I have discussed this issue of error estimation for normal forms extensively in chapter 5 of my book.
The other and more important reason for going to some higher degree is to achieve jet sufficiency with respect to a particular property of the flow (or, if possible, to achieve jet sufficiency with respect to topological equivalence). These issues are too technical for a short article.
The new reviewer also added references (Birkhoff, Bruno, and Nayfeh). Birkhoff and Bruno are valuable references but I would not send someone to them from a quick survey article. Birkhoff is historically valuable but very out of date, and Bruno is, shall we say, peculiar in his personal interests. (Much of his book is concerned with the question of when a normal form transformation converges. He is the acknowledged expert on this topic, but it is mostly irrelevant today although it matters greatly for a few purposes. A lot of the rest of his book deals with his power transform method, which is equivalent to a portion of the blowup method used for instance by Freddy Dumortier, and is not as powerful.) Nayfeh is popular but often wrong from a mathematical perspective.
Reviewer's comments (18 October 2006)
As for the text, I accept Murdock's response. After all, the differences are mainly questions of taste.
As for references: True, Birkhoff s ancient, but credit ought to be given when it is due; Bruno is a bit special, but is a seminal work, and deals with some special cases of non-semi-simple systmes; Nayfeh - let Murdock decide.
The following is thoughts I'd like you to forward to Murdock, as I'd like to share them with him. Let him decide what to do or not to do, as a result.
It is true that there have been susbtantial developments in the formal theory of Normal Forms. However, my impression from the text as is that the uniformed reader gets the notion of Normal Forms as some abstract mathematical construct, detached from "real life". After all, the main purpose for their development, as well as their main use as of today (already extended to PDE's) is in perturbation expansions. There is a whole crowd of applied maathematicians and theoretical physicists using normal forms as a tool in analyzong perturbed systems in areas such as:
Nonlinear wave equations; Classical Mechanics; Coupled systems of ODE's in Physics; Engineering, Mathematical Biology .....; Questions of asymptotic integrability; accelerator design, and more. None of this is coneveyd to the reader. My feeling is that an item in an encyclopedia ought to convey more than just the basic information about the entry. i.e., a definition is not enough. Some background material is also needed.
According to the author, having a semisimple linear part is the necessary condition to get the asymptotic approximation of an ODE by using the normal form method.
However, when the eigenvalues of the linear part are in the Poincare domain, we have (1)the normal form transformation is convergence. (2)the normal form can be chosen to be in a triangular form and therefore the normal form equation can be integrated directly.
Therefore, if the eigenvalues are all negative, in a neighborhood of the origin, we can get the series solution (convergence) of the ODE by using the normal form technique. It seems that having a semisimple linear part is not necessary?
Reply to question
When I talk about asymptotic expansion, I (usually) mean for time 1/epsilon where epsilon is a small parameter used to dilate the coordinates about the origin. (There are other cases where the approximation is for all future time, or is defined in a shadowing sense. The full discussion is in section 5.3 of my book.)
I don't think that convergence of a series solution is relevant. I suppose you mean a power series solution in time. It is true that sometimes this is convergent. But being a power series, it can only be convergent on a fixed interval of time, not an expanding interval of length 1/epsilon. Even if it is convergent for all t, the truncation error estimate is in terms of a power of t, not a power of epsilon.