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\def\giorno{4 November 96}
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\font \petit = cmr9
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{\nopagenumbers
~\vskip 3 truecm
{\bf POINCARE' RENORMALIZED FORMS}
\footnote{}{{\tt Revised and expanded version -- \giorno }}
\footnote{}{{\tt P.A.C.S. nos.: 02.30 , 03.20 , 46.10}}
\vskip 3 truecm
Giuseppe Gaeta$^*$
I.H.E.S., 35 Route de Chartres
91440 Bures sur Yvette (France)
{\tt gaeta@ihes.fr}
\vfill\parindent=0pt
{\bf Summary.} {In Poincar\'e Normal Form theory, one considers a series
of transformations generated by homogeneous polynomials obtained as
solution of the homological equation; such solutions are unique up to
terms in the kernel of the homological operator. Careful consideration
of the higher order terms generated by polynomials differing for a term
in this kernel leads to the possibility of further reducing the Normal
Form expansion of a formal power series, in a completely algorithmic way.
The algorithm is also applied to a number of concrete cases. An alternative
formulation, conceptually convenient but computationally unpractical, is
also presented, and it is shown that the discussion immediately extends to
the Hamiltonian case and Birkhoff normal forms.}
\vfill
{\petit $^*$ Also at Centre Emile Borel, I.H.P., UMS 839 CNRS/UPMC.
On leave from Department of Mathematics, Loughborough University,
Loughborough LE11 3TU (England)}
\vfill \eject}
\pageno=1
\parindent=10pt
\parskip=0pt
{\bf Introduction, motivation, overview.}
The theory of {\it Normal Forms} \ref{1-3} was created by Poincar\'e in
his Thesis, and is still a fundamental tool in our study and
understanding of Nonlinear Dynamics.
In this note we will consider systems of ODEs in $R^n$ defined by $\xd
= f(x)$ with $f(x)$ a (formal) vector power series having a zero in the
origin, and (formal) coordinate transformations given by (formal) power
series\footnote{$^1$}{This corresponds to the original Poincar\'e
theory; for the generalizations of normal forms theory to different
setting and/or more general class of transformations -- e.g. ${\cal
C}^k$ or topological ones -- see \ref{2}.}.
By a (generally, only formal) series of near-identity coordinate
transformations, it is possible to transform a system to its Normal Form
(NF in the following) up to any given order\footnote{$^2$}{In the following,
we will mean by NF the
``infinite order'' NF, and denote as ``partial NF (of order $m$) the NF
of order $m$.} $m$, and formally for $m =
\infty$; thus, the study of the local behaviour of ODEs around a
singular point can be reduced to the study of the local behaviour of ODEs
which are in NF.
It should be stressed that $A)$ the equivalence between the ``original
system'' and its NF is, in general, only formal: thus, in general a
system is {\it not} conjugated to its NF; however, for $m$ sufficiently
small (depending on the system), the system {\it is} conjugated to its
partial NF of order $m$; $B)$ For a given system, the reduction to NF
-- although obtained by means of a well defined algorithm -- becomes
computationally very difficult with the increasing of $m$: the required
computations can be set in terms of linear algebra, but they require at
order $m$ to consider a basis of homogeneous monomials of degree $m$ in
the variables $x_1 , ... , x_n$; calling $M(m,n)$ the dimension of this
(this is the number of partitions of $m$ as the sum of $n$ nonnegative
integers),
we have to then to consider -- and invert -- matrices of order $\[ M(m,n)
\]^n$.
Thus, in practice, when we analyze a {\it given} system by means of NF
techniques, we consider the partial NF of some order $m$, study the
truncation of this at order $m$ -- which is by construction in NF --
and then resort to other kinds of considerations, typically $m$-th
order averaging, to ensure that the trajectories of the truncated and
the full system are near enough (say with a distance less than $\eps$)
for long times (typically for $t < 1/ \eps^m$).
On the other side, if we want to consider the most general behaviour of
a system, we can operate a first reduction to (Jordan) normal form for
its linear part, i.e. for the matrix $A = (Df)(0)$, and then study the
most general NF compatible with this linear part (the determination of
this is also referred to as the problem of {\it unfolding} of the NF),
and the behaviour of the systems given by these NFs.
It is well known that the NF corresponding to a given system is in
general {\it not unique}\footnote{$^3$}{This is related to the fact that
the homological operator $\L_0$ has a nontrivial kernel, see below.};
correspondingly, the unfolding of NFs -- which can be described as the
most general {\it resonant} system\footnote{$^4$}{I.e. a system with
nonlinear part in $\ker (\L_0^+ )$, see below.} -- is exhaustive, but
in general not minimal: i.e., the Poincar\'e-Dulac theory provides a
list of NFs such that any system can be (formally) reduced to one of
the NFs of teh list, but we are not guaranteed that the NFs are
pairwise not conjugated.
It is thus of interest to provide a reduced classification of NFs, i.e.
a list of NFs (say, corresponding to a given linear part $A$) in which
the redundancies in the Poincar\'e-Dulac classification, or at least
some of them, have been eliminated.
This problem has been considered by several authors \ref{4-9}, mainly
in the language of Lie algebra filtration; indeed, several theoretical
result exist (see e.g. the section on ``further normalization'' in
\ref{6}), and it is actually also possible to define a {\it unique
NF} \ref{9}. However, all these results -- to the best of my
knowledge -- are of difficult concrete implementation.
The purpose of this note is indeed to propose a procedure of
``higher order normalization'' (which for the formal case $m = \infty$
I call ``renormalization'') which is completely algorithmic, and not
more difficult to implement than the standard Poincar\'e-Dulac
normalization. As the proof of this will be based on consideration of
the procedure itself, the discussion to follow will be completely
constructive (as in the very spirit of perturbation theory \ref{10}).
Actually, the procedure I propose below is nothing else than a direct
generalization -- or, more modestly, iteration (see sect.10) -- of the one proposed
by Poincar\'e; the main ``new'' ingredient will be to make use of the
freedom in the choice of the generating functions $h_k$ for the
near-identity coordinate transformations which comes from the
nontriviality of the kernel of $\L_0$, and a control of higher-order
effects in this transformation.
Not surprisingly if one considers that -- as mentioned above -- previous
results were obtained in the framework of Lie algebra, we
will find it convenient to consider Lie-Poincar\'e \ref{11,12} rather
than standard Poincar\'e transformations (once again, details are given
below). However, I will on purpose {\it not} discuss the Lie-theoretic
side of the procedure described here: on the one side, the Lie-theoretic
frame is clearly discussed by other authors; and on the other, I want
to focus attention on the computational side and show how -- as already
stated -- this higher order normalization is completely in the original
frame of the Poincar\'e approach.
\bigskip
{\it Plan of the paper.}
The paper is divided into three parts. In the
first part (sections 1-4) we will recall the classical
(Poincar\'e-Dulac) theory of NFs; this, although well known (see e.g.
\ref{1}), is discussed in some detail both to fix notation and to
point out some features which are usually not discussed but which will
be relevant for the discussion to follow. Thus, after briefly
introducing the general setting of NFs theory (sect.1) and the detailed
computation of the effect of a Poincar\'e transformation (sect.2), we
will recall the Poincar\'e algorithm for reduction to NF (sect.3), and
the formulation based on Lie-Poincar\'e, rather than standard
Poincar\'e, transformations (sect.4). As this is perhaps less well
known than the standard one, we also discuss it in more detail in the
Appendix A.
In the second part (sections 5-11) we introduce the ``higher
normalization'' (and the ``renormalization'' {\it tout court}),
prove constructively -- i.e. by giving a completely explicit
algorithm -- our main result, i.e. that any system can be transformed
into $m$-th normalized form for any $m$ (and formally into renormalized
form, i.e. for $m = \infty$) by a formal series of Poincar\'e
transformations, and discuss the relevance and limitation of this. In
more detail, we first introduce ``higher order homological operators''
(sect.5) and discuss how these are related to the non-unicity of the
standard NF (sect.6). We will then discuss as one can use ``higher order
effects" in the Poincar\'e transformations (i.e; effects on $f_{k+m}$
for the transformation with generator $h_k$) to further simplify the NF;
we will at first show explicitely and in some detail this for $m=1$ and
$m=2$ (sect.7); this will provide an obvious motivation for the
introduction of the required abstract functional setting (sect.8), in
terms of which the generalization of the discussion of section 7 will be
immediate, and we will give an explicit constructive -- and easy -- proof
of the general result (sect.9). We will also discuss a slightly different
approach, based on fully ``iterate'' the standard normalization several
times (sect.10); this could provide perhaps an easier conceptual
understanding, as it is an even more direct extension of the Poincar\'e
procedure, but is computationally very inconvenient, as it is based on
``successively sweeping all orders'' several times, each time considering
higher order effects. The discussion of several questions stemming from our
construction, and of some of its advantages and limitations, is then given
(sect.11).
The third part (sections 12-14) is devoted to examples; in all the
examples we consider one is actually able to determine the {\it full}
renormalized form (i.e. the unfolding can be determined up to $m =
\infty$), and in many of them this is dramatically simpler than the
standard Poincar\'e-Dulac NF. In particular, we consider the classical
two-dimensional resonant problem with linear part corresponding to
rotations (sect.12), i.e. with eigenvalues $\sigma = (\la , - \la )$;
the more general two dimensional problem with eigenvalues
$\sigma = (\la , - n \la )$
(sect.13); and a number of three-dimensional problems (sect.14).
We add then two appendices; in the first we discuss in more detail
the Lie-Poincar\'e transformation (app.A), while the second is devoted to the
extension of our approach and result to the Hamiltonian case and
Birkhoff normal forms (app.B).
A preliminary version of part of
this work appeared in preprint form as {\tt mp-arc 96-263}.
\bigskip
{\bf Acknowledgements.}
This paper was completed during my stay in I.H.E.S.; I would like to
thank the Director, prof. J.P. Bourguignon, for his invitation,
and the whole staff for the warm hospitality.
In this period, I also enjoyed the hospitality of the Centre Emile
Borel of the Institut Henri Poincar\'e, in the frame of the Integrable
Systems semester.
\vfill\eject
{\bf 1. General setting.}
The Poincar\'e theory of Normal Forms \ref{1-3} for dynamical systems,
i.e. for first order autonomous smooth ODEs of the form
$$ {\dot x} = f(x) \qquad \qquad , ~ x \in R^n ~,~ f : R^n \to R^n
\eqno(1.1) $$
which we also call {\it dynamical systems},
or equivalently for vector fields
$$ X = \sum_{i=1}^n \ f^i (x) {\pa \over \pa x^i} \ , \eqno(1.2) $$
is based on systematically employing near-identity changes of
coordinates with homogeneous vector polynomial functions as generator.
One is interested in $f$ being a formal power series, i.e.
$$ f(x) = \sum_{k=0}^\infty f_k (x) \eqno(1.3) $$
with $f_k (x)$ homogeneous of order $(k+1)$ in the $x$.
We denote by $V$ the set of vector formal power series $f:R^n \to R^n$
which have the origin as a fixed point, and by $V_k \ss V $ the set of
polynomial vector functions homogeneous of order $(k+1)$; obviously,
$$ V \ = \ \sum_{k=0}^\infty {}^\oplus \ V_k \ . \eqno(1.4) $$
It will be useful to define the bracket $\{ . , . \} : V \times V \to V$
given by
$$ \{ f , g \} = (f \cdot \grad ) g - (g \cdot \grad
) f \equiv f^i {\pa g \over \pa x^i} - g^i {\pa f \over \pa x^i} \ ;
\eqno(1.5) $$
this expresses the Lie commutator of vector fields when we look at the
component of vector fields in the $x$ coordinates; that is, for
$X=f^i \pa_i$ and $Y=g^i \pa_i$, we have $[X,Y] = h^i \pa_i$ with $h =
\{ f , g \}$. Notice that $$ \{ . , . \} \ : \ V_k \times V_m \ \to \
V_{k+m} \ . \eqno(1.6) $$
The (standard) homological operator $\L_0$ can be defined in terms of
this bracket, as $\L_0 (.) = \{ f_0 , . \}$; by (1.6), $\L_0 : V_k \to
V_k$.
In the following, we will need (linear) operators acting between the
spaces $V_k$, and in particular we will have to consider the
complementary sets of the ranges of such operators; it is thus
convenient to introduce a scalar product in $V$ (actually, in each of
the $V_k$), so that we can consider the adjoint operators.
It turns out that the convenient scalar product is defined as follows
\ref{1,13,14}. First of all, we notice that each of the $V_k$ is a
finite dimensional vector space. In each of these, we can choose a basis
$e_{\mu , j} (x)$ of functions which have components
$$ e_{\mu,j}^i (x) = x^\mu \de_{i,j} = (x^1)^\mu_1 ... (x^n)^\mu_n
\de_{i,j} \ ; \eqno(1.7) $$
we define then a scalar product in $V_k$ as
$$ \( e_{\nu , j } , e_{\mu , i} \)_k \ = \ < \mu , \nu > \ \de_{i,j} \ ,
\eqno(1.8) $$
where $<.,.>$ is a scalar product in the space of monomials (in $x^1
... x^n$); the customary -- and convenient -- choices are either $< \mu
, \nu > = \de_{\mu , \nu } \equiv \prod_{i=1}^n \de_{\mu_i , \nu_i }$
(this is the standard choice \ref{1}, or the Bargmann \ref{14,15} scalar
product\footnote{$^5$}{This has the advantage that, with $\L_0 = \{ Ax
, . \}$, the adjoint of this corresponds simply to the adjoint matrix,
i.e. $\L_0^+ = \{ A^+ x , . \}$.}
$$ < \mu , \nu > \ = \ \[ \pa_\mu x^\nu \]_{x=0} \ = \
\prod_{i=1}^n \ (\mu_i ! ) \ \de_{\mu_i , \nu_i } \ \ ; \eqno(1.9) $$
The scalar product in $V$ is then naturally defined in terms of these as
$\( f, g \) = \sum_k \( f_k , g_k \)_k $.
\bigskip
{\bf 2. Poincar\'e transformations.}
One considers then near-identity changes of coordinates of the form
$$ x = y + h_k (y) \quad \quad , \quad h_k \in V_k \ , \eqno(2.1) $$
also called Poincar\'e transformations. We denote by $\Ga$ the jacobian
of the change of coordinates, i.e. $\Ga^i_{~j} = {\pa h_k^i / \pa
y^j}$. Under the change of coordinates (2.1), our system
(1.1) is transformed into $$ {\dot y} = \[ I + \Ga \]^{-1} \ f \( y + h_k
(y) \) \ . \eqno(2.2) $$
For $y$ -- and therefore $x$ -- small enough, $\La = (I + \Ga )^{-1}$
does surely exist, and we can write it in a power series as $ \La \equiv
\( I + \Ga \)^{-1} = \sum_{m=0}^\infty \[ (-1)^m \( \Ga \)^m \]$.
Similarly, we can expand $f_m (y + h_k (y) ) $ as a power series; we
write $J = (j_1 , ... , j_n )$, $|J| = \sum_i j_i$. With this multiindex
notation, $\pa_J := \pa_1^{j_1} ... \pa_n^{j_n}$, and similarly $h^J_k
:= (h^1_k)^{j_1} ... (h^n_k)^{j_n}$. We define the operators
$ \Phi_h^r = (1 / r!) \sum_{|J|=r} \( h^J \cdot \pa_J \)$
(representing all the partial derivatives of order $|J|$), and in terms
of these the system (2.2) can then be written as
$$ {\dot y} \ = \
\sum_{m=0}^\infty \ \sum_{r=0}^\infty \ \sum_{s=0}^\infty \ \[ (-1)^s \
\Ga^s \Phi_{h_k}^{r-s} \] \ f_m (y) \ . \eqno(2.3) $$
Thus we see that, in a Poincar\'e transformation with generator $h_k
\in V_k$ each term $f_m$ is transformed into a new $\~f_m$
given\footnote{$^6$}{Here the square brackets in $[m/k]$ denotes integer
part.} by
$$ \~f_m \ = \ f_m \ + \ \sum_{p=1}^{[m/k]} \[ \sum_{s=0}^p \
(-1)^s \Ga^s \Phi_{h_k}^{p-s} \] f_{m-kp} \ . \eqno(2.4) $$
One could, in principle, also obtain explicit formulas for terms of all
degrees, but these become quickly too involved to be of practical use.
We notice, however, that the terms of degree smaller than $k$ are
not changed at all,
$$ \~f_m = f_m \quad \forall m < k \ , \eqno(2.5) $$
and the terms of degree $k \le m < 2k$ are changed according to
$$ \~f_{k+\nu} = f_{k+\nu} + \[ \Phi_h - \Ga \] f_\nu \quad (0 < \nu < k)
\ . \eqno(2.6) $$
\bigskip
{\bf 3. Transformation to Poincar\'e normal form.}
The transformation to Poincar\'e normal form is given by a well known
algorithm, which is just the same if we consider the Poincar\'e or the
Lie (see next section) form for the changes of coordinates. Indeed, in
both cases we have that for the transformation with generator $h_k \in
V_k$ it results $\~f_m = f_m $ for $mFrom (4.4) it is easy to derive formulas for the decomposition of $\~f$
into homogeneous factors, i.e. for $\~f = \sum_m \~f_m$. We introduce
the notation $\H (.) = \{ h , . \}$, and with this we have
$$ \~f_m \ = \ \sum_{s=0}^{[m/k]} {1 \over s !} \H^s \( f_{m - sk} \) \
. \eqno(4.5) $$
Notice that we have written $[m/k]$ for the integer part of $(m/k)$,
and defined $\H^0 (f) = f$.
\bigskip
{\bf 5. The homological operators.}
We will define a series of homological operators $\L_k$ associated to
$f$; the usual
homological operator, which we will denote by $\L_0$, will be the first
of these. This definition will suit our way of proceeding,
based on Poincar\'e-Lie transformations and thus on (4.4).
For $f \in V$, $f = \sum f_k$, we define the Lie operator $\F : V \to V$
associated to $f$ as $\F = \{ f , . \}$; clearly we can write
$$ \F \ = \ \sum_{k=0}^\infty \ \{ f_k , . \} \ \equiv \
\sum_{k=0}^\infty \ \L_k \ . \eqno(5.1) $$
The operators $\L_k = \{ f_k , . \}$ defined in (5.1) are called the
series of homological operators associated to $f$; the operator $\L_0$
coincides with the usual homological operator considered in Poincar\'e
Normal Form theory. Notice that, by (1.6), $\L_k : V_m \to V_{m+k}$. We
also denote by $\L_{k,m}$ the restriction of $\L_k$ to $V_m$.
Notice that when we operate a Poincar\'e transformation, the $f_k$ --
and thus the $\L_k$ -- change. However, at each normalization stage we
stabilize a new term $f_s$, and thus the corresponding $\L_s$.
It should be stressed that linear combinations of the homological
operators do not permit to
describe (4.4), or (4.5), in full generality: they are only related to
the first nontrivial term in (4.5). However, it will turn out that, in
the procedure we employ in the following, a suitable choice of the $h$
permits to analyze iterated Poincar\'e-Lie transformations in terms of
the $\L_k$ alone.
\bigskip
{\bf 6. Non-unicity of Poincar\'e normal forms.}
In the Poincar\'e procedure\footnote{$^{12}$}{Here, by this we mean
indifferently the usual Poincar\'e scheme, or the Poincar\'e-Lie one.},
shortly described above, one has no need to keep track of the effect of
the transformation generated by $h_k \in V_k$ on terms of higher order:
indeed, this will generate additional terms in $V_s$, in principle at
all the higher orders $s > k$, but these can then be disposed of by the
successive Poincar\'e transformations with generator $h_s$.
This point should be considered with some extra care: indeed, while the
terms generated by $\L_0 (h_s )$ are in $\ran (\L_0 ) \cap V_s = \ran
\( \L_{0,s} \)$, those appearing as ``higher order terms'' due to the
transformation generated by $h_k$ (with $k~~ k$.
Let us at first concentrate in particular on the effects at order $k+1$.
The term $f_{k+1}$ changes, due to (4.5), according to
$$ f_{k+1} \ \to \ f_{k+1} - \L_1 (h_k^{(1)} ) \ . \eqno(10.2) $$
This means that we can change $f_2^{(1)}$ into a
$$ f_2^{(2)} = f_2^{(1)} - \L_1 ( h_1^{(1)} ) \eqno(10.3) $$
without changing $f_1^{(1)}$ (because of $h_1^{(1)} \in \ker (\L_0 )$,
as we required above).
Notice that in this way we do not generate nonresonant
terms at higher orders, as the transformation operates by
iterated brackets between functions in $\ker (\L_0 )$ [equivalently,
commutators of vector fields which all commute with $X_0 = f_0 (x)
\pa_x$], see (4.4),(4.5), and this set is obviously closed under the
bracket operation\footnote{$^{16}$}{The same kind of considerations apply
when considering higher order normalizations; this is strictly related
to the Lie algebra filtrations mentioned in the Introduction, and is a
substantial advantage of the Lie-Poincar\'e approach in this context.}.
Proceeding in this way for $k=1,2,...$, we can, by a suitable choice
of the $h_k^{(1)}$, change the $f_m^{(1)}$ with $m \ge 2$ and
eliminate any term in the image of $\ker (\L_0 )$ under $\L_1$, i.e.
in the range of $\M_1$.
In this way we arrive to a ``second normalized form'' for our system:
$$ {\dot x} \ = \ f_0^{(0)} + f_1^{(1)} + \sum_{k=2}^\infty f_k^{(2)}
\ \ . \eqno(10.4) $$
Notice that here $f_0^{(0)} \in V_0$ is still the original one,
$f_1^{(1)} \in \ker (\L_0^+ )$ is the one obtained with the first
normalization, and all the $f_k^{(2)}$ are in $\ker (\M_1^+ )$.
We can now repeat the same procedure choosing $h_k^{(2)} \in \ker (
\L_0 ) \cap \ker ( \L_1 )$ and concentrating on the effect of the
Poincar\'e transformation generated by this on $f_{k+2}$, changing
all terms with $k \ge 3$ and producing in the end a ``third normalized
form''
$$ {\dot x} \ = \ f_0^{(0)} + f_1^{(1)} + f_2^{(2)} +
\sum_{k=3}^\infty f_k^{(3)} \ \ , \eqno(10.5) $$
and so on.
The general procedure is now clear; it is also clear that in
this way we can, operating $n$ successive ``higher
order'' normalizations, take the system into renormalized form up to
order $n$. We will also state the general result that can be obtained
in this way as follows.
{\bf Definition 2.} {\it The dynamical system
$ {\dot x} = \sum_{k=0}^\infty f_k $ (the vector field $X = f^i (x) \pa_i$,
the formal power series $f(x) = \sum f_k (x)$)
is said to be in {\rm n-th normal form } if $f_k \in F^{(p)}_k$ for
all $k$, with $p = \min (k,n)$.
When this condition is satisfied for all $n$ (with no upper limit),
we say that the system is in {\rm Poincar\'e renormalized form}.}
Notice that a system in $n$-th normal form is also in renormalized form
up to order $n$.
{\bf Proposition 2.} {\it Given a dynamical system
$ {\dot x} = \sum_{k=0}^\infty f_k^{(0)}$ (a vector field $X = f^i (x) \pa_i$,
a formal power series $f(x) = \sum f_k (x)$),
it is always possible to reduce it, by a sequence of $n$ formal
Poincar\'e normalizations, into the n-th normal form
$$ {\dot x} \ = \ \sum_{k=0}^n f_k^{(k)} \ + \ \sum_{s=n+1}^\infty
f_s^{(n)} \ , $$
where $f_k^{(k)} \in F^{(k)}_k$ for all $k \le n$.}
We can also say formally, considering infinite sequences of
Poincar\'e normalizations, that a system can always be taken into
renormalized form.
\bigskip
{\bf 11. Discussion.}
We will now briefly discuss some points related to the
results obtained in this paper.
{\it Remark 4.} First of all, we notice that the procedure introduced here is not only
completely constructive and algorithmic, but it can also be implemented
on a computer in the same way as the standard Poincar\'e
normalization. Thus, at least at the formal level (i.e. without
considering the convergence properties for the series entering the
procedure), taking a system into renormalized form is not really more
difficult than taking it into standard Poincar\'e normal form.
{\it Remark 5.} Notice also that the main computational difficulties in the computer
implementation of (formal) standard Poincar\'e normalization arise from
the large size of involved matrices; this size is however the same, if
we stop at the same order, in the renormalization proposed here, so that
the latter is also not more demanding than the standard one in what
concerns computational resources.
{\it Remark 6.} On the other side, the iteration of Poincar\'e normalizat
ion allows to
obtain a reduction in the normal form expansion, which could be quite
significant as it will be shown by some of the examples given below.
In this way, both the normal form classification and the study of
systems of ODEs (vector fields, power series) via their normal form can
be considerably simplified.
{\it Remark 7.} As briefly mentioned above, this ``further simplification
'' of
normal forms is closely related to the well known problem of the
non-uniqueness of Poincar\'e-Dulac normal forms for resonant systems:
indeed, our procedure amounts to obtain a classification of normal
forms in which some redundant ones have been eliminated; also, the
procedure can be seen as amounting to a careful choice of the terms
$\delta h_k \in \ker ( \L_{0,k} )$ which are not selected by the
standard Poincar\'e procedure. It should be stressed that in general
{\it neither the n-th normalized form, nor the renormalized form, will
be unique}: that is we remove only partially the non-uniqueness
inherent to the Poincar\'e procedure.
{\it Remark 8.} In this respect, as already mentioned in the introduction
and in section 6,
it is known
that one can define a {\it unique} normal form (see \ref{4-8}, and
particularly \ref{9}); thus one could wonder where is the advantage in
considering instead the renormalized, or the n-th normalized, forms
introduced here; the answer
is that the theory of unique normal forms is not easily implemented in
concrete computations, while the method proposed here is completely
constructive and moreover goes through computations which are of the
same kind as those required by the standard Poincar\'e normalization.
Thus, in a sense, these renormalized -- or n-th normalized -- forms
represent a useful ``compromise'' between the simplicity of the
algorithm leading to the standard Poincar\'e-Dulac normal form (at the
price of a redundant classification), and the simplicity inherent to
having a unique normal form (at the price of undergoing a difficult
construction).
{\it Remark 9.} It can also be remarked that in the course of our construction,
which aims at obtaining $f_k^* \in F^{(k)}_k$, the $f_k$ -- and thus
the $\L_k$ -- change: this means that the whole series of the
$H^{(p)}$, $\M_p$ and in particular of the $F^{(p)}$ does also change,
so that we are ``aiming at a moving target''. However, at each
normalization stage we stabilize a new term $f_s$, and thus the
$H^{(p)}$ and $F^{(p)}$ get successively stabilized, and at this point
we can successfully attain our goal of attaining an $f_k^* \in F^{(k)}_k$.
{\it Remark 10.} In our discussion, we have supposed to have fixed a scalar
product in each of the $V_k$ and thus -- considering these as
orthogonal subspaces -- in $V$. It should be stressed that the precise
form of this is not essential to our construction (but obviously it is
when we have to perform explicit computations).
{\it Remark 11.} We would like to recall that all of the above discussion is -- as
customary also in standard normal forms theory -- conducted at a
purely formal level\footnote{$^{17}$}{This however does not prevent it
from being useful: even in the standard theory, in most concrete cases
one performs the normalization up to some finite order $N$ and studies
the system truncated at order $N$, resorting then to different
considerations to ensure the equivalence between the truncated system
and the full one.}, i.e. without considering the convergence of the
power series determining the transformation into renormalized form.
It should be mentioned that recent results \ref{18-20} allow to infer
the convergence of the (standard) normalizing transformation from
suitable symmetry properties of the system, or of a system which is
formally equivalent to it; thus, if the renormalized form unfolding
(which is in some cases easy to determine, see the examples below)
displays appropriate symmetry properties, it can be used to guarantee the
convergence of the standard normalization.
\vfill\eject
\bigskip
{\bf 12. Planar vector fields with rotations as linear part (1:1
resonance).}
We will consider, as a first and meaningful example, the unfolding of Normal Forms
for vector fields in $R^2$ having linear part $f_0 (x) = Ax$ with
$$ A \ = \ \pmatrix{0&-1\cr1&~0\cr} \ \ . \eqno(12.1) $$
As it is well known, the Poincar\'e Normal Forms corresponding to this
can be written in the form
$$ f(x) \ = \ Ax \ + \ \sum_{k=1}^\infty \ \( x_1^2 + x_2^2 \)^k \[ a_k
I + b_k A \] x \ , \eqno(12.2) $$
where the $a_k , b_k $ are arbitrary real constants. Writing $r^2 =
(x_1^2 + x_2^2 )$, this reads
$$ f_k (x) = \cases{0 & for $k$ odd \cr
r^{2m} \[ a_m I + b_m A \] x & for $k=2m$.\cr} \eqno(12.3) $$
We will see the the renormalized normal form unfolding is remarkably
simpler. In order to do this, and to avoid trivial steps, we will
consider the renormalized form of a system which is already in
Poincar\'e normal form.
Thus, let us consider an $f$ which has already be taken into Poincar\'e
normal form, and proceed to its renormalization; we will consider only
the nontrivial transformations (but keep the indices notation
introduced above).
Thus, let us first consider the term $f_2^{(0)}$; since it is in normal
form (and since for the same reason $f_1 = 0$, i.e. $\L_1 \equiv 0$), we
cannot modify it by our algorithm, i.e. it will remain in the form given
above,
$$ f_2^{(2)} = f_2^{(0)} = r^2 (a_1 I + b_1 A ) x \ . \eqno(12.4) $$
We then have $f_3 = 0$, and
$$ f_4^{(0)} = r^4 (a_2 I + b_2 A ) x \ ; \eqno(12.5) $$
the generator $h_2^{(2)}$ of the transformation
$$ f_4^{(2)} = f_4^{(0)} - \M_2 \( h_2^{(2)} \) \eqno(12.6) $$
must be, for $h_2^{(2)} \in \ker (\L_0 )$, of the form
$$ h_2^{(2)} = r^2 (\a I + \b A ) x \ , \eqno(12.7) $$
and thus
$$ \M_2 \( h_2^{(2)} \) \ = \ 4 \( a_1 \b - b_1 \a \) Ax \ ; \eqno(12.8)
$$
it is then clear that, unless $a_1 = b_1 = 0$, we can always choose $\a
, \b$ so that
$$ f_4^{(2)} (x) = f_4^{(4)} (x) = r^4 a_2 x \ . \eqno(12.9) $$
Let us now consider $f_6$; now $h_4^{(2)} (x) = r^4 (\a I + \b A ) x$
(again for $h\in \ker (\L_0 )$), and
$$ \L_2 \( h_4^{(2)} \) \ = \ \[ (- 2 a_1 \a ) I + (2 a_1 \b - 4 b_1 \a
) A \] x \ . \eqno(12.10) $$
Thus, if $a_1 \not= 0$, we can eliminate completely $f_6$ in this way.
It is quite easy to get convinced that, under the same condition $a_1
\not= 0$, the same holds for all the $f_{2m}$. Indeed, for
$$ h_{2(k-1)}^{(3)} (x) = r^{2(k-1)} \[ \a I + \b A \] x \eqno(12.11) $$
(again, $h_{2(k-1)}^{(3)} \in \ker (\L_0 )$ requires this form for
$h$), we have
$$ \L_2 \( h_{2(k-1)}^{(3)} \) \ = \
- \[ \( 2(k-2) a_1 \a \) I + \( 2 (k-1) a_1 \b - 2 b_1 \a \) A \] x \ ;
\eqno(12.12) $$
thus, we can eliminate completely all the $f_{2m}$.
We have thus shown that:
{\bf Lemma 1.} {\it If in the Poincar\'e normal form (12.2) for $f(x)$ the
constant $a_1$ is nonzero, then the corresponding Poincar\'e
renormalized form is given by $$ f^* (x) \ = \ Ax + r^2 \( a_1 I + b_1
A\) x + r^4 a_2 x $$ and thus its unfolding depends on three real
parameters.}
\bigskip
We can also analyze what happens if the nondegeneracy condition $a_1
\not= 0$ is not satisfied. We assume now that $ a_1 = 0$ and $b_1 \not=
0$.
>From (12.8), it appears that, choosing $\a = - b_2 / (4 b_1 )$ in
$h_2^{(2)}$, we can still reduce $f_4$ to $r^4 a_2 x$
(no reduction at all would be possible if $a_1 = b_1 = 0$).
When it comes to considering $f_6$, (12.10) shows that choosing $\a = -
b_3 / (4 b_1)$ in $h_4^{(3)}$ we can arrive to $f_6^{(3)} = r^6 a_3 x$.
We can then proceed further in the renormalization; $\L_3 \equiv 0$,
and thus the next -- and last possible -- step will be
$ f_6^{(5)} = f_6^{(3)} - \L_4 \( h_2^{(4)} \) $,
where $h_2^{(4)} \equiv h = r^2 ( \a I + \b A ) x$, due to the
condition $h \in \ker (\L_0 )$. Recall however that we also have to ask
$h \in \ker (\L_2 )$: this condition is readily see to be equivalent to
$a_1 \b = b_1 \a$; with our present assumptions, this means that $\a =
0$. Thus, we cannot eliminate $f_6^{(3)}$.
We could then check explicitely that the higher order terms, i.e. the
$f_{2k}$ with $k \ge 4$, can be completely eliminated.
Rather than going on with discussion of more and more degenerate cases,
we will give a general criterion and an inductive proof of it.
{\bf Lemma 2.} {\it Let the vector formal power series $f : R^2 \to R^2$
be given by $f(x) = Ax + \sum_{k=1}^\infty r^{2k} \[ a_k I + b_k A
\] x$ and let $\mu$ be the lowest number such that $a_\mu \not= 0$, and
$\nu$ the lowest number such that $b_\nu \not=0$, so that $f(x)$ can be
written as
$$ f(x) \ = \ Ax \ + \ \sum_{k=\mu}^\infty \ r^{2k} a_k x \ + \
\sum_{k=\nu}^\infty \ r^{2k} b_k Ax \ . \eqno(12.13) $$
Then, the Poincar\'e renormalized form of $f$ up to any given order $n$
is given by
$$ f^* (x) \ = \ Ax + r^{2\mu} a_\mu x + r^{2\nu} \b A x + r^{4\mu} \a x
\ , \eqno(12.14) $$
where $a_\mu \not= 0$ is the same as in (12.13), and the $\a$, $\b$
could (possibly, but not necessarily) vanish. In particular, if $\nu >
\mu$, then $\b = 0$.}
{\tt Proof.} To see that this is true, it is convenient to use the
vector fields notation, with $X = f^i \pa_i = \sum_k X_k $, and $X_k =
f^i_k \pa_i$.
It is useful to consider the vector
fields $D$ and $R$ corresponding respectively to dilations and
rotations in $R^2$, i.e.
$$ D = x_1 \pa_1 + x_2 \pa_2 \quad , \quad R = -x_2 \pa_1 + x_1 \pa_2 \
; \eqno(12.15) $$
moreover, we consider the vector fields $ Z_k = r^k D$ and
$Y_k = r^k R$ (for $k$ even); these satisfy
$$ \[ a Z_k + b Y_k , \a Z_m + \b Y_m \] = (m-k) a \a Z_{(m+k)} + (m a
\b - k b \a ) Y_{(m+k)} \ . \eqno(12.16) $$
With the notation introduced above, the effect of $\L_k
(h_m )$ with $h_m \in \ker (\L_0 )$ can be computed via
$$ \[ a Z_k + b Y_k , \a Z_m + \b Y_m \] \ = \ (m-k) a \a Z_{k+m} \ + \
(m a \b - k b \a ) Y_{k+m} \ . \eqno(12.17) $$
First of all, we notice that we can eliminate all the terms $a_p Z_p$
in $f$, except the one for $p=2 \mu$: indeed, it suffices to choose
each time a $h_{p-\mu} = r^{2(p-\mu )} \( \a I + \b A \) x$ with $\a =
a_p / \( (p-2\mu ) a_\mu \)$. Notice that by a suitable choice of $\b$
(in particular, $\b = 0$ if $b_\mu = 0$) we can always manage to do
this without modifying the term $b_p Y_p$. Let us then assume we
eliminate first all the terms $a_p Z_p$ (except $p=\mu$ and possibly
$p=2\mu$) up to $p=n$.
Let us now look at the terms $b_p Y_p$ with $p$ greater than the
smaller of $\mu$ and $\nu$: it is clear, again by (12.17), that these
can be eliminated via the term $\L_\mu (h_{p-\mu} ) $ by choosing $\b =
b_p / \( (p-\mu ) a_\mu \)$ (if $\mu < \nu$), or via the term $\L_\nu
(h_{p-\nu} ) $ by choosing $\a = - b_p / \( (p-\nu ) b_\nu \)$ (if $\nu
< \mu$). Notice that if $\nu \le \mu$, the term $b_\nu Y_\nu$ cannot be
eliminated. $\odot$
{\it Remark 12.} In the symplectic case, i.e. when all the $a_k$ in (12.2) vanish,
this corresponds to a classical result \ref{21}, recently extended \ref{22}.
\bigskip
{\bf 13. A generalization (1:n resonances).}
\def\Z{{\bf Z}}
Consider, as a second example, the DS in $R^2$ -- where we have
coordinates $(x_1 , x_2) $ -- with linear part $f_0 (x) = Ax$ given by
$$ A \ = \ \pmatrix{ 1 & 0 \cr 0 & - n \cr} \eqno(13.1) $$
with $n$ a positive integer; as we already studied the case $n=1$,
we consider only $n>1$.
The resonance conditions for this $A$ are
$$ \mu_1 - n \mu_2 = 1 \ \ {\rm or} \ \
\mu_1 - n \mu_2 = - n \eqno(13.2) $$
which give resonant monomials of the form, respectively
$$ \pmatrix{ c_1 (x^n y)^k x \cr 0 \cr} \ \ {\rm and} \ \
\pmatrix{ 0 \cr c_2 (x^n y)^k y \cr} \eqno(13.3) $$
where $c_1 , c_2$ are arbitrary real constants, and $k \ge 1$ a positive
integer.
Thus, the Poincar\'e-Dulac normal form expansion is
$$ f(x) = \pmatrix{ x \ + \ \sum_{k=1}^\infty \sigma_k (x^n y)^k x \cr
- n y \ + \ \sum_{k=1}^\infty \rho_k (x^n y)^k y \cr} \eqno(13.4) $$
Let us now try to determine the corresponding PRF. It is clear that we
only have to care about the terms $f_m$ with $m=\nu k$, where $\nu =
n+1$ and $k \in \Z_+$,
as the other ones can be disposed of by the usual Poincar\'e normalization.
The first nonzero term $f_{\nu q}$ (i.e. the first nonzero resonant term)
cannot be eliminated, and we will write it in the form
$$ f_{\nu q} (x) = \pmatrix{ \alpha (x^n y)^q x \cr
\beta (x^{n-1} y)^q y \cr} \eqno(13.5) $$
(notice that at least one of $\a , \b$ must be nonzero).
The $H^{(0)} = \ker (\L_0 )$ is identified by (13.3) as well; in order to determine
$H^{(\nu q)}$ we have to apply $\L_{\nu q}$ on terms of the form given by (13.3); we
write these as
$$ h(x) \equiv h_{\nu k} (x) = \pmatrix{a_k (x^n y)^k x \cr
b_k (x^n y)^k y \cr } \eqno(13.6) $$
Applying $\L_{\nu q} = \{ f_{\nu q} , . \}$ on the $h_{\nu k} \in H^{(0)}$ we get
$$ \L_{\nu q} \( h_{\nu k} \) \ = \ \pmatrix{
(a_k \beta k + a_k \alpha k n - \alpha b_k q - a_k \alpha n q) (x^n y)^{k+q} x \cr
(b_k \beta k + \alpha b_k k n - b_k \beta q - a_k \beta n q) (x^n y)^{k+q} x \cr}
\eqno(13.7) $$
Thus, in order to eliminate the term $f_{\nu (k+q)}$ via the action of $\L_{\nu q}$
on $h_{\nu k}$, we have to choose $a_k ,b_k $ in such a way that
$$ \pmatrix{
\beta k + \alpha (k-q) n & - \alpha q \cr
- \beta n q & \beta (k-q) + \alpha k n \cr}
\ \pmatrix{ a_k \cr b_k \cr}
\ = \ \pmatrix{ \sigma_{k+q} \cr
\rho_{k+q} \cr}
\eqno(13.8) $$
This is a linear equation in $(a_k ,b_k )$, and can be solved whatever the
values of $\sigma_{k+q} , \rho_{k+q} $ if the
determinant of the matrix $M$ on the l.h.s. of (13.8) does not vanish.
This determinant is
$$ \Delta = (\alpha n + \beta )^2 k (k-q) \eqno(13.9) $$
and thus, unless $\beta = - \alpha n$, we can always eliminate all the
higher order resonant terms at the exception of $f_{2 \nu q}$.
Setting $k=q$ in $M$, it is immediate to observe that the terms with $\rho_{2q}
= - n \sigma_{2q}$ are in the range of $\L_{\nu q}$ and thus can be eliminated;
thus in this nondegenerate case (i.e. for $\b \not= - \a n$) the system is reduced to
$$ f(x) \ = \ \pmatrix{ ~~~ x & + \a (x^n y)^q x & + n \gamma (x^n y)^{2q} x \cr
- n y & + \b (x^n y)^q y & ~~~ + \gamma (x^n y)^{2q} y \cr} \eqno(13.10) $$
%%%% \bigskip
Let us consider the degenerate case in which $\Delta = 0$.
When $\beta = \alpha n$, the kernel of $M^+$ corresponds to $\sigma = n
\rho$, and thus -- eliminating terms which are in the range of $\M_{\nu q}$ --
we can still reduce to consider ``second normalized'' forms where the terms
$f_{\nu p}$ with $p>q$ are of the form
$$ f_{\nu p} (x) \ = \
\pmatrix{ n \rho_p (x^n y)^k x \cr
\rho_p (x^n y)^k y \cr} \eqno(13.11) $$
Let now $p$ be the smallest integer (greater than $q$) for which $\rho_p \not=
0$; to go beyond (13.11) we have to consider the action of $\L_{\nu p}$, but now
we cannot consider all the $h \in H^{(0)}$ acceptable: we have to restrict
ourselves to $\ker (\L_0 ) \cap \L_{(\nu q)}$. From the above formulas it is
easily seen that the $h_{\nu k} \in \ker \( \L_{\nu q } \)$ are those of the form
$$ h_{\nu k} \ = \ \pmatrix{a_k (x^n y)^k x \cr - n a_k (x^n y)^k y \cr} \eqno(13.12) $$
(i.e. $b_k = - n a_k$ in the notation used above).
Acting with $\L_{\nu p}$ on the $h_{\nu k}$ given by (13.12), we get, as it can be
seen either by direct computation, either from (13.7) by changing $q$ into $p$
and writing $\a = n \b$ and $b = - na$,
$$ \L_{\nu p} (h_{\nu k} ) = \sigma_p k (1+n^2 ) (h_{\nu k} )
\sigma_p k (1+n^2 ) a \pmatrix{1 \cr - n \cr} \ \ . \eqno(13.13) $$
Again by direct computation or considering the $M^+$ corresponding to the above
$M$ with $q$ replaced by $p$, $b_k = - n a_k$ and $\alpha = n \beta$, we have that
now $\ker (\M^+_{\nu p} ) \cap V_s$ reduces, for $s > \nu p$, to
$$ \pmatrix{n c (x^n y)^{2p} x \cr c (x^n y)^{2p} y \cr} \ \ . \eqno(13.14) $$
We have thus proved that in the degenerate case $\beta = \alpha n$ the PRF is given by
$$ f(x) \ = \ \pmatrix{ x & + n \beta (x^n y)^q x & + n \gamma (x^n y)^p x &
+ n \eta (x^n y)^{2p} x \cr
- n y & + \beta (x^n y)^q y & + \gamma (x^n y)^p y &
+ \eta (x^n y)^{2p} y \cr } $$
\bigskip
{\bf 14. Some three-dimensional examples.}
We will now consider a number of three-dimensional examples; in these,
the Poincar\'e NF will be a finite polynomial, but nevertheless our
procedure will produce a simplified PRF.
{\it A)}
Let us consider the VFs with linear part given by $f_0 (x) = Ax$, and
$$ A = \pmatrix{1&0&0\cr 0&2&0\cr 0&0&5\cr} \ . \eqno(14.1) $$
One can easily check that the PNF depend on four arbitrary constants,
and is given explicitely by
$$ f(x) \ = \ \pmatrix{ x \cr 2y + c_1 x^2 \cr
5z + c_2 x y^2 + c_3 x^3 y + c_4 x^5 \cr} \ . \eqno(14.2) $$
We want now to show how this can be reduced by our procedure. Let us
start by operating with $\L_1$ on $h \in H^{(1)} = \ker (\L_0 )$; now
$$ f_1 = \pmatrix{0\cr c_1 x^2 \cr 0\cr} \ ; \eqno(14.3) $$
obviously we get nothing by acting on $h_1^{(1)} \in H^{(1)} \cap
V_1$ (which would be a multiple of $f_1$), while acting on
$$ h_2^{(1)} = \pmatrix{0 \cr 0 \cr a x y^2 \cr} \ \ {\rm and} \ \
h_3^{(1)} = \pmatrix{0 \cr 0 \cr b x^3 y \cr} \eqno(14.4) $$
we can -- provided $c_1 \not= 0$ -- eliminate $f_3$ and $f_4$ by
choosing
$$ a = {c_3 \over 2 c_1} \ , \ b = {c_4 \over c_1} \ . \eqno(14.5) $$
When $c_1 = 0$, we can still act with $\L_2$; as
$$ f_2 = \pmatrix{0 \cr 0 \cr c_2 x y^2 \cr} \ , \eqno(14.6) $$
acting on $h_1^{(1)}$ (written as above), we can -- provided $c_2 \not=
0$ -- eliminate $f_3$ by choosing $ a = - c_3 / (2 c_2 )$; notice that
now we are not able to modify $f_4$ in any way, as $\L_1 = \L_3 = 0$,
and $\L_2 ( h_2^{(1)} ) = 0$.
Finally, if $c_1 = c_2 = 0$, we write
$$ f_3 = \pmatrix{0 \cr 0 \cr c_3 x^3 y \cr} \eqno(14.7) $$
and $\L_3 ( h_1^{(1)}$ eliminates $f_4$, if $c_3 \not= 0$, by choosing
$a = - c_4 / c_3$.
We summarize our discussion as follows: we have shown that the PRF for
$f(x)$ corresponding to the linear part $Ax$ ($A$ as above) is in one of
the following forms:
$$ \eqalign{
(0)~~~~~ & f(x) = \pmatrix{x \cr 2y \cr 5z \cr} \cr
(1)~~~~~ & f(x) = \pmatrix{x \cr 2y + \a x^2 \cr 5z + \b x y^2 \cr} \cr
(2)~~~~~ & f(x) = \pmatrix{x \cr 2y \cr 5z + \a x y^2 + \b x^5 \cr} \cr
(3)~~~~~ & f(x) = \pmatrix{x \cr 2y \cr 5z + \a x^5 \cr} \cr} $$
where $\a , \b$ are arbitrary constants, with $\a \not= 0$.
Obviously we could have a more compact notation unifying cases (0),(2)
and (3), if desired: that is, we can write the PRF as
$$ f(x) \ = \ \pmatrix{ x \cr 2 y + a_1 x^2 \cr 5 z + a_2 x y^2 + a_3 x^5 \cr} $$
where at least one of the $a_i$ can always be taken to be zero.
This means that if we are interested in the nonlinear behaviours
compatible with the assigned linear part, we could study these
two-parameters families rather than the five-parameters family of PNFs;
needless to say, this is a much lighter task.
{\it B)}
Let us consider the VFs with linear part given by $f_0 (x) = Ax$, and
$$ A = \pmatrix{1&0&0\cr 0&3&0\cr 0&0&9\cr} \ . \eqno(14.8) $$
One can easily check that the PNF depend on five arbitrary constants,
and is given explicitely by
$$ f(x) \ = \ \pmatrix{ x \cr 3y + c_1 x^3 \cr
9z + c_2 y^3 + c_3 x^3 y^2 + c_4 x^6 y + c_5 x^9 \cr}
\ . \eqno(14.9) $$
Let us now consider the action of $\L_2$, corresponding to
$$ f_2 = \pmatrix{0 \cr c_1 x^3 \cr c_2 y^3 \cr} \ , \eqno(14.10) $$
on $H^{(1)}$; we will write
$$ h_2^{(1)} = \pmatrix{0 \cr a x^3 \cr b y^3 \cr} \ , \
h_4^{(1)} = \pmatrix{0 \cr 0 \cr d x^3 y^2 \cr} \ , \
h_6^{(1)} = \pmatrix{0 \cr 0 \cr e x^6 y \cr} \ . \eqno(14.11) $$
With this notation, we have
$$ \L_2 (h_2^{(1)} ) = \pmatrix{0 \cr 0 \cr 3 (b c_1 - a c_2 ) x^3 y^2
\cr} \ ; \
\L_2 (h_4^{(1)} ) = \pmatrix{0 \cr 0 \cr 2 d c_1 x^6 y \cr} \ ; \
\L_2 (h_6^{(1)} ) = \pmatrix{0 \cr 0 \cr e c_1 x^9 \cr} \ . \eqno(14.12) $$
Thus, if $c_1 \not= 0$, we can eliminate $f_4 , f_6 $ and $f_8$ by
suitably choosing $a,b,d,e$. If $c_1 = 0$ and $c_2 \not= 0$, we can
still eliminate $f_4$, but neither $f_6$ nor $f_8$ (the latter can actually
be eliminated with successive steps in our procedure, if $f_6$ does not
vanish, see below).
When $c_1 = c_2 = 0$, we can still act with $\L_4$, corresponding to
$$ f_4 = \pmatrix{0 \cr 0 \cr c_3 x^3 y \cr} \ . \eqno(14.13) $$
With the same notation as above for $h^{(1)}_2$, we have
$$ \L_4 ( h_2^{(1)} ) = \pmatrix{0 \cr 0 \cr - 2 a c_3 \cr} \eqno(14.14) $$
and thus we can -- if $c_3 \not= 0$ -- eliminate $f_6$ in this way;
however $\L_4 (h_4^{(1)} ) = 0 $ and $f_8$ cannot be eliminated.
Finally, if $c_1 = c_2 = c_3 = 0$, with
$$ f_6 = \pmatrix{ 0 \cr 0 \cr c_4 x^6 y \cr} \eqno(14.15) $$
we have
$$\L_6 (h_2^{(1)} ) = \pmatrix{0\cr 0\cr - 2 a c_4\cr} \eqno(14.16) $$
and thus can eliminate $f_8$ (notice this also applies to the case
$c_1 = 0 \not= c_2$ considered above).
Summarizing, we have shown that the PRF correspond to one of the
following possibilities:
$$ f \ = \ f_0 \ + \ \cases{0 & \cr f_2 & \cr f_2 + f_6 & (with
$c_1 = 0$) \cr f_2 + f_8 & (with $c_1 = 0 $) \cr
f_4 + f_8 & \cr f_6 & \cr f_8 & \cr} \eqno(14.17) $$
{\it C)}
We want now to show that sometimes the procedure we propose here is
not able to operate simplification on the usual Poincar\'e normal form.
Let us consider the VFs with linear part given by $f_0 (x) = Ax$, and
$$ A = \pmatrix{2&0&0\cr 0&3&0\cr 0&0&8\cr} \ . \eqno(14.18) $$
One can easily check that the PNF depends on two arbitrary constants,
and is given explicitely by
$$ f(x) \ = \ \pmatrix{ 2 x \cr 3y \cr
8z + c_1 x y^2 + c_2 x^4 \cr} \ . \eqno(14.19) $$
In this case, our procedure cannot modify the form (14.19) of the system,
and thus we get no further simplification.
Similarly, let us consider the VFs with linear part given by $f_0 (x) = Ax$, and
$$ A = \pmatrix{1&0&0\cr 0&2&0\cr 0&0&5\cr} \ . \eqno(14.20) $$
One can easily check that again the PNF depends on two arbitrary constants,
and is now given explicitely by
$$ f(x) \ = \ \pmatrix{ 2 x \cr 3y \cr
10 z + c_1 x^2 y^2 + c_2 x^5 \cr} \ . \eqno(14.21) $$
As in the previous example, our procedure cannot modify the form (14.21) of
the system, and thus we get no further simplification.
\vfill\eject
{\bf Appendix A (Poincar\'e-Lie transformations).}
In this appendix, we shortly go over the Poincar\'e-Lie transformation,
and the derivation of (4.4); we will follow the discussion given in
\ref{12}.
We recall that in this case the change of coordinates is given by
(4.2), and that this transforms $X$ into $\~X$ given by (4.3) \ref{5}.
As mentioned in section 4, $\~X$ can now be explicitely computed by the
Baker-Campbell-Haussdorf formula \ref{12,17}, as
$$ \~X = \sum_{n=0}^\infty {(-1)^n \la^n \over n!} X^{(n)} \eqno(A.1) $$
where the $X^{(n)}$ are determined recursively by $X^{(n+1)} = \[
X^{(n)} , H_k \]$, with $X^{(0)} = X$.
We can thus consider a one-parameter family of vector fields $X_\la$,
where $X_0 = X$ and $X_1 = \~X$; this satisfies ${d X_\la / d \la } =
\[ H_k , X_\la \]$.
Correspondingly, we write $x_\la$ for $e^{-\la H_k} x$ (i.e. the
transformed coordinates, see (4.2), corresponding to $\la$), and $ X_\la
= f^i_\la (x_\la ) (\pa / \pa x_\la^i)$; the $f^i_\la$'s satisfy then $$
{d f^i_\la \over d \la } \ = \ \{ h_k , f_\la \}_\la^i \ , \eqno(A.2) $$
where $\{ .,. \}_\la$ is the bracket $\{.,.\}$ in the $x_\la$
coordinates, i.e. $\{ f,g \}_\la = f^j (\pa g / \pa x^j_\la ) - g^j
(\pa f / \pa x^j_\la )$.
If we consider the power series expansion of $f$, and writing for ease
of notation $f(x,\la ) = f_\la (x_\la )$ and $\{.,.\}$ for
$\{.,.\}_\la$, we have
$$ {\pa f^i_m (x , \la ) \over \pa \la } \ = \ \{ h_k , f_{m-k} \}^i \
. \eqno(A.3) $$
The $\~X = X_1$ is then written in the $\~x$ coordinates as $\~X = f^i
(x , 1 ) (\pa / \pa {\~x}^i ) \equiv \~f^i (\~x ) (\pa / \pa {\~x}^i
)$; the $\~f$ correspond to the solution of (A.2) for $\la = 1$.
These can be expressed by means of the BCH formula: indeed, from (A.1)
and the recursion relation for $X^{(n)}$, we have immediately that
$$ f (x, \la ) \ = \ \sum_{n=0}^\infty \ \[ {(-1)^n \la^n \over n!}
\ \varphi^{(n)} (x) \] \eqno(A.4) $$
with $\varphi^{(0)} (x) = f (x,0)$ and $\varphi^{(n+1)} = \{
\varphi^{(n)} , h_k \}$.
>From this, we have indeed, with $\H (.) = \{ h , . \}$,
$$ f_\la \ = \ \sum_{n=0}^\infty {\la^n \over n!} \ \H^n (f) \ \ ,
\eqno(A.5) $$
and for $\la=1$, i.e. for $\~f (x) = f (x,1)$, this is just (4.4).
\vfill\eject
{\bf Appendix B (The Hamiltonian case)}
\def\ad{{\tt ad}}
It is well known that the Poincar\'e theory of normal forms has a
counterpart in the Hamiltonian case, due to Birkhoff (and then extended
to the resonant case by Gustavsson, in the same way as the Poincar\'e
theory was extended by Dulac); in this case, one can deal directly
with the Hamiltonian and its normal form, rather than with the
hamiltonian vector field this generates.
It turns out that the procedure for the reduction of Poincar\'e NF
presented here can also be applied, pretty much in the same way, in
the case of Birkhoff NF; in this appendix we will shortly show this
extension (details will not be showed, as they are similar to those
for the discussion given above, provided the appropriate parallel are
done).
We will consider Hamiltonians defined in a neighbourhood of the origin
in the phase space $R^{2 \ell} \equiv R^n$, in which we will take
coordinates\footnote{$^{18}$}{We prefer to write all the coordinate
indices as lower ones, no matter if contravariant or covariant, to
simplify the notation.} $q_1,...,q_\ell ; p_1 , ... , p_\ell$; we also
denote these by $x_1,..., x_n$, where it is meant that $x_i = q_i$ for
$i\le \ell$, and $x_i = p_{i-\ell}$ for $i > \ell$, and having a
critical point in the origin. Notice that, by adding an inessential
constant $c = - H(0)$ to $H (x)$, we can always require that $H(0)=0$.
We consider scalar functions on $R^n$, and denote by $\S$ the space of
formal power series $s : R^n \to R$; we also denote by $\S_k$ the
space of homogeneous scalar functions of degree $(k-2)$ on $R^n$.
We can then write any Hamiltonian in $\S$ which has a critical point
in the origin, choosing $H(0)=0$ , as
$$ H (q,p) \ = \ \sum_{k=0}^\infty H_k (q,p) \eqno(B.1) $$
where $H_k \in \S_k$.
We define then in $\S$ the familiar antisymmetric Poisson bracket $\{ .
, . \} : \S \times \S \to \S$ as
$$ \{ F , G \} \ = \ \sum_{i=1}^\ell \ \[ {\pa F \over \pa q_i} \, {\pa
G \over \pa p_i} \ - \ {\pa F \over \pa p_i} \, {\pa G \over \pa q_i}
\] \ . \eqno(B.2) $$
With the above notation for $\S_k$, we have
$$ \{ . , . \} \ : \ \S_k \times \S_m \to \S_{k+m} \ . \eqno(B.3) $$
To any $H \in \S$ we associate an hamiltonian vector field $X_H : R^n
\to \T R^n$, defined by
$$ X_H \ = \ {\pa H \over \pa p_i } \, { \pa ~ \over \pa q_i} \ - \
{\pa H \over \pa q_i} \, {\pa ~ \over \pa p_i } \eqno(B.4) $$
so that $X_H (F) = \{ F , H \}$. It is immediate to check that
$$ \{ F , G \} = K \ \Longleftrightarrow \ \[ X_F , X_G \] = - X_K \
. \eqno(B.5) $$
To any $F \in \S$ we can associate its $\ad$-operator $\ad_F
\equiv \{ F , . \}$ and its Lie operator\footnote{$^{19}$}{This is
usually denoted by $\L_F$, but we want to keep the notation similar to
the one already employed and reserve $\L$ for the homological
operators.} $\P_F$, defined as the time-one action of the flow under
$(- X_F ) $; this is also written as
$$ \P_F \ = \ \exp \[ \ad_F \] \ = \ \sum_{r=0}^\infty \, {1 \over
r!} \, \( \ad_F \)^r \ \ . \eqno(B.6) $$
We will now consider changes of coordinates given by the action of
$\P_F$ for $F \in \S$, similarly to what we have done in the
Poincar\'e case. Notice that again, when we consider only first
nontrivial order terms -- i.e., the action on $H_m$ if $F \in \S_m$
-- this gives simply
$$ q_i \to q_i + \pa F / \pa p_i ~~~ , ~~~ p_i \to p_i - \pa F / \pa
q_i \ ; \eqno(B.7) $$
the advantage of considering Lie operators rather than the usual
near-identity changes of coordinates lies in the fact that, being
defined in terms of Poisson bracket, these are guaranteed to generate
canonical transformations at all orders; moreover, we are guaranteed
of the invertibility of some relevant operators \ref{11}.
Now, as it can be checked by direct computation \ref{11,23}, the
effect of the change of coordinates given by $\P_F$ is given by
$$ \eqalign{
H \ \to \ \~H \ = & \ \sum_{r=0}^\infty \sum_{r=0}^\infty \, {1 \over
r!} \, \( \ad_F \)^r (H) = \cr
= & H + \{ F, H \} + {1\over 2} \{ F , \{ F , H \} \} + {1 \over 6}
\{ F , \{ F , \{ F , H \} \} \} + ... \cr} \eqno(B.8) $$
which is formally analogous to (4.4). If we insert in this the
expansion (B.1) for $H$ and consider an $F \in \S_k$, we get the
exact analogue of (4.5), i.e.
$$ \~H_m \ = \ \sum_{r=0}^{[m/k]} {1 \over r !} \, (\ad_F)^r H_{m-rk}
\ \ . \eqno(B.9) $$
Then, to any series (B.1) we associate a series of homological
operators $\L_k$ given by
$$ \L_k \ = \ \{ H_k , . \} \ \equiv \ \ad_{H_k} \eqno(B.10) $$
which have the same formal properties of the $\L_k$ considered
before.
The computation will now easily proceed along the same lines, and
what's more using the same formal properties, as in the Poincar\'e
case; indeed, we could repeat word by word (with the same symbols --
which have now a different definition) the discussion given above,
arriving at the same results. We will not bore the reader with this
repetition, but only give the final result; in this it is meant that
again $F^{(k)} = \bigcap_{s=0}^{k-1} \ker (\L_k )$.
{\bf Definition 3.} {\it We say that the Hamiltonian $H = \sum_k H_k$ is
in {\bf Birkhoff renormalized form} up to order $n$ if $H_k \in
F^{(k)}_k$ for all $k \le n$.}
{\bf Proposition 3.} {\it Any Hamiltonian can be brought into Birkhoff
renormalized form up to any desired order $n$ by means of a formal
series of Lie transforms.}
It is maybe worth mentioning explicitely that we can also proceed by
iterated Bikhoff normalizations, exactly as we proceeded by iterated
Poincar\'e normalizations in sect.10, arriving at the same results.
We would then have explicitely:
{\bf Definition 4.} {\it A polynomial Hamiltonian
$ H = \sum_{k=0}^\infty H_k $
is said to be in {\rm n-th normal form } if $H_k \in F^{(p)}_k$ for
all $k$, with $p = \min (k,n)$.
When this condition is satisfied for all $k$ (with no upper limit),
we say that the system is in {\rm Birkhoff renormalized form}.}
Notice that a Hamiltonian in $n$-th normal form is also in renormalized
form up to order $n$, as it was the case for dynamical systems (vector
fields, formal power series).
{\bf Proposition 4.} {\it Given a Hamiltonian $H \in \S$, $ H =
\sum_{k=0}^\infty H_k^{(0)}$, it is always possible to reduce it, by
a sequence of $n$ formal Birkhoff normalizations, into the n-th
normal form $$ H \ = \ \sum_{k=0}^n H_k^{(k)} \ + \
\sum_{s=n+1}^\infty H_s^{(n)} \ , $$
where $H_k^{(k)} \in F^{(k)}_k$ for all $k \le n$.
Formally, considering infinite sequences of
Birkhoff normalizations, a Hamiltonian can always be taken into
renormalized form.}
\vfill\eject
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\bigskip\bigskip\bigskip\parskip=4pt
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\bye
~~