\magnification=\magstep1
\def\giorno{3 June 96}

\def\A{{\cal A}}
\def\B{{\cal B}}
\def\C{{\cal C}}
\def\D{{\cal D}}
\def\F{{\cal F}}
\def\G{{\cal G}}
\def\H{{\cal H}}
\def\I{{\cal I}}
\def\J{{\cal J}}
\def\K{{\cal K}}
\def\L{{\cal L}}
\def\M{{\cal M}}
\def\Q{{\cal Q}}
\def\R{{\bf R}}  %% reals
\def\T{{\rm T}}  %% tangent
\def\S{{\cal S}}
\def\V{{\cal V}}

\def\a{\alpha}
\def\b{\beta}
\def\g{\gamma}
\def\ga{\gamma}
\def\de{\delta}   %% NON ridefinire come \d !!!!
\def\eps{\varepsilon}
\def\phi{\varphi}
\def\la{\lambda}
\def\ka{\kappa}
\def\s{\sigma}
\def\z{\zeta}
\def\om{\omega}
\def\th{\theta}
\def\vth{\vartheta}

\def\Ga{\Gamma}
\def\De{\Delta}
\def\La{\Lambda}
\def\Om{\Omega}
\def\Th{\Theta}

\def\pa{\partial}
\def\pd{\partial}
\def\d{{\rm d}}       %% derivative
\def\xd{{\dot x}}
\def\yd{{\dot y}}
\def\grad{\nabla}     %% gradient
\def\lapl{\triangle}  %% laplacian
\def\ss{\subset}
\def\sse{\subseteq}
\def\Ker{{\rm Ker}}
\def\Ran{{\rm Ran}}
\def\ker{{\rm Ker}}
\def\ran{{\rm Ran}}
\def\iff{{\rm iff\ }}


\def\({\left(}
\def\){\right)}
\def\[{\left[}
\def\]{\right]}
\def\=#1{\bar #1}
\def\~#1{\widetilde #1}
\def\.#1{\dot #1}
\def\^#1{\widehat #1}
\def\"#1{\ddot #1}


\def\mapright#1{\smash{\mathop{\longrightarrow}\limits^{#1}}}
\def\mapdown#1{\Big\downarrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\def\mapleft#1{\smash{\mathop{\longleftarrow}\limits^{#1}}}
\def\mapup#1{\Big\uparrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}

\def\ref#1{[#1]}

\font \petit  = cmr9


{\nopagenumbers
~\vskip 3 truecm

{\bf POINCARE' RENORMALIZED FORMS}
\footnote{}{{\tt \giorno }}

\vskip 3 truecm
Giuseppe Gaeta

Department of Mathematics

Loughborough University

Loughborough LE11 3TU (England)

{\tt G.Gaeta@lboro.ac.uk}

\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 planar vector fields whose linear part
has eigenvalues $\la = \pm i$.}

\vfill\eject}
\pageno=1
\parindent=0pt
\parskip=10pt


{\bf Introduction.}

The theory of Normal Forms, first introduced by Poincar\'e, is in
many aspects central to the study of nonlinear dynamical systems; in
particular, its original version -- in which we are interested here --
deals with expansion of a system of ${\cal C}^\infty $ ODEs in $R^n$
(equivalently, of a ${\cal C}^\infty$ vector field on a $n$-dimensional
smooth manifold) in the neighbourhood of an equilibrium point, say the
origin $x=0$, and one aims at a classification of these up to
formal\footnote{$^1$}{By ``formal'', it is meant that these are defined
by series, and we do not consider the problem of convergence of such
series.} near-identity changes of coordinates.

It is well known \ref{1-3} that a system of ODEs (having the origin as
equilibrium point) can be taken into Normal Form by means of the
Poincar\'e algorithm, i.e. by performing a series of near-identity
changes of coordinates (Poincar\'e transformations) of the form $x' = x
+ h_k (x)$ where the $h_k (x)$ are homogeneous vector polynomials; these
are chosen as solution of a certain equation, the {\it homological
equation}, and such solutions are unique up to terms in the kernel of the
relevant {\it homological operator}.

In this way one manages to eliminate, order by order, all the {\it
nonresonant} terms. However, each of these changes of coordinates
produces new terms of higher order; those which are nonresonant will
then be disposed of by successive Poincar\'e transformations, while if
resonant terms are generated in this way, we are not able to eliminate
them by the Poincar\'e algorithm.

It should also be stressed that, while two vector polynomials $h_k (x) $
and $h'_k (x)$ which differ only by an element in the kernel of the
homological operator produce the same transformation on terms of order
$k$, they will in general give raise to different higher order terms;
in particular, we can in this way generate different higher order
resonant terms.

It is then clear that, unless we give a prescription
concerning the projection of $h_k (x)$ on the kernel of the homological
operator $\L_0$ (e.g. that $h_k (x) \in \[ \ker (\L_0 ) \]^\perp$), the
Normal Form is not uniquely defined.

Here we argue that this should not be seen as a drawback, but as an
advantage: indeed, a careful exploitation of the higher order terms
generated in each Poincar\'e transformation can lead to a remarkable
simplification of the Normal Form unfolding; this simplification is
specially important for systems satisfying resonance
relations\footnote{$^2$}{Here the $\la_k$ are the eigenvalues of the
matrix describing the linearization of the system around the equilibrium
point, and $m_k$ are nonnegative integers.} $\sum_k m_k \la_k = \la_r$
in a large or even infinite (as e.g. in the classical case where $\la_k =
\pm i$) number.

In order to present the argument, and the completely defined algorithm
which implements it, we need to consider the usual Poincar\'e scheme,
or actually the Lie-Poincar\'e one \ref{4,5}, paying special attention
to the higher order terms generated in each change of coordinates. Thus,
we will shortly go over the Poincar\'e and Lie-Poincar\'e methods
providing the relevant detailed formulas, in sections 1-5.
We do have a special advantage in considering Lie-Poincar\'e
transformations, as for these we have closed formulas expressing
transformed vector fields in terms of iterated commutators, essentially
the Baker-Campbell-Hausdorff formula; although this is a classical
result, we reproduce it in the appendix for completeness.

We briefly discuss the problem of non-unicity of Poincar\'e Normal Forms
(in section 6), and in sections 7-9 we implement this discussion by
providing an algorithm to reduce a system to a form which amounts
essentially to applying iteratively the Poincar\'e normalization
procedure first on the system, then on its normal form, then on this
``normalized normal form'', and so on (through $m$ normalizations if we
want to reduce the terms of order up to $m$), each time taking into
account terms of higher order; due to this, we call the final form thus
obtained the {\bf Poincar\'e renormalized form} for the system. It
should be stressed that in actual computation one would prefer to
proceed order by order, and this can actually be done; thus, we present
our algorithm directly in this form, i.e. we take sequentially the terms
of order 1,2,... to their renormalized form.

Finally, in section 10 we apply the algorithm to the case, mentioned
above, of planar vector fields having the generator of rigid rotations
as linear part: in this case the relevant matrix $A$ has eigenvalues
$\la_k = \pm i$, and thus an infinite number of satisfied resonance
relations and of terms appearing in the Poincar\'e normal form
unfolding, parametrized by two infinite sequences of real numbers.
However, the renormalized form unfolding constains only three nonlinear
terms -- i.e. three free real parameters -- at most.



%\vfill\eject
\bigskip
{\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
$$ {\dot x} = f(x) \qquad \qquad , ~ x \in R^n ~,~ f : R^n \to R^n
\eqno(1.1) $$
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{6,7}. 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
$$ \( e_{\nu , j } , e_{\mu , i} \) \ = \ < \mu , \nu > \ \de_{i,j} \ ,
\eqno(1.8) $$
where $<.,.>$ is the Bargman \ref{7,8} scalar product
$$ < \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 \( f_k , g_k \)$.

\vfill\eject
\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{$^3$}{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) $$

\vfill\eject
\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 $m<k$, and
$$ \~f_k = f_k + \{ h_k , f_0 \} = f_k - \L_0 (h_k ) \ ; \eqno(3.1) $$
the expression for higher order terms are (at least for $m\ge 2k$) more
involved, and differ slightly in the two approaches.

We consider then sequentially the terms in $V_k$ for $k=1,2,3,...$ (up to
any desired finite order $n$, or formally for $n \to \infty$), and choose
suitable generators $h_k$; in this way we can normalize sequentially the
terms $f_k$, and successive changes of coordinates of higher order leave
these unchanged\footnote{$^4$}{In general, the series of changes of
coordinates and of Poincar\'e transformations defined in this way is only
formal (i.e. does not converge in any neighbourhood of the origin), even
for finite $n$; here we will not be concerned by this problem, as we
deal with formal power series.}.

The ``suitable'' generator $h_k$ mentioned here is obtained as solution
to the homological equation; we first define the projection operator
$$ P_0 : V \to \ran (\L_0 ) = \[ \ker (\L_0^+ ) \]^\perp \eqno(3.2) $$
(this could be properly defined in each of the $V_k$ considering the
restriction $\L_{0,k}$ of $\L_0$ to $V_k$, and the projection operators
$P_0^{(k)}: V_k \to \ran \( \L_{0,k} \)$; with these, $P_0 = \sum^\oplus
P_{0,k}$), and then require that $h_k$ solves the homological equation
$$ \L_0 (h_k ) \ = \ P_0 \( f_k \) \ . \eqno(3.3) $$ The solutions to
this are given by $$ h_k \ = \ \L_0^+ \( P_0 f_k \)  \ + \ \ell_k \ ,
\eqno(3.4) $$ where $\ell_k (x)$ is an arbitrary
function\footnote{$^5$}{This freedom in choosing $\ell_k$, i.e. $h_k$,
does also come into play when we have to take into normal form not a
single vector field, but a Lie algebra of vector fields \ref{9}.} in
$\ker (\L_0 )$.

In this way, $\~f_k \in \[ \ran (\L_0 ) \]^\perp = \ker \( \L_0^+ \)$.
As it is well known, proceeding in this way one finally arrives to a
system ${\dot x} = \^f (x)$ which is in normal form (up to any given
order $n$), i.e. such that all the $\^f_k (x) $ with $k\ge 1$ are in
$\ker ( \L_0^+ )$, and this means that all the nonlinear terms (up to
any given order $n$) are resonant with the linear part of the system.


\bigskip
{\bf 4. Lie transformations.}

A slightly different way of approaching Normal Forms is based on Lie --
rather than Poincar\'e -- transformations \ref{4,5}. In this case, the
change of coordinates is given by the time-one
action\footnote{$^6$}{That is, the time-one flow for the one-parameter
group of diffeomorphisms of $R^n$ generated by the vector field referred
to.} of a vector field $H_k$ given by
$$ H_k \ = \ h_k^i(x) {\pa \over \pa x^i } \eqno(4.1) $$
so that the change of coordinates is written as
$$ \~x \ := y \ = \ \[ e^{- \la H_k} x \]_{\la = 1} \ . \eqno(4.2) $$

This has several advantages: first of all, we do not have to worry
about the domain of existence of the inverse change of coordinates
\ref{4}; second, we are dealing with actions of vector fields and we can
use Lie group theory; and finally, we have a representation of the vector
field $X$ in the new coordinates which is easier to handle.

In this way, $X$ is transformed into
$$ \~X \ = \ \[ e^{\la H_k} \ X \ e^{-\la H_k} \]_{\la = 1} \ \ ,
\eqno(4.3) $$
and this can be explicitely computed by the Baker-Campbell-Haussdorf
formula \ref{5}.

We will just recall the final result, i.e. that $\~X$ can
be written as $\~X = \~f^i (x) \pa_i$, with
$$ \~f \ = \ f + \{ h , f \} + {1 \over 2} \{ h , \{ h,f \} \} +
{1\over  6} \{ h, \{ h, \{ h, f \} \} \} + ... \ \ \ . \eqno(4.4) $$
Full details of the derivation of this formula are given in the appendix.

>From (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
$f$ in a way slightly different from the customary one; the usual
homological operator, which we will denote by $\L_0$, will however be
just the standard one. 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 $V_m$.

It should be stressed that 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.



%\vfill \eject

\bigskip
{\bf 6. Non-unicity of Poincar\'e normal forms.}

In the Poincar\'e procedure\footnote{$^7$}{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 $k<s$) cannot be guaranteed to
be (and in general, are not) in the same space; thus, not all of these
can then be eliminated by suitably choosing $h_s$.

Notice in particular that even the action of higher order homological
operators (corresponding to the first nontrivial term in the expansion
(4.4) -- or the equivalent one for standard Poincar\'e procedure)
generate terms which are in $\ran \( \L_{s-k} \) \cap V_s$. It should be
stressed that this is true also if $s-k < k$, i.e. if the relevant
homological operator is associated to a term which is already in
Poincar\'e normal form.

However, this feature should not be necessarily seen as a drawback in
the Poincar\'e procedure: if on the one side this shows that we could
introduce resonant terms which were not initially present in our
system, on the other side we could use the same mechanism to eliminate
(some of the) resonant terms initially present, or generated as higher
order terms by previous changes of coordinates. More in general, we could
use these higher order effects of the Poincar\'e transformations to
further normalize -- we will use the term {\it renormalize} -- the
Poincar\'e normal form.

Indeed, it is well known that the Poincar\'e normal form is by no means
unique, i.e. that two different Poincar\'e normal forms can be
conjugated. The idea of further normalization is also not new, and has
been considered by several authors \ref{10-15}, mainly in the context of
Lie algebras filtration; here we mention in particular the work of
Broer \ref{13}, and the ``unique normal forms'' studied by Baider
\ref{15}. However, such an approach seems (at least to the present
author) of difficult implementation for the study of concrete dynamical
systems. Here, we want to study the same problem from a direct point of
view, i.e. considering the properties of Poincar\'e transformations and
the explicit higher order effects, see (2.4) and (4.5). This allows to
give a well definite algorithm, which -- once a basis is chosen in each
of the $V_k$, e.g. the one given by the $e_{\mu , j} (x)$ considered in
section 4 -- only requires linear algebra computations. Moreover, these
reduce to consideration of the action of (higher order) homological
operators and to solution of relevant (higher order) homological
equations, thus representing a natural and straightforward
generalization of the Poincar\'e scheme.

%\vfill\eject
\bigskip
{\bf 7. Poincar\'e renormalization -- I.}

We can use the considerations of the previous section, and the general
formulas obtained for the higher order action of a Poincar\'e
transformation, to devise a scheme of Poincar\'e normalization which
includes iterated normalization (renormalization) and leads to a
simplified normal form unfolding. We choose to work in the Lie-Poincar\'e
scheme; the reason for this preference will be clear in the following.

We sketch below the construction of such renormalized forms, first for
orders two and three (to let the reader grasp an intuitive
understanding), and then giving the general algorithm.

We start from a system like (1.1), (1.3), and denote by $A = (Df)(0)$
the matrix corresponding to the linear part of $f$ (which will not be
modified by our transformations). We rewrite the system, for further
reference, as
$$ {\dot x} \ = \ \sum_{k=0}^\infty f_k^{(0)} (x) \ ; \eqno(7.1) $$
the upper index on $f_k$ (and similar ones from now on) refers
to the number of Poincar\'e transformation of order $k$ applied so far.

We can, if we prefer, preliminarly operate a linear change of coordinates
to take $A$ into (real or complex) canonical form; however, this would
not be of great use: even if $A$ is semisimple, and thus can be
diagonalized, our present considerations on higher order homological
operators would not benefit from this fact.

We can then, by means of the usual Poincar\'e transformation with
generator $h_1^{(0)}$ chosen as solution to the homological equation for
$k=1$, take the term $f_1^{(0)}$ into NF, i.e. transform it to $$
f_1^{(1)} \ = \ f_1^{(0)} - \L_0 \( h_1^{(0)} \) \ \in \ \ker \( \L_0^+
\) \cap V_1 \ . \eqno(7.2) $$

Let us now pass to consider the term in $V_2$ (which is now already
changed, but that we still denote by $f_2^{(0)}$); by the usual
Poincar\'e procedure, i.e. choosing $h_2^{(0)}$ as solution to the
homological equation at order $k=2$, this is changed into
$$ f_2^{(1)} \ = \ f_2^{(0)} - \L_0 \( h_2^{(0)} \) \ \in \ \ker \(
\L_0^+ \) \cap V_2  \ \ . \eqno(7.3') $$

However, we know that we can still change this by a transformation with
generator $h_1^{(1)} \in V_1$. It is clear that now this $h_1^{(1)}$
cannot be completely arbitrary, if we want to keep the already
normalized term $f_1^{(1)}$ in its present -- and satisfactory
-- form; thus, to leave this untouched, we are obliged to require
$$ h_1^{(1)} \ \in \ \ker (\L_0 ) \cap V_1 \ . \eqno(7.3'') $$
It is clear that in this way we can only eliminate terms which
are in $\ran ( \M_1 )$, where we have defined
$$ \M_1 = \[ \L_1 \]_{\ker (\L_0 )} \ \ . \eqno(7.4) $$
Indeed, notice that (7.3') implies that in (4.5) we are reduced to
$\~f_2 = f_2 + \{ h_1,f_1 \}$. It is easy to see that this same remark,
suitably generalized, will also hold in general for our proposed
procedure: the transformations will be given by action of the series of
homological operators. This represents the main advantage, in the present
discussion, of considering Lie transformations.

Thus, this second normalization of the term in $V_2$ will transform
$f_2^{(1)}$ into
$$ f_2^{(2)} \ = \ f_2^{(1)} - \L_1 \( h_1^{(1)} \) \ = \ f_2^{(1)} -
\M_1 \( h_1^{(1)} \) \ , \eqno(7.5) $$
and by suitable choice of $h_2^{(1)}$ we can have
$$ f_2^{(2)} \ \in \ \[ \ker \( \L_0^+ \) \cap V_2 \] \cap \ker \(
\M_1^+ \) \ . \eqno(7.6') $$
If we denote by $P_1$ the operator of projection on $\ran ( \M_1 )$,
this ``suitable choice'' amounts to
$$ h_1^{(1)} \ = \ \M_1^+ \ P_1 \ f_2^{(1)} \ \ . \eqno(7.6'') $$

We could then repeat the same procedure for the term $f_3^{(0)}$, by
applying Poincar\'e transformations with generators $h_3^{(0)}$
(solution of the homological equation at order $k=3$), $h_2^{(1)}$, and
$h_1^{(2)}$; this time, however, we should not only make sure that the
$h_2^{(1)}$ does not modify the already established $f_2^{(2)}$, i.e.
that
$$ h_2^{(1)} \ \in \ \ker \( \L_1 \) \ , \eqno(7.7') $$
but also that $h_1^{(2)}$ does not modify neither the $f_2^{(2)}$ nor
the $f_1^{(1)}$, i.e. that
$$ h_1^{(2)} \ \in \ \ker \( \L_0 \) \cap \ker \( \L_1 \) \ .
\eqno(7.7'') $$
Again, these requirements will guarantee that in (4.5) only the term
$\L_1 (h_2^{(1)} ) $ and $\L_2 (h_1^{(2)} )$ are effectively giving a
contribution to the transformed $f_3$.

The general scheme of construction should at this point be quite clear,
and we can pass to describe it in abstract terms.

%\vfill\eject
\bigskip
{\bf 8. Functional setting.}

We will consider some chains of spaces and operators, together with the
chain of spaces $V_k \ss V$ and the chain of operators $\L_k : V \to V$
introduced above. These definitions will be based on a formal power
series $f = \sum f_k$ (as it was already the case for the $\L_k$).

We define the spaces $H^{(p)} \sse V$ by $H^{(0)} = V$ and, for $p \ge
1$,
$$ H^{(p)} \ = \ \ker (\L_0 ) \cap ... \cap \ker
(\L_{p-1} ) \ = \ \bigcap_{s=0}^{p-1} \ker \( \L_s \) \eqno(8.1) $$
It is obvious that $ H^{(p+1)} \sse H^{(p)}$, so that the
$H^{(p)}$ realize a filtration of $V$.

We define then the operators $\M_k$ as the restriction of $\L_k$ to $\ker
(\L_0 ) \cap ... \cap \ker (\L_{k-1} )$, i.e. $\M_p = \L_p
\big|_{H^{(p-1)}}$. Clearly,
$$ \ker \( \M_p \) \ = \ \bigcap_{s=0}^{p} \ker \( \L_s \) \ = \
H^{(p+1)} \ . \eqno(8.2) $$

Next, we define the spaces $F^{(p)}$ by $ F^{(0)} = V$ and
$$ F^{(p)} = \bigcap_{k=0}^p \[ \ran \( \M_k \) \]^\perp =
\bigcap_{k=0}^p \ker \( \M_k^+ \) \ ; \eqno(8.3) $$
hence the $F^{(p)}$'s satisfy $F^{(p+1)} \ \sse \ F^{(p)}$ and realize a
filtration of $V$, as it was already the case for the $H^{(p)}$'s.
It should be noticed that $F^{(p+1)} =
F^{(p)} \backslash \[ \ran \( \M_p \) \cap F^{(p)} \]$.

We can also define projection operators for each of these spaces; we
denote them by:
$$ \eqalign{
\pi_s : V \to \ker (\L_s ) \quad , & \quad {\^\pi}_s : V \to \ran
(\L_s^+ )  \cr  \chi_s : V \to \ker (\L_s^+ ) \quad ,
& \quad {\^\chi}_s : V \to \ran (\L_s )  \ . \cr } \eqno(8.4) $$

It will be useful to define the composition of projection operators
given by
$$ \mu_s = \pi_{s-1} \circ \pi_{s-2} \circ ... \circ \pi_0
\qquad (s \ge 1) \eqno(8.5) $$ (and $\mu_{s,k}$ as the restriction of
this to $V_k$). Later on we will use the notation $\mu_0$: this should
be meant as the identity operator.

We also consider the projection on the range of $\M_s$, i.e.
$$ P_s : V \to \ran \( \M_s \)
\eqno(8.6) $$

In this way, we can redefine our spaces and operators using the
projection operators; in particular we have $H^{(s)} = \mu_s \[ V \]$ and
$\M_s = \L_s \circ \mu_s$.

{\tt Remark 1.} Notice that we could consider the decomposition of the
spaces and operators introduced above according to (1.4), i.e.
considering their intersection with, and restriction to, the spaces
$V_k$. As in each of the $V_k$ (considered as vector spaces) we have a
finite dimensional basis, the relevant operators -- in particular, the
$\M_{p,k} = \M_p\big|_{V_k}$ -- can be written in matrix form using
these bases. $\odot$





\bigskip

{\bf 9. Poincar\'e renormalization -- II.}

We can now, with the notation introduced in the previous section,
describe in general and abstract terms the procedure sketched in
section 7 above.

{\bf Definition.} {\it We say that the dynamical system ${\dot x} = f(x)
= \sum f_k (x)$ (the vector field $X = f^i (x) \pa_i = \sum f^i_k (x)
\pa_i$; the formal power series $f(x) = \sum f_k (x)$) is in {\bf
Poincar\'e renormalized form} up to order $n$ if $f_k \in F^{(k)}$ for
all $k \le n$.}

{\bf Proposition.} {\it Any dynamical system (vector field, formal
power series) can be brought into Poincar\'e renormalized form up to
any desired order $n$ by means of a formal series of Lie-Poincar\'e
transformations.}

{\tt Proof.} We will prove constructively the above proposition by
giving a well defined algorithm for the transformation to Poincar\'e
renormalized form.

We operate sequentially for $k=1,2,...,n$ in the following way. If
$f_k^{(0)}$ is the term of order $k$ after performing the required
transformation at orders up to $k-1$, we operate then a series of
Lie-Poincar\'e transformations with generators $h_k^{(0)} ,
h_{k-1}^{(1)} , ... , h_1^{(k-1)}$, where $h_p^{(s)} \in H_p^{(s)}$;
this condition guarantees, see (4.5), that at each step the
transformation generated by $h_{k-p}^{(p)}$ will transform $f_k^{(p)}$
into
$$ f_k^{(p+1)} \ = \ f_k^{(p)} \ - \ \M_p \( h_{k-p}^{(p)} \) \ .
\eqno(9.1) $$

The $h_k^{(0)}$ is chosen as the solution to the standard homological
equation (3.3), i.e. as
$$ h_k^{(0)} = \L_0^+ \ P_0 \ f_k^{(0)} \equiv \M_0^+ \ P_0 \
f_k^{(0)} \ ; \eqno(9.2) $$
the $h_{k-p}^{(p)}$ should be chosen as the projection on $H^{(p)}_{k-p}$
of the solution to the (higher order) homological equations
$$ P_p f_k^{(p)} - \M_p \( h_{k-p}^{(p)} \) = 0 \ , \eqno(9.3) $$
which means, explicitely,
$$ h_{k-p}^{(p)} \ = \ \mu_p \circ \M_p^+ \circ P_p \ \( f_k^{(p)} \)
\eqno(9.4) $$
(with $\mu_0 = I$, the (9.2) is included in this formula as well).

Clearly, in this way we arrive in the end -- i.e. after applying the
procedure to terms of order $k=2,3,...n$ -- at a system
$$ {\dot x} \ = \ f^* (x) \ \sum_{k=0}^\infty f^*_k (x) \eqno(9.4) $$
in which, for $k \le n$,
$$ f^*_k (x) \ = \ f_k^{(k)} \ \in \ F_k^{(k)} \ . \eqno(9.5) $$
The proof is thus complete. $\odot$

{\tt Remark 2.}
We would like to point out that one can also work following a different
order; i.e., once we have fixed the order $n$ up to which we want to
put the system in Normal Form, we can first proceed to the usual
Poincar\'e normalization, i.e. consider the transformations generated
on the $h_k^{(0)}$, and corresponding to the action of $\L_0$, for
$k=1,...,n$; we can then consider the transformations, generated by
$h_k^{(1)}$, corresponding to the action of $\L_1$ (or more precisely
of $\M_1$), i.e. ``second-normalize the normal form'', and so on.
$\odot$

{\tt Remark 3.}
We would also like to remark explicitely that, although we have
preferred to avoid a cumbersome notation, one could consider
restriction of operators to the relevant $V_k$ subspaces, and thus
transform -- by means of bases in each of the $V_k$ -- the above
equations in algebraic ones. $\odot$

\vfill\eject
\bigskip
{\bf 10. Planar vector fields with rotations as linear part.}

We will consider, as a meaningful example, the unfolding of Normal Forms
for vector fields in $R^2$ having linear part $f_0 (x) = Ax$ with
$$ A \ = \ \pmatrix{0&-1\cr1&~0\cr} \ \ . \eqno(10.1) $$

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(10.2) $$
where the $a_k , b_k $ are arbitrary real constants. Writing $r^2 =
(x_1^2 + x_2^2 )$, this reads
$$ f_k (x) = \cases{0 & for $k$ odd \cr
r^{2m} \[ a_m I + b_m A \] x & for $k=2m$.\cr} \eqno(10.3) $$

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,
$$ f_2^{(2)} = f_2^{(0)} = r^2 (a_1 I + b_1 A ) x \ . \eqno(10.4) $$

We then have $f_3 = 0$, and
$$ f_4^{(0)} = r^4 (a_2 I + b_2 A ) x \ ; \eqno(10.5) $$
the generator $h_2^{(2)}$ of the transformation
$$ f_4^{(2)} = f_4^{(0)} - \M_2 \( h_2^{(2)} \) \eqno(10.6) $$
must be, for $h_2^{(2)} \in \ker (\L_0 )$, of the form
$$ h_2^{(2)} = r^2 (\a I + \b A ) x \ , \eqno(10.7) $$
and thus
$$ \M_2 \( h_2^{(2)} \) \ = \ 4 \( a_1 \b - b_1 \a \) Ax \ ; \eqno(10.8)
$$
it is then clear that, unless $a_1 = b_1 = 0$, we can always choose $\a
, \b$ so that
$$ f_4^{(2)} (x) = f_4^{(4)} (x) = r^4 a_2 x \ . \eqno(10.9) $$

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(10.10) $$
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(10.11) $$
(again, $h_{2(k-1)}^{(3)} \in \ker (\L_0 )$ requires this form for
$h$), we have
$$ \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(10.12) $$
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 (10.2) 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 (10.8), 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 $f_6$, (10.10) 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$; 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
$$ f(x) \ = \ Ax \ + \ \sum_{k=\mu}^\infty \ r^{2k} a_k x \ + \
\sum_{k=\nu}^\infty \ r^{2k} b_k Ax \ . \eqno(10.13) $$
Then, the Poincar\'e renormalized form of $f$ up to any given order $n$
is given by
$$ f^* (x) \ = \ Ax + r^{2\mu} a_\mu x + r^{2\nu} \b A x + r^{4\mu} \a x
\ , \eqno(10.14) $$
where $a_\mu \not= 0$ is the same as in (10.13), 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.
$$ D = x_1 \pa_1 + x_2 \pa_2 \quad , \quad R = -x_2 \pa_1 + x_1 \pa_2 \
; \eqno(10.15) $$
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(10.16) $$

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(10.17) $$

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 (10.17), 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. $\odot$

\vfill\eject


{\bf Appendix.}

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{5}.

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{5,16}, 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

~\bigskip\bigskip
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\bigskip\bigskip\bigskip\parskip=6pt

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\bye