% plain TeX \def\giorno{4 November 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\P{{\cal P}} \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 \font \ptt = cmtt9 {\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 {\bf References} \parindent=10pt \bigskip\bigskip\bigskip\parskip=4pt \item{1.} V.I. Arnold, ``Geometrical methods in the theory of differential equations'', Springer, Berlin 1988 \item{2.} V.I. Arnold and Yu.S. Il'yashenko, ``Ordinary differential equations''; in {\it Encyclopaedia of Mathematical Sciences - vol. I, Dynamical Systems I}, (D.V. Anosov and V.I. Arnold eds.), p. 1-148, Springer, Berlin 1988 \item{3.} A.D. Bruno, ``Local methods in nonlinear differential equations'', Springer, Berlin 1989 \item{4.} M. Kummer, {\it Nuovo Cimento B}, {\bf 1} (1971), 123 \item{5.} M. Kummer, {\it Comm. Math. Phys.} {\bf 48} (1976), 53 \item{6.} J.C. van der Meer, ``The Hamiltonian Hopf bifurcation'', {\it Lecture Notes in Mathematics} {\bf 1160}, Springer, Berlin 1985 \item{7.} H.W. Broer, {\it Bifurcations of singularities in volume preserving vector fields}, Ph.D. Thesis, Groningen 1979; {\it Formal normal form theorems for vector fields and some consequences for bifurcations in the volume preserving case}, Springer, Berlin 1981 \item{8.} A. Baider and R.C. Churchill, {\it Proc. R. Soc. Edinburgh A} {\bf 108} (1988), 27 \item{9.} A. Baider, {\it J. Diff. Eqs.} {\bf 78} (1989), 33 \item{10.} G. Gallavotti, ``Perturbation theory'', {\tt mp-arc 93-318} \item{11.} G. Benettin, L. Galgani and A. Giorgilli, {\it Nuovo Cimento B} {\bf 79} (1984), 201 \item{12.} Yu. A. Mitropolsky and A.K. Lopatin, ``Nonlinear mechanics, groups and symmetry'', Kluwer, Dordrecht 1995 \item{13.} C. Elphick, E. Tirapegui, M.E. Brachet, P. Coullet, and G. Iooss, {\it Physica D} {\bf 29} (1987) 95 \item{14.} G. Iooss and M. Adelmeyer ``Topics in bifurcation theory and applications, World Scientific, Singapore 1992 \item{15.} V. Bargmann, {\it Comm. Pure Appl. Math.} {\bf 14} (1961), 187 \item{16.} G. Cicogna and G. Gaeta, {\it Journ. Phys. A: Math. Gen.} {\bf 27} (1994), 461 \& {\bf 27} (1994), 7115 \item{17.} L.D. Landau and I.M. Lifshitz, ``Quantum Mechanics'', Pergamon, London, 1959 \item{18.} L.M. Markhashov, {\it J. Appl. Math. Mech.} {\bf 38} (1974), 788 \item{19.} A.D. Bruno and S. Walcher, {\it J. Math. Anal. Appl.} {\bf 183} (1994), 571 \item{20.} G. Cicogna, {\it J. Math. Anal. Appl.} {\bf 199} (1996), 243 \item{21.} C.L. Siegel and J.K. Moser, ``Lectures on celestial mechanics'', Springer, Berlin 1955 (reprinted in {\it Classics in Mathematics}, 1995) \item{22.} E. Forrest and D. Murray, {\it Physica D} {\bf 74} (1994), 181 \item{23.} G. Cicogna and G. Gaeta, in preparation. \bye