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\def\giorno{12 June 96}

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\def\B{{\cal B}}
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\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}}
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\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\ }}


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{\nopagenumbers
~ \vskip 2.5 truecm
\centerline{\bf Algorithmic reduction of Normal Forms}
\footnote{}{{\tt Last modified \giorno}}
\footnote{}{PACS numbers: 02.30, 03.20, 46.10}
\vskip 3 truecm
\centerline{Giuseppe Gaeta}
\bigskip
\centerline{\it Department of Mathematics, Loughborough University}
\centerline{\it Loughborough LE11 3TU, England}
\smallskip
\centerline{\tt G.Gaeta@lboro.ac.uk}
\vskip 5 truecm
{\bf Abstract} {We present an algorithmic procedure to further reduce
(Poincar\'e) Normal Forms; we give both general results and methods,
and a concrete application.}
\vfill\eject}
\pageno=1

~\vskip 3 truecm
{\bf 1.}
In the Poincar\'e theory of Normal Forms\ref{1}, one considers
systems of $C^\infty$ ODEs in $R^n$, having a fixed point -- say, the
origin -- and aims to a classification of perturbative expansions
(formal power series) for the systems around the fixed point modulo
formal $C^\infty$ changes of coordinates. One obtains then the {\it
Poincar\'e-Dulac theorem} according to which any system can be
formally put into Poincar\'e Normal Form up to any given order $n$ by
means of a formal series of Poincar\'e transformations.

Concretely, let us denote by $V_k$ the set of homogeneous vector
polynomials of degree $k+1$; if we write the system as
$$ {\dot y} \ = \ \sum_{m=0}^\infty f_m (y) \ , \eqno(1) $$
where $f_m \in V_m$, we then consider near-identity changes of
coordinates of the form
$$ y \ = \ x + h_k (x) \eqno(2) $$
with $h_k \in V_k$; as it can be readily checked
by direct computation, \er{2} changes \er{1} into a system ${\dot x}
= \sum_m \~f_m (x) $, where $\~f_m (x) = f_m (x)$ for $m<k$,
$$ \~f_k = f_k - \L_0 (h_k ) \ , \eqno(3) $$
and terms with $m>k$ are changed in a more complicate way; in \er{3}
the $\L_0$ denotes the linear {\it homological operator}: if we define
$$ \{ f , g \} := (f \cdot \grad ) g - (g \cdot \grad ) f \ ,
\eqno(4) $$
then the $\L_0 : V_k \to V_k$ associated to $f$ is simply given by
$$ \L_0 (h) = \{ f_0 , h \} \ . \eqno(5) $$


It is clear that, denoting by $\pi$ the projection on the range of
$\L_0$, by choosing $h_k$ as the solution of the {\it homological
equation}
$$ \pi f_k = \L_0 (h_k ) \eqno(6) $$
we can eliminate all terms in $V_k \cap \ran (\L_0 )$; we call {\it
resonant} the terms in $[ \ran (\L_0) ]^c$ (by introducing a scalar
product in each of the $V_k$, this is $\ker (\L_0^+ )$; the
appropriate scalar product is the Bargman one\ref{2}), and {\it
non-resonant} those in $\ran (\L_0 )$. We can thus consider
transformations like (2) successively for $k=1,2,...,n$, and eliminate
all the non-resonant terms up to order $n$. When this is done, i.e.
when in \er{1}
$$ f_k \in \[ \ran (\L_0 ) \]^c \ \ \forall k \ge 1 \ , \eqno(7) $$
we say the system is in {\it Poincar\'e Normal Form}.

\bigskip
{\bf 2.}
Some points should be noted in the above construction: {\tt (i)} the
solution to \er{6} is not unique, being only unique up to a $\de h_k
\in \ker (\L_0 )$; {\tt (ii)} the $h_k$ and $\~h_k = h_k + \de h_k$
produce the same transformation on the term $f_k$, but (in general)
different ones on terms $f_m$ with $m>k$; {\tt (iii)} the
transformation \er{2} can, and in general will, produce resonant
terms in $V_m$ for $m>k$.

These simple considerations suggest that one could try to use the
freedom in choosing $h_k$ in order to produce resonant higher order
terms, and thus eliminate from the Normal Form some of the resonant
terms (not eliminable by the standard Poincar\'e algorithm described
above).

This is indeed possible; moreover -- what is even more relevant -- it
can be done in a completely algorithmical way, as for the Poincar\'e
procedure. We do now discuss the algorithm and the final result,
together with a relevant example; a more complete discussion is given
elsewhere\ref{3}.

\bigskip
{\bf 3.}
Let us consider a formal power series $f(x) = \sum_k f_k (x)$, $f_k
\in V_k$ (we denote the linear space of these by $V$); to this we
associate a series of linear {\it homological operators} $\L_k : V_m
\to V_{m+k}$ by
$$ \L_k (h) = \{ f_k , h \} \ ; \eqno(8) $$
also, we denote by $\pi_k$ the projection on the range of $\L_k$.

We consider now a first chain of subspaces $H^{(p)} \sse V$, with
$H^{(0)} = V$, and defined for $p>0$ by
$$ H^{(p+1)} = H^{(p)} \cap \ker (\L_p ) \ . \eqno(9) $$
We can then consider $\M_p$, the restriction of $\L_p$ to $H^{(p)}$;
we define then a second chain of subspaces $F^{(p)} \sse V$, with
$F^{(0)} = V$, and defined for $p>0$ by
$$ F^{(p+1)} = F^{(p)} \cap \[ \ran (\M_p ) \]^c \eqno(10) $$
(having introduced the scalar products in $V_k$, and thus in $V$, we
would have $F^{(p+1)} = F^{(p)} \cap \ker (\M_p^+ )$ ).

Our main result is then: {\it Given any system \er{1}, it is possible
to formally reduce it, by a formal double series of Poincar\'e
transformations, to a system
$$ {\dot x} = f^* (x) = \sum_{m=0}^\infty f^*_m (x) $$
where $f^*_k \in F^{(k)}$ for all $k=1,...,n$, for arbitrary given
$n$.}
We say then that the system is in Poincar\'e {\it renormalized} form
(PRF for short).

This result can be proven constructively, giving the concrete algorithm
of reduction to renormalized form.

We notice that it would be possible to first reduce the system to the
usual Poincar\'e normal form, than take it into a ``second normalized
form'' using the effect of $h_k$ ($k=1,2,...$) on $V_{k+1}$, then into
a third one using the effect of $h_k$ ($k=1,2,...$) on $V_{k+2}$, and
so on up to an ``n-th normalized'' which is our renormalized form up
to order $n$; this is the reason for the name we adopt.

However, for computational purposes it appears preferable to proceed
in a different way, i.e. order by order, taking $f_k$ into $f_k^*$
before passing to operate on $f_{k+1}$; we illustrate the algorithm in
this approach.

\bigskip
{\bf 4.}
We will use Lie-Poincar\'e transformations, i.e. we will write the
change of coordinates as
$$ y \ = \ \[ e^{- \la H_k} x \]_{\la = 1} \ , \eqno(11) $$
where $H_k$ is a vector field with components $h_k$, i.e. $H_k =  h_k
(x) \cdot \grad $. The expression of $X = f(x) \cdot \grad$ in the
new coordinates is given by $\~X = e^{H_k} X e^{- H_k}$, which can be
explicitely computed by means of the Baker-Campbell-Haussdorf
formula\ref{4}; in this way the $f_m$ with $m<k$ are unchanged, and
$f_k$ follows \er{3}, as in the usual Poincar\'e transformations.
Referring to Ref.3 for details, we introduce the operator $\H = \{
h_k , . \}$ and just mention that with \er{11} the $f_m$ are transformed
into $$ \~f_m \ = \ \sum_{s=0}^{[m/k]} \, (1/k!) \, \H^s (f_{m-sk} ) \
, \eqno(12) $$ where the square bracket denotes integer part, and
$\H^0$ is the identity operator.

Let us now suppose the series $f = \sum_m f_m$ is in PRF up to
$f_{k-1}$, and let us try to transform $f_k \equiv f_k^{(0)}$ into
$f_k^*$ by means of transformations like \er{11}. In doing this, we
should make sure we do not modify the terms $f_1 , ... , f_{k-1}$
which are already in PRF. If we use $h_k^{(0)} \in V_k$, this is no
restriction, and in this way we can eliminate all terms in $[\ran
(\L_0 )] \cap V_k$, as from \er{12} we get now $f_k^{(0)} \to
f_k^{(1)} = f_k^{(0)} - \L_0 (h_k^{(0)} )$. Let us now choose a
$h_{k-1}^{(1)} \in V_{k-1}$; in order to leave $f_{k-1}$ unchanged, we
require that $h_{k-1}^{(1)} \in \ker ( \L_0 ) = H^{(1)}$; from \er{12}
we have $f_k^{(1)} \to f_k^{(2)} = f_k^{(1)} - \L_1 (h_{k-1}^{(1)} )$
and thus in this way we can eliminate all terms in the image of
$H^{(1)}$ under $\L_1$, i.e. in $\ran (\M_1 )$. We can proceed
like this for $h_{k-s}^{(s)} \in V_{k-s}$ for all $s=2,3,...k-1$;
each time we have to ask, in order to leave the $f_m$ with $m<k$
unchanged, that $h_{k-s}^{(s)} \in H^{(s)}$, and with this
requirement \er{12} just gives
$$ f_k^{(s-1)} \to f_k^{(s)} = f_k^{(s-1)} - (1/s!) \L_{k-s}
\( h_{k-s}^{(s)} \) \ ; \eqno(13) $$
thus, in this step we can eliminate terms in $\ran ( \M_s )$. In this
way, after performing sequentially these transformations for
$s=0,1,..., k-1$ we arrive at a $f_k^{(k)} = f_k^* \in F^{(k)}$, as
claimed above.

With the notation introduced above -- and with the adjoint being
meant with respect to the Bargman scalar product\ref{2} -- the
$h_{k-s}^{(s)}$ to be used at each step are obtained as solution to
the higher homological equations
$$ h_{k-s}^{(s)} \ = \ \mu_s \circ \M_s^+ \circ P_s \( f_k^{(s)}
\) \ . \eqno(14) $$

Clearly, once we introduce a given orthogonal basis in each of the
$V_k$, \er{14} amounts to a linear algebra problem, and the whole
procedure can be implemented by solving at each stage {\it finite
dimensional linear algebraic equations}, as in the standard
(Poincar\'e) Normal Form reduction/unfolding procedure.

\bigskip
{\bf 5.}
In order to illustrate the reduction of Normal Forms obtained here by
a significant example, we consider the unfolding of Normal Forms for
two dimensional systems with linear part $f_0 (x) = Ax$, with $A$ the
rotation matrix
$$ A = \pmatrix{ 0 & -1 \cr 1 & 0\cr} \quad . \eqno(15) $$
It is well known that, with $r^2 = (x_1^2 + x_2^2 )$, the Poincar\'e
Normal Form corresponding to this is
$$ f(x) \ = \ Ax \ + \ \sum_{k=1}^\infty \ r^2 \[ a_k I + b_k A \] x
\ , \eqno(16) $$
with $a_k , b_k$ arbitrary real constants.
By the procedure proposed here, we can reduce the system to PRF,
whose general form is
$$ f^* (x) \ = \ Ax \ + \ \[\(  r^{2 \mu} a_\mu  + r^{4 \mu} \a \) I +
r^{2 \nu} \beta A \] x \ , \eqno(17) $$
where $\mu , \nu$ are respectively the smallest integers such that
$a_\mu \not= 0$, $b_\nu \not= 0$, and $\a , \b$ are real constants
(with $\b = 0$ if $\nu > \mu$); the computation of \er{17} is
immediate\ref{3}.

\bigskip

{\bf References}

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\bigskip

\item{1.} V.I. Arnold, {\it Geometrical methods in the theory of
differential equations}, Springer, Berlin 1988;
V.I. Arnold and Yu.S. Il'yashenko, {\it 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{2.} V. Bargmann, {\it Comm. Pure Appl. Math.} {\bf 14} (1961),
187;
C. Elphick, E. Tirapegui, M.E. Brachet, P. Coullet, and G.
Iooss, {\it Physica D} {\bf 29} (1987) 95;
G. Iooss and M. Adelmeyer {\it Topics in bifurcation theory
and applications}, World Scientific, Singapore 1992

\item{3.} G. Gaeta, ``Poincar\'e renormalized forms'', Preprint 1996
{\tt[ mp-arc 96-263]}; ``Reduction of Poincar\'e Normal Forms'',
Preprint 1996

\item{4.} L.D. Landau and I.M. Lifshitz, {\it Quantum Mechanics},
Pergamon, London, 1959;
Yu. A. Mitropolsky and A.K. Lopatin, {\it Nonlinear
mechanics, groups and symmetry}, Kluwer, Dordrecht 1995;
G. Benettin, L. Galgani and A. Giorgilli, {\it Nuovo Cimento
B} {\bf 79} (1984), 201

\bye