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\def\giorno{13/1/97}
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\title{Reduction of Normal Forms}
\author{Giuseppe Gaeta \cite{add}}
\address{I.H.E.S., 91440 Bures sur Yvette (France)}
\date{\giorno}
\maketitle
\begin{abstract}
We give an algorithm to simplify Poincar\'e
normal forms, making use of higher order
effects; this leads to a simplification in
the local study of nonlinear systems. The
procedure is easily extended to the
hamiltonian case and Birkhoff normal forms.
\end{abstract}
\pacs{PACS numbers: 02.30.Hq , 03.20.+i , 46.10.+z}
\section{Introduction and motivation}
Poincar\'e Normal Forms \cite{Arn2,EMS1,Bru,Ver} are among the
most powerful tools to study the local behaviour of Nonlinear Systems
around a singular point (the behaviour around a nonsingular point is
trivial, as follows from the rectification theorem \cite{Arn1}).
In general, given a nonlinear system of differential equations
$$ {\dot \xv} = f (\xv ) \ , \eqno(1) $$
where we suppose $f: \R^n \to \R^n$ is a formal power series (e.g. the
Taylor series corresponding to expansion around $\xv_0 = 0$) such that
$f(0) = 0$, we can transform this, by a sequence of formal changes of
coordinates, collectively called the normalizing transformation, into a
system which is in Normal Form (NF) up to any desired
order [in the following, we will mean by NF the ``infinite
order'' NF, and denote as ``partial NF (of order $m$)'' the NF of
order $m$.] $N$, and formally for $N = \infty$. 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.
Two points should be stressed: {\it (i)} The equivalence between the
original system and its NF is, in general, only formal (due to lack of
convergence of the normalizing transformation; this is in turn due to small
denominators): thus, in general a system is {\it not} conjugated to its NF;
however, for $N$ sufficiently small (depending on the system), the system
{\it is} conjugated to its partial NF of order $N$; {\it (ii)} For a given
system, the reduction to NF -- although obtained by means of a well defined
algorithm -- becomes computationally very demanding with the increasing of
$N$: the computations can be set in terms of linear algebra, but
they require to consider -- and invert -- matrices of order rapidly growing
with $N$.
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 $t < 1/ \eps^m$) \cite{Arn2,EMS3}.
On the other side, if we want to consider the most general behaviour of
systems with given linear part, 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 NF
expansion -- or in mathematical literature, {\it unfolding} of the NF
-- for $A$). For short enough times, the study of the NF expansion
gives the most general local behaviour (within suitable error) of
systems with given linear part.
It is well known that the NF corresponding to a given system is in
general {\it not unique}; correspondingly, the complete expansion
(unfolding) of the NF for $A$ 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
the 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 for $A$ in which the redundancies in the Poincar\'e-Dulac
classification, or at least some of them, have been eliminated: this
automatically simplifies the local analysis of nonlinear systems.
This problem has been considered by several authors \cite{Kum,Bai}
in a mathematically sophisticated language (based on
filtration of Lie algebras); several theoretical result exist, and it
is actually also possible to define a {\it unique NF} \cite{Bai}.
However, all these results are of difficult concrete implementation.
The purpose of this note is indeed to propose a procedure of
``further reduction'' of NFs (which
I call ``renormalization'') which does not lead to a unique reduced
NF, but yields a significant simplification and -- I would like to
stress -- is completely algorithmic, and not more difficult to
implement in practice than the standard Poincar\'e-Dulac normalization.
Actually, the procedure I propose below is nothing else than a direct
generalization of the one proposed by Poincar\'e; the main ``new''
ingredient will be to make use of the well known freedom in the choice of
the generating functions for the coordinate transformations,
and a control of higher-order effects in this transformation.
The theory esposed here does also apply to the hamiltonian
case, i.e. to Birkhoff normal forms \cite{Ver,EMS3,Arn3},
as briefly shown below.
\section{Change of coordinates formulas}
We consider changes of coordinates $x \to \~x$ defined by the time-one
action of a smooth vector field
$$ X_h = h^i (\xv ) \ \pa / \pa x^i \equiv h^i \pa_i \ , \eqno(2) $$
also known as {\it Lie transform} \cite{BGG,ML}. Thus, we have
formally $$ \~{\xv} = \[Êe^{\la X_h} \xv \]_{\la = 1} \ ; \eqno(3) $$
in this way, $X_0 = f^i (\xv ) \pa_i$ is transformed into $\~X =
e^{X_h} X_0 e^{- X_h}$; this can be
computed by means of the Baker-Campbell-Haussdorf formula
\cite{ML,LL}, and its effect is that $f(\xv )$ changes into
$$ \~f (\xv ) =
\sum_{n=0}^\infty \[ \( (-1)^n / n! \) \phi_n (\xv ) \] \ , \eqno(4)
$$ where $\phi_0 = f$ and $\phi_{n+1} = \{ \phi_n , h \}$, having
defined the bracket as $\{ \phi , \psi \} = (\phi \cdot \grad ) \psi
- (\psi \cdot \grad ) \phi$.
Let us denote by $V_k$ ($k\ge 0$) the vector functions whose components are
homogeneous polynomials of order $(k+1)$ in the $x^i$; these will be
seen as orthogonal subspaces of a space $V$ (direct sum of the $V_k$'s)
and equipped with a scalar product
\cite{Arn2,Ioo} for the sake of later considerations.
With $f_m \in V_m$, we rewrite (1) as $$ {\dot \xv} = \sum_{m=0}^\infty
f_m (\xv ) \ . \eqno(5) $$
If we choose the $h$ generating the change of coordinates to be $h_k \in
V_k$, we have that the $f_m$ with $m < k$ are unchanged, while for higher
$m$ we have (with square bracket denoting integer part)
$$ \~f_m = \sum_{s=0}^{[m/k]} (1/s!) \H^s \( f_{m-sk} \) \ , \eqno(6) $$
where we have defined $\H (.) = \{ h_k , . \}$.
Notice that, as well known, $\~f_k = f_k - \{ f_0 , h_k \}$.
\section{Normal forms and further reduction}
In the usual Poincar\'e normalization, we proceed by considering a sequence
of changes of coordinates with generators $h_k$ with $k=1,2,...$; these
$h_k$ are chosen by solving the homological equations
$$ \{ f_0 , h_k \} = \pi_0 \cdot \^f_k \ , \eqno(7) $$
where $\pi_0$ is the projection on the range of the homological operator
$\L_0 = \{ f_0 , . \}$ (once we have chosen a scalar product in the $V_k$,
which we suppose to have done, this is a projection on the kernel of
$\L_0^+$), and $\^f_k$ is the term in $V_k$ after considering the
transformation generated by $h_1, ... , h_{k-1}$. In this way we can
eliminate any term in $\ran (\L_0 )$; we also say that we remain with {\it
resonant} terms \cite{EMS1,Bru,Ver,Arn1}. Notice that the solution
$h_k$ to (7) is only determined up to a term $(\de h_k) \in \ker
(\L_0 )$.
Notice however that, due to higher order effects [i.e. $s > 1$ in
(6)], in this process we can -- and in general will -- generate terms
in $[\ran (\L_0 ) ]^\perp$ which were not present in the original
system; different choices of $(\de h_k)$ will give different higher
order terms. By the same mechanism of ``higher order effects'', it
would also be possible to eliminate (some of the) resonant terms
initially present: this is unlikely to happen by random choice, but
can be obtained by the procedure we are going to describe.
Suppose to have already taken into NF the terms $f_1$ and $f_2$, so that
$\L_0^+ (\~f_1 ) = \L_0^+ (\~f_2 ) = 0$. We consider now a change of
coordinates with generator $h_1^{(1)}$ such that $\~f_1$ is not changed,
which is the case if $h_1^{(1)} \in \ker (\L_0 )$; according to (7),
$\~f_2 \equiv f_2^{(1)}$ (we also write $\~f_1 \equiv f_1^{(1)}$) is then
changed into $$ f_2^{(2)} = f_2^{(1)} - \{ \~f_1 , h_1^{(1)} \} \ ;
\eqno(8) $$ to conform with the general notation introduced below, we
define the ``higher homological operator'' $\L_1 = \{ f_1^{(1)} , . \}$, and
$\M_1$ as the restriction of $\L_1$ to $\ker (\L_0 )$, so that (8) reads
$$ f_2^{(2)} = f_2^{(1)} - \M_1 \( h_1^{(1)} \) \ . \eqno(9) $$
In this way, we can eliminate from $f_2$ not only the terms in $\ran (\L_0
)$, but also those in $\ran (\M_1 )$. With the choice of a scalar product,
we obtain a term $f_2^{(2)} \in \ker (\L_0^+ ) \cap \ker (\M_1^+ )$. Notice
that $h_1^{(1)}$ is determined by an ``higher order homological equation''
$$ \M_1 \( h_1^{(1)} \) = \pi_1 \cdot f_2^{(1)} \ , \eqno(10) $$
where $\pi_1$ is the projection on the range of $\M_1$.
This construction can then be generalized to higher order terms $f_p$ as
follows.
For a given sequence $f_1 , f_2 , ... $ with $f_k \in V_k$, we define $L_p
= \{ f_p , . \}$; we define then the spaces $H^{(p)} \sse V$ by $H^{(0)} =
V$ and, for $p \ge 1$, $$ H^{(p)} = H^{(p-1)} \cap \ker
(\L_{p-1} ) \ ; \eqno(11) $$
obviously $ H^{(p+1)} \sse H^{(p)}$.
Next, we define the $\M_k$ as the restriction of $\L_k$ to $H^{(p-1)}$, and
$\pi_k$ as the projection on $\ran (\M_k )$. Finally, we define the spaces
$F^{(p)}$ by $ F^{(0)} = V$ and $$ F^{(p)} = F^{(p-1)} \cap \ker \( \M_p^+
\) \ ; \eqno(12) $$ hence these satisfy $F^{(p+1)} \ \sse \ F^{(p)}$.
With this notation, we can repeat the construction considered for
$f_2^{(2)}$ at all orders: suppose to have already operated for $f_1
, f_2 , ... f_{p}$, so that for $k \le p$ we have $f_k^{(k)} \in
F^{(k)} \cap V_k \equiv F^{(k)}_k$; we can then consider changes of
coordinates with generators $h_{p-t}^{(t)} \in H^{(t)} \cap V_{p-t}
\equiv H^{(t)}_{p-t}$ (with $t=1,...,p$): these will not affect the
terms $f_1^{(1)} , ... , f_{p-1}^{(p-1)}$, but permit to eliminate
terms in $f_{p+1}$, reducing this via
$$ f_{p}^{t+1} = f_{p}^{t} - \M_t \( h_{p-t}^{(t)} \) \eqno(13) $$
to a $f_{p}^{(p)} \in F^{(p)}_{p}$.
We say that the formal power series $f(\xv ) = \sum_{k=0}^\infty f_k$
(where $f_k \in V_k$) is in {\bf Poincar\'e renormalized form} (PRF) up to
order $N$ if $f_k \in F^{(k)}_k$ for all $k \le N$; if this is
satisfied for all $k$, we just say that $f$ is in Poincar\'e
renormalized form \cite{Gae}.
The above discussion shows (constructively) that: {\it any formal
power series can be formally transformed into Poincar\'e renormalized
form up to any given order by a formal series of changes of
coordinates.} This transformation is concretely achieved by the
algorithm described above. By analogy with the Poincar\'e normalizing
transformation, the sequence of changes of coordinates considered
here will be called the ``Poincar\'e renormalizing transformation''.
Notice that points {\it (i),(ii)} mentioned in the introduction
still apply, and in practice one will consider partial PRFs up
to a suitable order $m$, as for standard NFs.
Thus, we can consider renormalized forms instead than standard
Poincar\'e normal forms. The advantage of doing so is that {\it (a)}
the renormalized forms can present a substantial simplification, as
the following examples will show; {\it (b)} the computation of PRF
is not mor difficult than that of standard NF (going up to the same
order), as it also amounts to solving linear algebraic equations (of
the same order as those to be solved for NF at the same order). We
also notice that, in considering convergence problem, the
Bruno-Markhashov-Walcher theory \cite{BMW} allows to affirm
convergence based on symmetry considerations, making use of a
system possessing higher symmetry than,
but formally equivalent to, the original one: the PRFs
are natural candidates for such a system.
The present theory can also be easily extended to Birkhoff normal
forms, i.e. to the hamiltonian case \cite{Ver,EMS3,Arn3}, going
essentially through the same construction: now we would consider a
Hamiltonian $H$, $V_k$ would be the space of functions from the phase
space to $\R$ homogeneous of degree $(k+2)$, so that $H = \sum_k
H_k$, $\{.,.\}$ would be the usual Poisson bracket, the role of $A$
would be played by the quadratic part $H_0$ of the Hamiltonian, and
$\L_k = \{ H_k , . \}$; with this notation, the discussion would be
formally identical to the one presented above (see \cite{Gae} for
details).
\section{Examples}
We consider here some examples, comparing the expansion of PRF and
of standard NFs for some systems with given
linear part $f_0 (\xv ) = A \xv$; we only report results, referring for
details on these -- and in general for a complete discussion of PRFs -- to
\cite{Gae}.
{\it Example 1.} Consider $\R^2$ and $A=\{ \{ 0,-1\},\{1,0\}\}$ the standard
symplectic matrix. The standard NF expansion corresponding to this is
$$ f(\xv ) = A \xv + \sum_{k=1}^\infty |\xv |^{2k} \[ a_k I + b_k A \] \xv
\eqno(14) $$ with $a_k$, $b_k$ two arbitrary sequences of real numbers.
The expansion of PRF is, with $\mu , \nu$ suitable positive integers
(which can be explicitely determined \cite{Gae}), $$ f (\xv ) = A \xv +
\[ |\xv |^{2k\mu} c_1 I + |\xv |^{4\mu} c_2 I + |\xv |^{2\nu} c_3 A
\] \xv \ , \eqno(15) $$ and thus depends only on three arbitrary real
constants. When (14) is hamiltonian (i.e. $a_k = 0$ for all $k$), this
corresponds to a well known result \cite{SM}.
{\it Example 2.} Consider $\R^2$ with coordinates $(x,y)$, and $A$ diagonal
with entries $\{ 1 , -n \}$ (with $n>1$, as $n=1$ is equivalent to example
1 and $n \le 0$ is trivial).
The standard NF expansion is (sums are for $k = 1,..., \infty$)
$$ f(\xv ) = \pmatrix{ ~~ x + \sum \sigma_k (x^n y)^k x \cr - ny +
\sum \rho_k (x^n y)^k y \cr} \ ; \eqno(16) $$
again this depends on two arbitrary sequences of real numbers.
The expansion of PRF is, with $p,q$ suitable positive integers
(which can be explicitely determined \cite{Gae}),
$$ f(\xv ) = \pmatrix{ ~~x + n \[ c_1 (x^n y)^q + c_2 (x^n y)^p + c_3
(x^n y)^{2p} \] x \cr - n y + ~ \[ c_1 (x^n y)^q + c_2 (x^n y)^p +
c_3 (x^n y)^{2p} \] y \cr} \eqno(17) $$
and thus it depends again on three real numbers.
{\it Example 3.} Consider $\R^3$ and $A$ diagonal with entries
$\{1,2,5\}$; the standard NF expansion depends on four real parameters,
while the PRF expansion depends on three real parameters, of which at
least one can be chosen to be zero.
{\it Example 4.} Consider $\R^3$ and $A$ diagonal with entries
$\{1,3,9\}$; the standard NF expansion depends on five real parameters,
while for the PRF expansion there are seven possibilities, each one
depending at most on two real parameters.
\section*{Acknowledgements}
This note was written during my stay in IHES; I would like to express my
hearty thanks to its Director,
prof. J.P. Bourguignon, for his invitation,
and to all the staff for the warm
hospitality.
\hfill\eject
\begin{references}
\bibitem[*]{add} On leave from Department of Mathematical Sciences,
Loughborough University,
Loughborough LE11 3TU (Great Britain); {\tt G.Gaeta@lboro.ac.uk}
\section*{References}
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differential equations'', Springer, Berlin 1982
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differential equations''; in {\it Encyclopaedia of Mathematical
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dynamical systems''; Springer, Berlin 1990, 1996$^2$
\bibitem{Arn1} V.I. Arnold, ``Ordinary differential equations'',
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\bibitem{EMS3} V.I. Arnold, V.V. Kozlov and A.I. Neishtadt,
``Mathematical aspects of classical and celestial mechanics''; in
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\bibitem{Kum} M. Kummer, {\it Nuovo Cimento B}, {\bf 1} (1971), 123;
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J.C. van der Meer, ``The Hamiltonian Hopf bifurcation'',
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\bibitem{Bai} A. Baider and R.C. Churchill, {\it Proc. R. Soc.
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\bibitem{Arn3} V.I. Arnold, ``Mathematical methods of classical
mechanics'', Springer, Berlin 1978, 1989$^2$
\bibitem{BGG} G. Benettin, L. Galgani and A. Giorgilli, {\it Nuovo
Cimento B} {\bf 79} (1984), 201
\bibitem{ML} Yu. A. Mitropolsky and A.K. Lopatin, ``Nonlinear
mechanics, groups and symmetry'', Kluwer, Dordrecht 1995
\bibitem{LL} L.D. Landau and I.M. Lifshitz, ``Quantum Mechanics'',
Pergamon, London, 1959
\bibitem{Ioo} G. Iooss and M. Adelmeyer, ``Topics in bifurcation
theory and applications'', World Scientific, Singapore 1992
\bibitem{BMW} L.M. Markhashov, {\it J. Appl. Math. Mech.} {\bf 38}
(1974), 788; A.D. Bruno, {\it Sel. Math.} {\bf 12} (1993), 13;
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243
\bibitem{Gae} G. Gaeta, preprint
IHES/P/96/74 (available through {\tt http://www.ma.utexas.edu/mp\_arc}
as preprint {\tt 96-584})
\bibitem{SM} C.L. Siegel and J.K. Moser, ``Lectures on celestial
mechanics'', Springer, Berlin 1955 (reprinted in {\it Classics in
Mathematics}, 1995).
See also E. Forrest and D. Murray, {\it Physica D} {\bf 74} (1994), 181
\end{references}
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