\def\NF{normal form }
\def\DS{dynamical system }
\def\VF{vector field }
\def\NFs{normal forms }
\def\DSs{dynamical systems}
\def\VFs{vector fields }
\def\PD{Poincar\'e-Dulac }
\def\pa{\partial}
\def\C{{\cal C}}
\def\L{{\cal L}}
\def\R{{\bf R}}
\def\S{{\cal S}}
\def\phi{\varphi}
\def\s{\sigma}
\def\grad{\nabla}
\def\Ker{{\rm Ker}}
\def\Ran{{\rm Ran}}
\def\({\left(}
\def\){\right)}
\def\wt#1{{\widetilde #1}}
\def\titleb#1{{\bigskip \bigskip {\bf #1} \bigskip}}
\font\petit = cmr9
\magnification 1200
\newcount\notenumber
\def\clearnotenumber{\notenumber=0\relax}
\def\fnote#1{\advance\notenumber by 1
\footnote{$^{\the\notenumber}$}{\petit #1}}
\parindent=0pt
\parskip=10pt
{\nopagenumbers
~ \vskip 3 truecm
\centerline{\bf Normal forms and nonlinear symmetries}
%\footnote{}{9/10/93 -- version 1.2}
\vskip 2 truecm
\centerline{Giampaolo Cicogna}
\centerline{\it Dipartimento di Fisica, Universita' di Pisa}
\centerline{\it Piazza Torricelli 2, I-56126 Pisa (Italy)}
\centerline{\tt cicogna@ipifidpt.difi.unipi.it}
\bigskip
\centerline{Giuseppe Gaeta}
\centerline{\it Dipartimento di Fisica, Universita' di Roma}
\centerline{\it Piazzale A. Moro 2, I-00185 Roma (Italy)}
\centerline{\tt gaeta@roma1.infn.it}
\vskip 3 truecm
{\bf Abstract.}
We present some results concerning Poincar\'e-Dulac normal forms of
dynamical systems which are symmetric under (possibly nonlinear) Lie point
transformations. We show in particular that the vector fields defining the
dynamical system and the Lie symmetry can be put into a "joint normal
form".
\vfill \eject}
\pageno=1
\titleb{1. Introduction}
The theory of Poincar\'e-Dulac normal forms [1-4] provides a
classification of smooth dynamical systems (vector fields) in the
neighbourhood of a fixed point (zero), up to $\C^\infty$~-~equivalence
\fnote{Other kinds of normal forms do also exist,
corresponding to different kind of equivalence, e.g. Shoshitashvili
normal forms for $\C^0$-equivalence [1,2]. In this note, by normal
forms we will always mean the \PD ones.}.
If the dynamical systems - or \VFs - are moreover known to be
symmetric under a
{\it linear} transformation, this fact is reflected into their normal form,
and simplifies the \NF unfolding for equivariant setting [1,2,5,6].
In this note, we present some results concerning the case the \DS is
symmetric under a general - i.e. possibly {\it nonlinear} - Lie-point
transformation, i.e. a diffeomorphism. A more complete exposition of
the results given here and extensions thereof, are presented
elsewhere [7].
In section 2 below we fix notation and recall the Poincar\'e-Dulac
procedure for transforming a \DS or a \VF into its normal form, and in
section 3 we quickly reproduce geometrically some known results for the
case of linear symmetries. In section 4 we deal with nonlinear
diffeomorphisms, and obtain our main result as the final theorem ~3.
\titleb{2. Normal forms}
By a (smooth) dynamical system (DS) we will mean a system of first
order autonomous ODEs in\fnote{Since we are going to deal with a
local problem, considering a general smooth manifold $M \subseteq
\R^n$ would not add any generality to our considerations.} $\R^n$,
$$ {\dot x} = f (x) ~~~~;~~~ x \in M \equiv \R^n ~,~ f: M \to TM
\eqno(1) $$
where $f$ is a smooth ($\C^\infty$) function; equivalently, a DS is
identified by a (smooth) vector field (VF) on $M \equiv \R^n$, i.e.
for (1)
$$ \phi = f(x) \pa_x \equiv f^i (x) {\pa \over \pa x^i}
\eqno(2) $$
We are in particular interested in the case (1) admits a fixed point
$x_0$, that can be taken to be the origin of $\R^n$; i.e. we assume
from now on
$$ f(0) = 0 \eqno(3) $$
We want then to study, by means of formal perturbative expansions,
the flow of (1) in the vicinity of the fixed point.
We expand $f$ in a series of homogeneous terms, dropping that of
order zero due to (3)
$$ f(x) = Ax + \sum_{k=2}^\infty F_k (x) \eqno(4) $$
where $F_k $ is homogeneous of order $k$, and $A$ is an $n \times n$
(real) matrix\fnote{By means of linear transformations, $x \to Tx$,
$A \to TAT^{-1}$, we can classify the linear part of (1) in a
standard way; if the fixed point is hyperbolic [4,8,9] this provides a
topological ($\C^0$) classification of the flows, but even in this case
does not give a $\C^\infty$ classification. In the sequel we will
leave $A$ unchanged.}.
The \PD procedure shows that, by a sequel of nonlinear near- identity
formal changes of coordinates, the DS (1) can be reduced to a
$\C^\infty$ equivalent DS, its {\it normal form} (NF),
$$ {\dot x} = g(x) = Ax + \sum_{k=2}^\infty G_k (x) \eqno(5) $$
where the $G_k$ are all {\it resonant} with $A$ [1-4].
The changes of coordinates involved are of the form
$$ x \to \wt{x} = x + h_k (x) ~~~~~~ k \ge 2 \eqno(6) $$
and $h_k : \R^n \to \R^n$ is homogeneous of degree $k$. Under such a
change of coordinates, the terms $F_k$ of (4) will be changed to
$\wt{F_k}$, and it is straightforward to see that $\wt{F_m} = F_m$
for $m < k$, and
$$ \wt{F_k} = F_k - \L_A ( h_k ) \eqno(7) $$
where $\L_A$ is the {\it homological operator} associated with $A$,
given in terms of Poisson brackets by
$$ \L_A (f) = \{ Ax , f(x) \} \equiv \big( Ax \cdot \pa_x \big)
f(x) - \big(f(x) \cdot \pa_x \big) Ax \eqno(8) $$
Therefore, we can proceed sequentially and simplify $F_k$ as much as
possible by opportunely choosing $h_k$: the changes of coordinates
(6) for $k' > k$ will not affect the terms $F_k$ which have already
been simplified. This gives a procedure to {\it normalize} $f$ up to
any desired order (we will take this to be formally infinite).
If $\pi$ is the projection on the range\fnote{The considerations
involving kernel and ranges, and projections to these, of homological
operators can be made rigorous by considering spaces of homogeneous
polynomial \VFs [7].} of $\L_A$, the "opportune" choice of $h_k$
mentioned above corresponds to solving the {\it homological equation}
$$ \L_A (h_k ) = \pi ( F_k ) \eqno(9) $$ after which we will be left
with, cf. (7), $$ \wt{F_k} = \( I - \pi \) F_k \eqno(10) $$ It should
be stressed that the $G_k$ {\it cannot} be obtained simply as $G_k =
(I - \pi ) F_k$, as the transformation (6) at order $k$ does also
change, in a very complicate way, the $F_m$ with $m>k$.
Let us now make the simplifying assumption - which will be taken for
granted in the following - that $A = (Df)(0)$ is a {\it normal} matrix,
i.e.
$$ [ A , A^+ ] = 0 \eqno(11) $$
This ensures [5] that $\L_A^+ = \L_{A^+}$ and, in particular, that
$\Ran (\L_A ) $ and $\Ker (\L_A )$ are complementary
subspaces\fnote{Indeed, if (11) holds, then $\Ker (\L_A^+ ) =
\Ker (\L_{A^+} )$.}, so that (10) reads $\wt F_k \in \Ker (\L_A )$. In
other words, the NF satisfies $g \in \Ker (\L_A )$ or, in geometrical
terms, $$ \{ Ax , g(x) \} = 0 \eqno(12) $$
which expresses in particular the property of the $G_k(x)$ of being
resonant with $A$.
%\vfill \eject
\titleb{3. Linear symmetries}
Let us now consider the case (1) admits a linear symmetry, i.e.
$$ f(Sx) = S f(x) \eqno(13) $$
with $S$ an $n \times n$ matrix. In geometrical
terms, this means that there is a VF $\s = (Sx) \pa_x$ which commutes
with the VF (2), i.e.
$$ \big[\s,\phi\big]\ = \ \{ Sx, f(x) \}\ \pa_x = 0 \eqno(14) $$
The geometrical relation (14) is independent of the choice of
coordinates, and must therefore also hold if $\phi = f(x) \pa_x$ is
expressed in NF, $\phi = g(x) \pa_x$. Since $\s$ contains only linear
terms, its expression is not affected by the normalizing
transformation (6). Therefore the NF will still admit the same
symmetry, i.e. we are granted to have
$$ g (Sx) = S g(x) \eqno(15) $$
Relation (14) can also be written as $\L_S (f) = 0$;
therefore the above discussion\fnote{We have recast in geometrical
terms known results, see [5] for the algebraic approach.} can be
summarized by
{\bf Proposition.} If $f(x)$ satisfies (3) and (13), it can be
reduced to a NF $g(x)$ such that $g \in [ \Ker (\L_A ) \cap \Ker
(\L_S ) ]$.
Notice also that (12) shows that the VF $\alpha=(Ax) \pa_x$ is a (linear)
symmetry\fnote{If (11) is not satisfied, a weaker result holds, see
[5,7].} of the NF.
\titleb{4. Nonlinear symmetries}
If (1) admits a general, i.e. possibly nonlinear, time-independent
Lie-point symmetry $\s$, this fact is still expressed by the
commutation relation (14) i.e. $[\s,\phi]=0$, but now the representation
of $\s$ will contain nonlinear terms, i.e. we will have
$$ \s = s(x) \pa_x ~~~~,~~~s(x) = Sx + \sum_{k=2}^\infty B_k (x)
\eqno(17) $$
where we assume that $S=(Ds)(0)$ is a normal matrix (cf. also [7]).
Now the $B_k$ will be changed by the normalizing transformation (6),(9)
according to the equivalent of (7), i.e. $\wt{B_m} = B_m $ for $m