\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 $m<k$ and 
$$ \wt{B_k} = B_k - \L_S (h_k) \eqno(18) $$
We will write the commutation property $[\phi,\s]=0$ in the 
original coordinates by expanding $\phi = f(x) \pa_x $ and 
$\s = s(x) \pa_x$ in homogeneous polynomial VF; it results
$$ \{ f(x) , s(x) \} = \sum_{k=0}^\infty \( C_k - u_k \) \eqno(19) $$
where $C_0=\{f_0,s_0\},\ u_0 = u_1 = 0$ and, with
\fnote{We resort to this notation in order to conform to the 
notation of [7].} 
$f_k = F_{k+1}$, $s_k = B_{k+1}$, $f_0 = Ax$, $s_0 = Sx$, 
$$ \eqalign{ C_k \equiv & \{ f_0 , s_k \} + \{ f_k , s_0 \} \cr
u_k \equiv & - \sum_{j=1}^{k-1} \{ f_j , s_{k-j} \} \cr } \eqno(20) $$
It is easy to check that (6) at order $k$ does leave $u_m$
invariant for $m \le k$ and $C_k$ invariant for $m<k$, acting on
$C_k$ by
$$ \wt{C_k} = C_k + \{ \{ s_0 , f_0 \} , h_k \} \eqno(21) $$
Notice that  (14)  necessarily implies
$$ \{ f_0 , s_0 \} = 0 {\rm  \quad or,\ which \ is\ the\ same,\quad } 
[A,S] = 0 \eqno(22) $$
so that  $C_k$ is also invariant, as it should. 

We have then the

{\bf Theorem 1.} Let $\phi$ be now expressed in NF as $\phi = f(x) \pa_x$,
with $f$ satisfying (3) and (11); then, any $\s$ commuting with $\phi$
is expressed in the same coordinates as $\s = s(x) \pa_x$ with $s
\in \Ker (\L_A )$, where $A = (Df) (0)$.

{\it Proof:} Indeed, (14) implies $C_k = u_k$ $\forall k$; for $k=0$
we get (22), and for $k=1$
$$ \{ f_0 , s_1 \} + \{ f_1 , s_0 \} = 0 $$
Apply $\L \equiv \L_A$ to both sides, and recall $\L = \{ f_0 , .
\}$. Using Jacobi identity and the fact that $\{ f_0 , f_k \} = 0$
since $f(x)$ is in NF, we get $\L ( \L (s_1 )) = 0$; if (11) is
satisfied, this implies $s_1 \in \Ker (\L ) $.
We can then proceed similarly and recursively for all $k$: indeed,
$\{ f_k , s_0 \} \in \Ker (\L )$, since both $f_k$ and $s_0$ commute
with $f_0$; if $s_j \in \Ker (\L )$ $\forall j < k$, then $u_k \in
\Ker (\L )$; using again (11), $C_k = u_k $ is then solved only if
$$ \L (s_k ) = 0 ~~~;~~~ \L_S (f_k ) \equiv \S (f_k ) = u_k \eqno(23)
$$
which proves the theorem. $\bullet \bullet$

Notice that in the above theorem there seems to be an asymmetry in the
roles of $\phi$ and $\s$, as both $f$ and $s$ are in $\Ker (\L )$, but
we are not granted that neither $f$ nor $s$ belong to $\Ker (\S )$.
It is indeed possible to restore a symmetry of roles between $\phi$
and $\s$, as we do now prove. 

{\bf Theorem 2.} Let $\phi = f(x) \pa_x$ be in NF, and let $\s = s(x)
\pa_x$ commute with $\phi$. Then it is possible to choose coordinates
such that the representation of $\phi$ is not changed and that $s \in
[ \Ker (\L ) \cap \Ker (\S ) ]$. Moreover, necessarily $f \in [ \Ker
(\L ) \cap \Ker (\S ) ]$.

{\it Proof:} The homological equation (9) selects $h_k$ only up to a
$\delta h_k \in \Ker (\L )$; by opportunely choosing this $\delta
h_k$, we can eliminate terms of $B_k$ in $\Ker(\L ) \cap \Ran(\S )$
(recall $S$ is normal), so that due to theorem 1 we are left with
terms $s \in [ \Ker (\L ) \cap \Ker (\S ) ]$ alone. 
To prove $f \in \Ker (\S )$, notice that if $f_j \in \Ker (\S )$
$\forall j < k $, then $\{ f_j , s_m \} \in \Ker (\S )$ by Jacobi
identity and $s_m \in \Ker (\S )$ (as we have just proved); but $f_0
\in \Ker (\S )$, so indeed $u_k \in \Ker (\S )$. Applying $\S$ to $(
C_k - u_k ) = 0 $ we get therefore $\S ( \S (f_k ) ) = 0$; since $S$
is normal, we have necessarily $f_k \in \Ker (\S )$. $\bullet \bullet$

The above theorem suggests to introduce the following

{\bf Definition.} Let the VF $\phi = f(x) \pa_x$ and $\s = s(x)
\pa_x$ have a common fixed point, $f(x_0 ) = s(x_0 ) = 0$, and let
their linearization at the fixed point be given by normal matrices
$ A = (Df) (x_0 )$ and $ S = (Ds) (x_0 )$. The VF are in {\bf joint
normal form} if and only if both $f$ and $s$ are in $[ \Ker (\L ) \cap
\Ker (\S ) ]$.

Our discussion and results will then be recast as the

{\bf Theorem 3.} Let the vector fields $\phi$ and $\s$ $i)$ have a
common fixed point $x_0$, $ii)$ have normal linearizations at $x_0$,
and $iii)$ do commute. Then they can be put in joint normal form by
means of Poincar\'e- Dulac transformations.

It should be stressed that the above theorem is also particularly
useful in the unfolding of equivariant NFs for problem with assigned
symmetry: in this case, indeed, the allowed terms for the DS in NF
are not all those in $\Ker (\L )$, as it would be expected on the
basis of standard NF theory, but only those in $[ \Ker (\L ) \cap \Ker
(\S ) ]$.

The problem of joint NF unfolding can be further simplified (i.e. the
general form of joint NF further restricted) by an extension of the
above theory, given in [7]. Roughly speaking, this consists in
considering not only $\L = \{ f_0 , . \}$ (and the like for $\s$),
but the sequence of operators $\L^{[k]} = \{ \sum_{m=0}^k f_m , .
\}$; this gives a filtration of the Lie algebra of vector fields in
$\R^n$. Further details, extensions, examples and a more complete 
discussion can be found in [7].

%\vskip 2 truecm

\vfill\eject

\titleb{References}

[1] V.I. Arnold, "Geometrical methods in the theory of differential
equations"; Springer, Berlin, 1982

[2] V.I. Arnold and Yu. S. Il'yashenko, "Ordynary differential
equations"; in {\it Encyclopaedia of Mathematical Sciences - vol.I
(Dynamical Systems I)}, D.V. Anosov and V.I. Arnold eds., pp. 1-148;
Springer, Berlin, 1988

[3] A.D. Bruno, "Local methods in nonlinear differential equations";
Springer, Berlin, 1989

[4] F. Verhulst, "Nonlinear differential equations and dynamical
systems"; Springer, Berlin, 1990

[5] C. Elphick, E. Tirapegui, M.E. Brachet, P. Coullet and G. Iooss;
{\it Physica D} {\bf 29} (1987), 95

[6] G.R. Belitsky, "Normal forms, invariants, and local mappings" (in
russian); Naukova Dumka, Kiev, 1979

[7] G. Cicogna and G. Gaeta, "Poincar\'e normal forms and Lie-point
symmetries"; preprint 1993

[8] V.I. Arnold, "Ordinary differential equations"; Springer, Berlin,
1993 (2nd ed.)

[9] M. Hirsch and S. Smale, "Differential equations, dynamical systems,
and linear algebra"; Academic Press, New York, 1978

\bye