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\vskip 3truecm }
\def\titleb#1{ \bigskip \bigskip {\tbfont {#1} } \bigskip}
\parindent=0pt
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\def\G{{\cal G}}
\def\L{{\cal L}}
\def\Lk{{\L_{(k)} }}
\def\S{{\cal S}}
\def\M{{\cal M}}
\def\C{{\cal C}}
\def\F{{\cal F}}
\def\phi{\varphi}
\def\eps{\varepsilon}
\def\d{\delta}
\def\D{\Delta}
\def\ga{\gamma}
\def\Ga{\Gamma}
\def\sig{\sigma}
\def\al{\alpha}
\def\la{\lambda}
\def\pa{\partial}
\def\x{\times}
\def\o+{\oplus}
\def\Ker{{\rm Ker }}
\def\Ran{{\rm Ran }}
\def\ad{{\rm ad}}
\def\Lra{\Longrightarrow}
\def\LLRA{\Longleftrightarrow}
\def\sse{\subseteq}
\def\ss{\subset}
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\def\Poi{Poincar\'e }
\def\~#1{{\widetilde #1}}
\def\^#1{{\widehat #1}}
\def\dsum{{ \sum_{k=0}^\infty }^\o+ }
{\nopagenumbers
\titlea{Poincar\'e normal forms and Lie-point symmetries}
\centerline{Giampaolo Cicogna}
\centerline{\it Dipartimento di Fisica, Universit\`a di Pisa}
\centerline{\it P.zza Torricelli 2, I-56126 Pisa (Italy)}
\centerline{cicogna@ipifidpt.difi.unipi.it}
\vskip 1 truecm
\centerline{Giuseppe Gaeta}
\centerline{\it C.P.Th., Ecole Polytechnique}
\centerline{\it F-91128 Palaiseau (France)}
\centerline{gaeta@orphee.polytechnique.fr}
\vskip 2 truecm
\titleb{Abstract}
{\sl We study \Poi normal forms of vector fields in the presence of
symmetry under general - i.e. not necessarily linear - diffeomorphisms. We
show that it is possible to reduce both the vector field and the symmetry
diffeomorphism to normal form by mean of an algorithmic procedure similar
to the usual one for \Poi normal forms without symmetry; this double
normal form can be given a simple geometric characterization.}
\vfill\eject}
\pageno=1
\titleb{Introduction}
The Poincar\'e-Dulac theory [1-3] of analytic normal forms (NF) of
an analytic Ordinary Differential Equation \big(equivalently, vector field
(VF)\big) in the
vicinity of an isolated fixed point is a venerable topic, but also a
powerful tool in the study of dynamical systems.
Here we limit ourselves to the (properly speaking, Poincar\'e) case
in which the linear operator $A$, giving the linearization of the ODE at
the fixed point (see below), commutes with its adjoint, i.e.
$[ A , A^+ ] = 0$; in other words, we treat the case $A$ does not
contain Jordan blocks, or still the algebraic and geometric
multeplicities of its eigenvalues are equal (we denote this condition
as "Assumption A").
Some of the results we will obtain remain valid (in some cases,
if suitably modified) also if this assumption is not verified, as we
shall occasionally indicate: the extension goes along the lines of the
extension of the \Poi to the Poincar\'e-Dulac theory, see e.g. [1].
In the case of generic \Poi NF, these are nicely characterized [2,4,5]
by the fact that the linear part of, and the full evolution operator, do
commute; in other words, the resonant vectors are those which commute
with the linear operator $A$. In the case of linearly equivariant \Poi
NF, i.e. if the equations admit a linear symmetry, the general form of
the \Poi NF unfolding is restricted by another commutation relation,
i.e. only the equivariant resonant terms can appear [4].
Here we generalize this result to the case of nonlinear symmetries.
We also obtain some result concerning the properties of nonlinear
symmetries admitted by dynamical systems in NF, and the connections
existing between these symmetries and the \Poi procedure for
transforming the system into NF.
We would like to warmly thank prof. H. Duistermaat for convincing us
normal forms are not only useful, but also a very nice subject in
itself, and for introducing us to the particularly convenient
geometrical approach to NF we use here.
\titleb{1. Geometrical setting and notation}
Let us consider the space $M \sse \R^N$, and let $\M $ be
the space of analytical vector fields in $M$.
Elements of $\M$ are in correspondence with elements of $V$, the
space of analytical functions $f:M \to \R^N$ such that $f(x) \in T_x
M$; with $x \in M$ we write in component expansion (we will use greek
letters for elements of $\M$, latin ones for elements of $V$)
$$ \phi = f(x) \pa_x \equiv f^i (x) {\pa \over \pa x^i} \eqno(1.1) $$
Let us define $V_k \ss V$ as the space of homogeneous polynomial
functions of order $k$ in $V$, and let $\M_k$ be the corresponding
subset of $\M$.
In the set $\M$ is naturally defined a bilinear antisymmetric
operation $[.,.]$, the Lie commutator of vector fields, with which
$\M$ becomes a Lie algebra. This induces
a corresponding Lie-Poisson bracket $\{ .,. \} : V \x V \to V$.
Indeed, if $\phi = f(x) \pa_x$, $\psi = g(x) \pa_x$, then
$$ [\phi , \psi ] = \{ f,g \} \pa_x \eqno(1.2) $$
$$ \{ f , g \}^i = f^j \pa_j g^i - g^j \pa_j f^i \eqno(1.3) $$
By means of these we define the (linear) {\it adjoint action} of
$\phi \in \M$ on $\M$ itself, respectively of $f \in V$ on $V$, by
$$ \eqalign{ \ad_\phi (.) =& [ \phi , . ] \equiv L_\phi (.) \cr
\ad_f (.) =& \{ f , . \} \equiv L_f (.) \cr } \eqno(1.4) $$
It is clear that $\M$, $V$ can be decomposed as
$$ \M = { \sum_{k=0}^\infty }^\o+ \M_k ~~;~~ V = { \sum_{k=0}^\infty
}^\o+ V_k \eqno(1.5) $$
and it is equally clear that
$$ \eqalign{
\phi \in \M_m \Lra & L_\phi : \M_k \to \M_{k+m-1} \cr
f \in V_m \Lra & L_f : V_k \to V_{k+m-1} \cr } \eqno(1.6) $$
In particular, for $m=1$ this shows that the decomposition (1.5) is
a decomposition in invariant spaces under $\ad_\phi $, $\ad_f$ for
$\phi \in \M_1$, $f \in V_1$.
Let us now consider the flow induced in $M$ by the vector field
$\phi$ given by (1.1); this is described by the equation
$$ {\dot x} = f(x) ~= \phi \cdot x ~~~;~~~x \in M ~,~ f: M \to TM
\eqno(1.7) $$
A VF $\sig \in \M$ will be called a (time independent) Lie-point (LP)
symmetry of $\phi$ if and only if the flows of $\sig$ and $\phi$
commute; that is,
$$ \eqalign{
[ \sig , \phi ] = 0 ~~;~~ & L_\phi (\sig ) = 0 = L_\sig (\phi ) \cr
\{ f , s \} = 0 ~~;~~ & L_f (s ) = 0 = L_s (f) \cr } \eqno(1.8) $$
where the second line is in component notation and we write
$\sig = s(x) \pa_x $, here and in the following.
The Lie algebra of LP symmetries of $\phi$ (respectively of $f$) will
be denoted by $\G_\phi \sse \M$ (respectively $\G_f \sse V$); notice
that
$$ \G_\phi = \Ker ( \ad_\phi ) ~~;~~\G_g = \Ker ( \ad_f ) \eqno(1.9) $$
$$ \sig \in \G_\phi \LLRA \phi \in \G_\sig ~~;~~ s \in \G_f \LLRA f \in
\G_s \eqno(1.10) $$
{\bf Remark 1.} Notice that if $x = x_0$ is an isolated fixed point for
$\phi$ (an isolated zero for $f$), then it must also be a fixed
point for $\sig$ (a zero for $s$), on the account of (1.8). From now
on we will assume this to be the case, and set $x_0 = 0$.
It is therefore natural to consider the linearization of (1.7) at $x =
x_0$; this is given by
$$ {\dot x} = A x = f_0 (x) ~~~~~~~~~~~A = (Df)(x_0 ) \eqno(1.7') $$
The linear operator $A$ will play a central role in the following;
we will make a fundamental assumption on it to simplify our work:
{\bf Assumption A.} {\it The linear operator $A = (Df)(x_0)$ commutes
with its adjoint. }
In order to avoid unnecessary duplication of equations, from now on
we will use only the setting in $V$, and leave to the reader the
translation of our statements to the setting in $\M$.
Let us now expand $f,s$ in terms of the decomposition (1.5); we write
$$ \eqalign{
f(x) = \sum_{k=0}^\infty f_k (x) ~~~;&~~~~ f_k \in V_{k+1} \cr
s(x) = \sum_{k=0}^\infty s_k (x) ~~~;&~~~~ s_k \in V_{k+1} \cr }
\eqno(1.11) $$
so that by (1.6)
$$ \ad_{f_k} : V_m \to V_{m+k} ~~;~~ \ad_{f_k} : V_m \to V_{m+k}
\eqno(1.12) $$
We will consider in particular the linear operator
$$\ad_{f_0} \equiv L_{f_0} \equiv \L \eqno(1.13)$$
which is now decomposed as
$$ \L = \dsum \Lk ~~;~~ \Lk : V_k \to V_k ~~;~~ \Lk = \L
\vert_{V_k} \eqno(1.14) $$
so that in particular
$$ \Ker ( \L ) = \dsum \Ker ( \Lk ) ~~;~~\Ker (\Lk ) = \Ker (\L )
\cap V_k \eqno(1.14') $$
{\bf Definition 1.} A function $w \in V_k$ is called a {\it k-resonant
vector} if and only if $w \in \Ker ( \Lk ) \sse V_k$.
{\bf Remark 2.} Written explicitly, the condition $w\in \Ker(\L)$
becomes
$$0 = \{Ax,w\}^i = A^{jk}x_k\pa_jw^i-A^{ij}w_j=
(Ax)^j\pa_jw^i-(Aw)^i\equiv D_Aw^i $$
where $D_A\equiv(Ax)\cdot\pa-A$ is the well known homological operator
associated to $A$. If $A$ is diagonalized (thanks to Assumption A) with
eigenvalues $\al_1,\ldots,\al_n$, and $w^i$ is a monomial
$$w^i=x_1^{m_1}x_2^{m_2}\cdot\ldots\cdot x_n^{m_n}$$
the above condition acquires the familiar form [1]
$D_Aw^i=m_j\al_j-\al_i=0$\ .
Notice that under Assumption A, one has \footnote{$^1$}{These
decompositions can be easily verified on the basis of
the monomials $w^i$ introduced in Remark 2; one could
also introduce a scalar product in the space $V$, see [4], but this
is not necessary for our present purposes.}
$$ \eqalign{
V =& \Ker ( \L ) \o+ \Ran ( \L ) \cr
V_k =& \Ker ( \Lk ) \o+ \Ran ( \Lk ) \cr } \eqno(1.15) $$
Let us now consider $h \in V$, and $f,s \in V$ such that (1.8) is
satisfied; by the Jacobi identity, we then have
$$ \{ s , \{ f,h \} \} = \{ f , \{ s,h \} \} \eqno(1.16) $$
which also reads
$$ L_s \cdot L_f = L_f \cdot L_s \eqno(1.16') $$
so that (1.8) implies in particular
$$ \ad_s : \Ker ( \ad_f ) \to \Ker ( \ad_f ) \eqno(1.17) $$
Notice also that
$$ \{ f , s \} = 0 \Lra \{ f_0 , s_0 \} = 0 \eqno(1.18) $$
so that when (1.8) is satisfied,
$$ \S \equiv \ad_{s_0} : \Ker (\Lk ) \to \Ker ( \Lk ) \eqno(1.19) $$
In terms of the expansion (1.11), the condition (1.8) reads
$$ \sum_{j=0}^\infty \{ f_j , s_{k-j} \} = 0 ~~~~ \forall k \ge 0
\eqno(1.20) $$
\vfill \eject
\titleb{2. Poincar\'e normal forms}
In the (Poincar\'e) theory of normal forms, one considers
dynamical systems of the form (1.7) and proves that, if Assumption A
is satisfied, by means of formal changes of coordinates they can be
taken to the form
$$ {\dot x} = g(x) ~~= \sum_{k=0}^\infty g_k (x) \eqno(2.1) $$
where $g_k \in V_{k+1}$ and
$$ g_0 = f_0 ~~;~~g_k \in \Ker (\L_{(k+1)} ) ~~~~~ k < k^*
\eqno(2.2) $$
for $k^*$ arbitrarily large.
{\bf Remark 3.} Notice that $f_0 \in \Ker (\L_{(1)} )$ by definition.
{\bf Remark 4.} If Assumption A is not satisfied, $\L$ should be
substituted by its adjoint, and the above Remark would fail; this is
actually the main reason to consider Assumption A.
We will take formally $k^* = \infty$, so that the \Poi- Dulac
theorem will read
{\bf Theorem 1.} (Poincar\'e - Dulac) {\it By means of formal changes
of coordinates, it is possible to take the system {\rm (1.7)} to the
form {\rm (2.1)}, where $g \in \Ker ( \L ) = \G_{f_0}$.}
In this sense, the \Poi- Dulac procedures makes explicit the
symmetry of the dynamical system.
{\bf Remark 5.} Since $g_0 = f_0$, the operator $\L$ is well defined
and independent of the form, (2.1) or (1.7), of the system.
{\bf Remark 6.} If the system (2.1) satisfies (2.2) with $k^* = n$, we
say that it is in Poincar\'e normal form up to order $n$; when taking
the formal limit $n \to \infty$, we speak of Poincar\'e normal form,
tout court.
The changes of coordinates needed to transform the system to NF are
of the form
$$ x = y + h_k (y ) + R_{k+1} (y) \eqno(2.3) $$
where $h_k \in V_{k+1}$ and $R_k \in {\sum^\o+}_{m=k+2}^\infty V_m$;
notice that this can be seen as corresponding to the (time-one) flow
under the VF $\chi = h_k (x) \pa_x$.
Under (2.3), the system
$$ {\dot x} = \sum_m f_m (x) ~~;~~ f_m \in V_{m+1} \eqno(2.4) $$
is changed into
$$ {\dot y} = \sum_m \~f_m (y) ~~;~~ \~f_m = f_m ~~~~ m < k
\eqno(2.4') $$
where
$$ \~f_k = f_k - \{ f_0 , h_k \} \equiv f_k - \L (h_k ) \eqno(2.5) $$
so that if $\pi$ is the projection $\pi : V \to \Ran (\L )$, $\pi
f_k$ can be eliminated by judicious choice of $h_k$. The appropriate
$h_k$ for this, can be determined by solving the equation (also called
"homological equation")
$$ \L h_k = \pi f_k \eqno(2.6) $$
{\bf Remark 7.} Notice that $h_k$ is only defined up to elements of
$\Ker (\L_{(k+1)})$; in other words, the changes of coordinates
determined by $h_k$ and by
$$ {h'}_k = h_k + \d h_k ~~;~~ \d h_k \in \Ker ( \L_{(k+1)} )\eqno(2.7) $$
lead to the same $\~f_k$.
By this Remark and by (1.15) we can, under Assumption A, decide to
choose
$$ h_k \in \Ran ( \L_{(k+1)} ) \eqno(2.8) $$
Once $h_k$ has been fixed, any $s(x) = \sum_m s_m (x)$ will be
changed according to the same (2.4),(2.5) above; i.e.
$$ \~s_m = s_m ~~~~ m < k ~~~;~~ \~s_k = s_k - \{ s_0 , h_k \}
\eqno(2.9)$$
It is maybe worth stressing that geometrical objects, such as $\phi ,
\sig \in \M$, are not changed by (2.3), which affects only their
coordinate representation. In particular, $\G_\phi$ remains
unchanged, so that since $\phi = f(x) \pa_x = g(y) \pa_y$ and $f_0
\in \G_g$, then there must be a VF $\sig = f_0 (y) \pa_y \in
\G_\phi$, which will be represented as $\sig = s(x) \pa_x$ in the $x$
coordinates; if $g(y) \not= f_0 (y) \equiv g_0 (y)$, then $\sig \not=
\phi$, and the VF $\phi$ has at least a nontrivial symmetry.
Given a linear VF $\phi_0$, one can ask to classify (the local flow
of) all the VF which admit $\phi_0$ as linear part; in terms of the
dynamical system (1.7), this amount to classify (the local behaviour
of solutions of) all the systems $f$ which have the same
linearization $(Df)(x_0) = A$ at the fixed point $x_0$, $f_0 (x) = Ax$.
The problem of classifying all the $f$ as above up to formal analytic
transformations, reduces to the problem of classifying the most
general $f(x)$ with linear part $f_0 (x)$, upon reduction to
\Poi NF.
If the above classification is meant up to equivalence by formal
analytic transformations, the Poincar\'e NF is a convenient tool; it
should be stressed that if one is satisfied with a classification up
to transformation in a different class, e.g. up to topological or
$C^k$ equivalence, this would lead to different kind of NF and
NF reduction [1]. In the present paper, by NF we will always mean the
Poincar\'e NF.
\titleb{3. Symmetries and normal forms}
We want now to consider the relations between symmetry properties of
eq.(1.7) and its (reduction to) NF (2.1). The symmetry properties of
systems in NF have already been considered by some authors, see e.g.
[4,6]; in particular Elphick et al.\footnote{$^2$}
{In [2], p.67, this theorem is quoted from [5]; unfortunately this book is
not
available (to our knowledge) in the western literature.} [4] have
characterized the NF by means of the commutation properties between
the full VF describing time evolution and its linear part at the fixed
point $x_0$. Indeed, with Assumption A we have from [4]:
{{\bf Theorem 2.}}
{\it
Let the VF $\Phi \in \M$ be written in the $x$ coordinates as $\Phi =
f(x) \pa_x = \Phi_0 + \Phi_1$, where $\Phi_0 = f_0 (x) \pa_x$, $\Phi_1
(x) = [f(x) - f_0 (x)] \pa_x$, and $f_m \in V_{m+1}$. Let Assumption
{\rm A} be satisfied. Then $\Phi$ is in Poincar\'e NF if and only if
$[\Phi , \Phi_0 ] = 0$}
{\bf Corollary 1.} {\it $\Phi$ is in Poincar\'e NF if and only if
the following conditions, all equivalent, are verified: }
\parskip 0pt
$i)$ $\{ f,f_0 \} =0$; $ii)$
$\Phi \in \Ker(\ad_{\Phi_0}) = \G_{\Phi_0}$;
$iii)$ $ \Phi_0 \in \Ker(\ad_{\Phi})=\G_\Phi$.
\parskip 10 pt
The proof can easily be obtained from the discussion of sects.1 and
2; indeed, in the present notation this amounts to a corollary of
the \Poi- Dulac theorem as given in sect.2.
{\bf Remark 8.} From the point of view of symmetry
properties, an interesting result comes from $iii)$ of the Corollary
above, which states that the linear part $f_0(x)=Ax$ determines a linear
LP symmetry $\Phi_0=Ax\pa_x$ for the full problem $\dot x=f(x)$ [4,6].
{\bf Remark 9.} We notice that if Assumption A is not satisfied, the
above theorem would be stated with the commutator condition $[ \Phi_1 ,
\Phi_0^+ ] = 0$ (and correspondingly modified conditions in the
corollary). In this case, the statement of Remark 8, would be
substituted by the weaker result that the linear operator
$\Phi_0^+=A^+x\pa_x$ is a linear symmetry for the nonlinear part $\dot
x=\Phi_1\cdot x$ (and not for the full problem) .
We want to consider here the symmetries of the original system (1.7),
and how these are reflected into the NF coordinates. The motivation
for this comes from the following obvious but interesting fact (see
later discussion).
Let $\phi , \sig \in \M$, expressed in two systems of coordinates
$x$ and $y$ in $M$ as
$$ \eqalign{
\phi = f(x) \pa_x =& \~f (y) \pa_y \equiv g(y) \pa_y \cr
\sig = s(x) \pa_x =& \~s (y) \pa_y \equiv t(y) \pa_y \cr } \eqno(3.1) $$
The relation $[ \sig , \phi ] = 0$, equivalent to $\phi \in \G_\sig$,
$\sig \in \G_\phi$, is independent of the coordinate choice, so that
$$ \{ f,s \} = 0 \LLRA \{ \~f , \~s \} = 0 \eqno(3.2) $$
Therefore, the presence of a symmetry for eq.(1.7) will pose some
restriction to the NF (2.1): while in general the NF satisfies only
$$ \~f \in \Ker ( \ad_{f_0} ) \equiv \Ker ( \L ) \eqno(3.3) $$
in the presence of the symmetry we will also have from (3.2) that
$$ \~f \in \Ker ( \ad_{\~s } ) \eqno(3.4) $$
The combination of (3.3) and (3.4) can lead to a
simplification of the NF unfolding; see sections 5,6.
It could be worth checking explicitely (3.2) in the following way: let
us rewrite (1.20) at order $k$ as
$$\eqalign {\C_k=&\ u_k \qquad (k=0,1,2,\ldots) \qquad {\rm where} \cr
\C_k\equiv & \{ f_0 , s_k \} + \{ f_k , s_0 \} \quad {\rm and}
\quad
u_k\equiv -\sum_{j=1}^{k-1} \{ f_j , s_{k-j} \}
\quad , \quad {\rm with} \quad u_0=u_1=0 }\eqno (3.5)$$
and let us consider a transformation (2.3): under this
$$ \C_k \to \~\C_k = \C_k - \{ f_0 , \{ s_0 , h_k \} \} - \{ \{ f_0 ,
h_k \} , s_0 \} $$
or, using Jacobi identity,
$$ \~\C_k = \C_k + \{ \{ s_0 , f_0 \} , h_k \} \eqno(3.6) $$
so that $\{ s_0 , f_0 \} = 0 \Lra \~\C_k = \C_k$; the r.h.s. of (3.5)
remains unchanged in the change of coordinates, since it contains
only terms of degree smaller than $k$, and so the whole equation (3.5)
is invariant.
{\bf Remark 10.} Notice that if $S$ is a linear operator such that
$[A,S]=0$ and if $A$ satisfies Assumption A, then also $[A,S^+]=0$;
which implies that $S+S^+$ and $S-S^+$ commute with $A$. Then, we can
assume that the linear part $S\equiv (Ds)(x_0)$ of the symmetry $\sig$
satisfies Assumption A. In the following we will use freely the fact
that both $S$ and $A$ satisfy Assumption A, and denote $\ad_{s_0}$ by
$\S$. It should be stressed that if Assumption A is not verified, the
results stated in Theorem 3 below fail to be true, in general.
We will find it useful to have the following Lemma, which easily follows
from Jacobi identity.
{\bf Lemma 1.} {\it Let $v, w\in V$ such that $v,w\in \Ker(\L)$. Then }
$\{v,w\}\in\Ker(\L).$
We also have the following
{\bf Theorem 3.}
{\it Let $\Phi \in \M$ be expressed in Poincar\'e NF as $\Phi = f(x)
\pa_x$; then any $\sig$ such that $[\sig , \Phi ] =0$ is expressed in
the $x$ coordinates as $\sig = s(x) \pa_x$ where $s \in \Ker (\L )$.
In other words, all LP symmetries $\sig$ of a dynamical system in NF,
which are obtained as formal series expansion,
are also LP symmetries of the linearized system} $\dot x=f_0(x)=Ax$.
{\it Proof.} We can proceed recursively using the set of equations
(3.5). For $k=0$, $\{f_0,s_0\}=0$ is satisfied, see (1.18); for $k=1$ we
have $$\{f_0,s_1\}+\{f_1,s_0\}=0\eqno(3.7)$$
Applying $\L$ to this equation, we obtain
$$\L(\L(s_1))\equiv \{f_0,\{f_0,s_1\}\}=0\eqno(3.8)$$
being $\{f_0,\{f_1,s_0\}\}=0$. Using (1.15) one has
$$\L(s_1)=0 \qquad {\rm or} \qquad s_1\in\Ker(\L)\eqno(3.9)$$
The argument can be repeated recursively for each $k$: indeed,
$\{f_k,s_0\}\in\Ker(\L)$ for Lemma 1, and similarly
if $s_j\in\Ker(\L) ,\ \forall j