\font\tafont = cmbx10
\font\tbfont = cmbx9
\magnification =1200
\font\sixrm = cmr6
\font\pet = cmr10
\def \pn{\par\noindent}
\def\titlea#1#2{{~ \vskip 2 truecm
\tafont {\centerline {#1}} %\medskip
{\centerline{#2}} }
%\vskip 1truecm
}
\def\titleb#1{\pn \bigskip \bigskip\parindent=0pt {\tbfont {#1} } \bigskip
\parindent=16pt}
\def\Ref{\pn{\bf References} \parskip=3 pt\parindent=0pt}
\hfuzz=2pt
\tolerance=500
\abovedisplayskip=2 mm plus6pt minus 4pt
\belowdisplayskip=2 mm plus6pt minus 4pt
\abovedisplayshortskip=0mm plus6pt minus 2pt
\belowdisplayshortskip=2 mm plus4pt minus 4pt
\predisplaypenalty=0
\clubpenalty=10000
\widowpenalty=10000
%\parindent=14pt
%\parskip=0pt
\def \pd{\partial}
\def \ka{\kappa}
\def \ep{\varepsilon}
\def \phi{\varphi}
\def \s{\sigma}
\def \z{\zeta}
\def \th{\vartheta}
\def \G{{\cal G}}
\def \F{{\cal F}}
\def \A{{\cal A}}
\def \B{{\cal B}}
\def \H{{\cal H}}
\def \a{\alpha}
\def \b{\beta}
\def\C{{\cal C}}
\def\L{{\cal L}}
\def\R{{\bf R}}
\def\S{{\cal S}}
\def\V{{\cal V}}
\def\Ker{{\rm Ker}}
\def\Ran{{\rm Ran}}
\def\({\left(}
\def\){\right)}
\def\wt#1{{\widetilde #1}}
\def\=#1{\bar #1}
\def\~#1{\widetilde #1}
\def\.#1{\dot #1}
\def\^#1{\widehat #1}
\def\"#1{\ddot #1}
\baselineskip 0.57 cm
{\nopagenumbers
%\pageno=1
\titlea{Normal forms and nonlinear symmetries}
{ }
\bigskip \bigskip
\centerline{Giampaolo Cicogna}
\centerline{\it Dipartimento di Fisica, Universit\`a di Pisa}
\centerline{\it Piazza Torricelli 2, I-56126 Pisa (Italy)}
\centerline{{\tt E-Mail: cicogna@ipifidpt.difi.unipi.it}}
\bigskip
\centerline{Giuseppe Gaeta}
\centerline{\it Centre de Physique Th\'eorique, Ecole Polytechnique}
\centerline{\it F - 91128 Palaiseau (France)}
\centerline {{\tt E-Mail: gaeta@orphee.polytechnique.fr}}
\vskip 3 truecm
\pn
%{\tt PACS n. 03 20, 02 20 }
%\vfill\eject
\titleb{Abstract}
\pn
We give some general theorems, and extensions of previous results,
concerning the problem of transforming an algebra of vector fields
into Poincar\'e normal form. By means of an unifying algebraic language,
we show the possibility of obtaining either a "parallel" or "joint"
normal form of the vector fields in a well definite way, which simplifies
the construction of normal forms, providing a precise restriction on
their structure. The application to the
finite dimensional dynamical systems and to their Lie point symmetries is
also discussed.
\vfill\eject }
\pageno =1
\titleb{1. Introduction and notations}
\pn
The problem of transforming a vector field (or an algebra of vector
fields) into normal form (in the sense of Poincar\'e - Dulac - Birkhoff)
is an old and important topic [1-9], not only for its algebraic aspects,
but also for its applications in the theory of dynamical systems,
especially in connection with symmetry properties [1-11].
Quite different approaches and points of view ("algebraic", "analytical",
or "dynamical") for this problem can be found in the literature, and it is
not rare that the differences in the language make it difficult even the
comparison of (apparently unrelated but strongly connected) results.
In this paper we want to propose some general results and extensions
of previous statements, in a (essentially) self-contained
presentation, using a geometrical approach similar to that in [11],
with an abstract and
"unifying" algebraic language, but avoiding as far as possible any
technicality (Sect. 2). The applications to dynamical systems and their
symmetries (Lie point symmetries) are discussed in Sect. 3.
Let us recall some basic definitions and fix some notations. Let $u\in
M\subseteq R^n$, where $M$ is a smooth neighbourhood of the origin in $R^n$,
and consider the space $\V$ of analytical vector fields
(VF) $\phi:M\to TM$ in $R^n$: they are in one-to-one correspondence with the
elements of the space $V$ of analytical functions $f:M\to R^n$;
in component expansion we shall write (here and in the following,
sum over repeated indices, unless otherwise stated)
$$\phi\equiv f(u)\pd_u\equiv f_i(u){\pd\over{\pd u_i}}\qquad (i=1,
\ldots,n) \eqno(1.1)$$
We assume that $u=0$ is an isolated fixed point for $f(u)$; $f(u)$ will
be also written as a series expansion in the form:
$$f\equiv Au+\~f\equiv\sum^\infty_{j=1}f_{(j)}\eqno(1.2)$$
where $Au\equiv f_{(1)}$ is the linear part of $f$, $\~f$ the
nonlinear part, and $f_{(j)}\in V_{(j)}$, the subspace of the
homogeneous polynomial functions in $V$ of degree $j$.
Given two VFs $\phi=f\pd_u$ and $\psi=g\pd_u$ in $\V$, the notion of Lie
commutator $[\phi,\psi]$ in $\V$ induces a Lie-Poisson bracket $\{f,g\}$
in $V$:
$$[\phi,\psi]=\{f,g\}\pd_u \qquad \{f,g\}_k=f_i\pd_ig_k-g_i\pd_ig_k
\qquad\Big(\pd_i\equiv{\pd\over{\pd u_i}}\Big) \eqno(1.3)$$
Also, a notion of scalar product can be introduced
in each subspace $V_{(j)}$ [8,11].
Given a $n\times n$ matrix $A$, we denote by $\A:V\to V$ the homological
operator associated to $A$ (which is also the Lie derivative $\L_A$
along the VF $Au\pd_u$):
$$\A(h_k)=(A u)_i\pd_ih_k-(Ah)_k\eqno(1.4)$$
where $h=h(u)\in V$.
According to the classical Poincar\'e - Dulac - Birkhoff definition
[1], a (nonlinear) term $h(u)$ is said to be {\it resonant with} $A$ if
(see also [8])
$$\A^+(h)\equiv\{A^+u,h\}=0\ ,\eqno(1.5)$$
and a VF $\phi=(Au+\~f)\pd_u$ is said to be in normal form (NF) if all
nonlinear terms are resonant with $A$, i.e. $\~f\in\Ker\A^+$. If $A$ is
diagonal, with eigenvalues $\a_1,\ldots,\a_n$, a monomial
$h_k(u)=u_1^{m_1}\cdot \dots\cdot u_n^{m_n}$ of degree $j$ (with $m_i$
integer numbers such that $\sum_i m_i=j, \ m_i\ge 0$)
is resonant if $m_i\a_i=\a_k$, which
is the usual "resonance condition" for the eigenvalues [1]. As well
known, the relevance of the above definitions is essentially due to the
fact that, given a VF $\phi$, all nonresonant terms can be removed by
means of a coordinate transformation. As usual in NF theory, these
transformations are expressed by means of {\it formal} series, i.e. no
assumption is made on their convergence (cf. [1]).
Notice that for both operators $\A$ and $\A^+$ one has, for each $j$,
$$\A:V_{(j)}\to V_{(j)} \quad {\rm and}\quad \A^+:V_{(j)}\to V_{(j)}
\ . \eqno(1.6)$$
If $\A=\L_A$ and $\B=\L_B$ are the homological operators associated to two
$n\times n$ matrices $A$ and $B$, the operator $\A\B-\B\A$ is just the
homological operator $\C=\L_C$ associated to the matrix commutator $C=[A,B]$.
In particular, the three statements $\A\B=\B\A \ $, $[A,B]=0$, and
$\{Au,Bu\}=0$ are equivalent.
Finally, if $A$ is any $n\times n$ matrix, we shall denote by
$$A=A_s+A_n\eqno(1.7)$$
its (unique) decomposition into commuting semisimple (diagonalizable)
and nilpotent part.
\vfill\eject
\titleb{2. The algebraic approach.}
\pn
The main results will be obtained as a consequence of a series of simple
lemmas, which can be of some independent interest: even if some of
these are not new, it is convenient to give
a complete list of all of them, together with a sketch of their proof.
\medskip\pn
{\sl Lemma 1}. If $[A,B]=0$ then $[A_s,B]=[A,B_s]=0\ .$
\pn
{\sl Proof}. This easily follows once the matrix $A$ (or resp. $B$) is
written in its Jordan form.
\medskip\pn
{\sl Lemma 2}. Given the matrix $A$, the set $\Ker \A^+$ of
terms $h(u)$ resonant with $A$, is given by $K(\kappa(u))u$, where $K$ is
the most general matrix such that $[K,A^+]=0$ and its entries $K_{ij}$
are functions of the time independent analytical constants
of motion $\kappa=\kappa(u)$ of the linear system $\.u=A^+u$ [8,11].
\pn
{\sl Proof}. This follows writing explicitly the equation $\A^+ (h)=0$ as a
first order PDE, and applying standard procedures [12].
\medskip\pn
{\sl Lemma 3}. $\Ker \A\subset \Ker \A_s$ ; $\Ker \A^+\subset \Ker \A_s$,
where $\A_s$ is the homological operator associated to the semisimple part
$A_s$ of $A$.
\pn
{\sl Proof}. According to Lemma 1, if $K$ commutes with $A^+$, then it
also commutes with $A_s\ (=A^+_s)$. On the other hand, the solutions of
the linear systems $\.u=A_su$ and $\.u=A^+u$ are respectively
$u(t)=\exp(A_st)u(0)$ and $u(t)=\exp(A^+t)u(0)$; it is
easy to be convinced that the (time-independent) constants of motion,
which can be expressed in analytic form (polynomial, in this case) of
the second system are also constants of
motion of the first one (but the converse is not true). The statement of
Lemma 3 then follows from Lemma 2.
\medskip\pn
{\sl Lemma 4}. If $\phi=(Au+\~f)\pd_u$ and $\psi=(Bu+\~g)\pd_u$ form a
2-dimensional algebra, then it is possible to perform a (formal) coordinate
transformation which takes the nonlinear terms $\~f$ into normal form with
respect to $A$, and $\~g$ with respect to $B$ ("{\it parallel} normal form").
\pn
{\sl Proof}. Up to a linear transformation, any 2-dimensional algebra
satisfies the commutation rule
$$[\phi,\psi]=c \psi\eqno (2.1)$$
where $c$ is any constant (including $c=0$). First of all, we can always put
$\~g$ into NF with respect to $B$, so, let us assume (without changing
notations)
$$\~g\in \Ker\B^+$$
where $\B^+$ is the homological operator associated to $B^+$.
Now, from (2.1), $[A,B]=cB$, or $[A^+,B^+]=-\=c B^+$, which implies, in
terms of the homological operators,
$$\B^+\(\A^+(\~g)\)=\A^+\(\B^+(\~g)\)-\=c\ \B^+(\~g)=0$$
i.e. $\A^+(\~g)\in \Ker\B^+$, or $\A^+:\Ker \B^+\to\Ker\B^+$. This
ensures the possibility of performing another transformation, leaving
invariant the space $\Ker\B^+$ of the terms resonant with $B$, in such a way
to change the terms $\~f$ into NF with respect to $A$. Then we can choose
coordinates in such a way that:
$$\~f\in \Ker \A^+\qquad {\rm and} \qquad \~g\in \Ker \B^+ \ . \eqno(2.2)$$
\bigskip
We can now state the first main result.
\medskip\pn
{\sl Theorem 1}. Let $\phi=f(u)\pd_u=(Au+\~f)\pd_u,\ \psi=g(u)\pd_u=(Bu+
\~g)\pd_u$ satisfy
$$[\phi,\ \psi]=0\eqno (2.3)$$
then, by means of a formal coordinates transformation, $\~f,\ \~g$ can be
taken into a "{\it joint} normal form" (JNF) of this type:
$$\~f\in \Ker\A^+\cap \Ker\B_s \quad {\rm and}\quad
\~g\in \Ker\A_s\cap \Ker\B^+ \ . \eqno(2.4) $$
\pn
{\sl Proof}. From Lemmas 3 and 4, we get
$$\~f\in \Ker\A^+\subset\Ker\A_s\ ; \quad \~g\in \Ker\B^+\subset \Ker\B_s
\eqno(2.5)$$
Let us write now (2.3) step by step, with
$$f(u)=Au+\sum^\infty_{j=2}f_{(j)}\quad {\rm and}\quad
g(u)=Bu+\sum^\infty_{j=2}g_{(j)}\eqno(2.6)$$
We have first
$$[A,B]=0\eqno(2.7)$$
and
$$\{Au,g_{(2)}\}-\{Bu,f_{(2)}\}=0\quad {\rm or}\quad \A(g_{(2)})=\B(f_{(2)})
\eqno(2.8)$$
whereas, for $k>2$,
$$\{Au,g_{(k)}\}-\{Bu,f_{(k)}\}=\sum^{k-1}_{j=2}\{f_{(j)},g_{(k-j+1)}\}
\eqno(2.9)$$
Applying the operator $\A_s$ to (2.8), and using Lemma 1, we obtain, thanks
to (2.5)
$$\A_s\(\A(g_{(2)})\)=\B\(\A_s(f_{(2)})\)=0$$
and also $\A_s^2(g_{(2)})=0$, which implies
$$\A_s(g_{(2)})=0\ ;$$
in fact, being $A_s$ a diagonalizable matrix, we can choose coordinates
such that $\Ker\A_s$ is the orthogonal complement to $\Ran\A_s$ in the
space $V_{(2)}$ . Repeating the argument for the operator
$\B_s$ applied to (2.8), we get similarly
$$\B_s(f_{(2)})=0\ .$$
An immediate application of the Jacobi identity shows that if
$$f_{(j)},\ g_{(i)} \in \Ker\A_s\cap\Ker\B_s\quad \forall\ i,j=2,
\ldots ,k-1$$
then the same is true for
$$\{f_{(j)},g_{(i)}\}\quad {\rm and}\quad \sum^{k-1}_{j=2}\{f_{(j)},
g_{(k-j+1)}\} \eqno(2.10)$$
This allows us to proceed inductively: applying the operators $\A_s$ and
$\B_s$ to (2.10), we can conclude, for all $j$, that $f_{(j)}\in \Ker\B_s$
and $g_{(j)}\in \Ker\A_s$, which, together with (2.5), gives the result.
\bigskip
The possibility of extending of the above results (namely, Lemma 4 and
Theorem 1) to algebras of dimension $d>2$ is clearly related to the
specific commutation properties of the algebra.
We consider here some special cases.
\medskip\pn
{\sl Theorem 2}. Let us consider a $d-$dimensional algebra $\G$ of VFs
spanned by $\phi_a=f_a\pd_u=(A_au+\~f_a)\pd_u\ (a=1,\ldots,d)$. Then:
\pn
i) If the algebra $\G$ is {\it solvable}, then all the nonlinear terms
$\~f_a$ can be put in parallel NF, namely
$$\~f_a\in \Ker\A_a^+ \qquad {\rm for\ each}\ a=1,\ldots,d\ .\eqno(2.11)$$
ii) If the algebra $\G$ is {\it nilpotent} (in particular: abelian), then
one can put all $\~f_a$ into a JNF, precisely (with obvious notations):
$$\~f_a\in \Big(\bigcap_{b\ne a}\Ker\A_{b,s}\Big)\cap\Ker\A^+_a
\qquad {\rm for\ each}\ a=1,\ldots,d \eqno(2.12)$$
iii) In any solvable (resp.: nilpotent, or in particular abelian)
{\it subalgebra} of a generic algebra $\G$, all nonlinear terms can be
put parallel NF as in (2.11) (resp.: in JNF as in (2.12)).
\pn
{\sl Proof}. If the algebra is solvable, let us consider the sequence of
commutators terminating in $0$
$$[\phi,\phi]=\phi^{(1)},\ [\phi^{(1)}, \phi^{(1)}]=\phi^{(2)},\
\ldots,\ [\phi^{(m)},\phi^{(m)}]=0\eqno(2.13)$$
In the ideal $\G^{(m)}$ spanned by $\phi^{(m)}$ all non linear terms of
the VFs can be taken in NF (or even in JNF if dim$\ \G^{(m)}>1$:
in this case indeed this subalgebra is abelian and Theorem 1 can
be directly applied).
Using $[\phi^{(m-1)},\phi^{(m-1)}]=\phi^{(m)}$ and
$\G^{(m-1)}\supseteq\G^{(m)}$ , we can repeat the argument of Lemma 4
to show that also in $\G^{(m-1)}$ parallel NFs can be obtained,
and so on. If now the algebra $\G$ is nilpotent,
let us consider the sequence of commutators terminating in $0$
$$[\phi,\phi]=\phi^{[1]},\ [\phi,\phi^{[1]}]=\phi^{[2]},\
\ldots,\ [\phi,\phi^{[m]}]=0\eqno(2.14)$$
The first part of this Theorem ensures (since nilpotency implies
solvability) that all $\~f_a$ can be taken in their respective
NF: $\~f_a\in\Ker\A_a^+$; on the other hand, the last commutator
in (2.14) says that all fields in the abelian ideal $\G^{[m]}$ spanned by
$\phi^{[m]}$ commute with all the $\phi_a \in \G$.
Then the procedure followed in the proof of Theorem 1 can
be repeated for each $\phi_a\in\G$, using the last commutator in (2.14)
in order to obtain (2.12). Statement iii) is an immediate consequence.
\bigskip\pn
{\sl Remark 1}. The result in Theorem 1 and its extension in Theorem
2.ii) are generalizations of Theorem 2.2
\footnote{$^1$}
{Any 2-dimensional nilpotent algebra is in fact abelian. Unfortunately,
Ref. [7] came to our knowledge only after our paper [11]
- which contains results already obtained in [7], although by different
methods - was published. }
of Ref. [7], which gives in fact $\~f_a\in\bigcap_b \Ker\A_{b,s}$.
Notice that actually condition $\~f\in \Ker\A^+$ is a rather stronger
restriction for $\~f$ than $\~f\in \Ker\A_s$, the space $\Ker\A^+$ being
in general
considerably smaller than the space $\Ker\A_s$, as simple examples can
easily show. Notice also that, in general, it is not possible to
extend the result in Theorem 1 (and in 2.ii) as well) to have also e.g.
$\~f\in\Ker\B$ or $\~f\in\Ker\B^+$. The case of solvable algebras is
quite different: e.g., if $[\phi,\psi]=\psi$, then $[A,B]=B$, but this
implies $B_s=0$, and therefore one gets in this case $\B_s=0$.
\medskip\pn
{\sl Remark 2.} In NFs theory it is usual to give special attention to
the case of VFs with {\it normal} linear part, i.e. $[A,A^+]=0$ [11].
Here, we will not consider this restriction: the general results given
here can of course be specialized to this case (obtaining among others
some of the results given in [11]). For instance, eq. (2.12) becomes
immediately, with this restriction,
$$ \~f_a\in \bigcap_b\Ker\A_b \ . \eqno(2.12')$$
\medskip
%\vfill\eject
\titleb{3. Applications to dynamical systems and their symmetry properties}
\pn
Let us now apply the above algebraic results to the case of (finite
dimensional) dynamical systems (DS). With $u=u(t)\in M\subseteq R^n$, let
$$\.u=f(u)=Au+\~f(u)\eqno(3.1)$$
be a DS, where $\.u=du/dt$ and $f(0)=0$.
%\parindent=16pt
Denoting by $\phi\equiv f\pd_u$ the VF expressing the dynamical flow of
this DS, any VF $\psi=g\pd_u$ such that
$$[\phi,\psi]=0\eqno(3.2)$$
is the generator of a Lie-point time-independent (LPTI) symmetry of this
DS [13-15] (see also [10-11,16-20] and Ref. therein). Therefore, according to
Theorem 1, one can choose coordinates in
such a way that both VFs $\phi$ and $\psi$ are in JNF (2.4).
In concrete cases, once the DS is given, a typical problem is that of
finding its LPTI symmetries: then, the set of eq.s (2.7-9)
may be used in practice in order to construct recursively step by step
the symmetry field, and the JNF condition (2.4) determines the nonlinear
terms which may be removed, both in the DS and in the VF describing
the symmetry.
\medskip\pn
{\sl Remark 3.} Clearly, it is not granted, in general, that all LPTI
symmetries of a DS can be written as a series expansion (even if formal;
for some examples of "singular" LPTI symmetries, see e.g. [17-18]); however,
the method of proceeding step by step may be useful to construct
"approximate" symmetries, i.e. "up to the a given (finite) order" [16]).
Some sufficient conditions ensuring the existence of (polynomial) LPTI
symmetries, and some explicit examples can be found
in Ref [10,11,17-20]. A (linear) symmetry which is always
present (unless $A_s$, the semisimple part of $A$, is $=0$) is given by
the following Proposition.
\medskip\pn
{\sl Proposition 1.} Any DS (3.1) which is in NF, i.e. with
$\~f\in\Ker\A^+$, admits the linear symmetry generated by
$$\s=A_su\pd_u\eqno(3.3)$$
If $A_s$ is diagonalized, with real eigenvalues $\a_i\ (i=1,\ldots,n)$, this
symmetry generates the scaling $u_i\to u_i\ \exp(\ep \a_i)\ (\ep\in R)$.
\pn
{\sl Proof.} The symmetry condition (3.2) is certainly satisfied by this
$\s$ (3.3), indeed
$$\{Au+\~f, A_su\}=\{\~f,A_su\}=0$$
as a consequence of the resonance assumption and Lemma 3.
\medskip
This generalizes Proposition 4 of Ref. [10], where a
diagonalizable $A$ was assumed, and another result contained in Ref. [8]
which - in the present language - may be stated as follows: if the DS (3.1)
is in NF, then $(A^+u)\pd_u$ is a symmetry for the {\it nonlinear} part
of the DS $\ \.u=\~f$ (but not necessarily for the {\it full } DS $\.u=f$).
Another property of LPTI symmetries and NFs, which is a direct consequence
of (2.4), is given by the following Proposition.
\medskip\pn
{\sl Proposition 2.} If the DS (3.1) admits a LPTI symmetry
$\psi=g\pd_u=(Bu+\~g)\pd_u$, and
the fields $\phi,\ \psi$ are in JNF, then $\psi$ is also a symmetry for
the linear semisimple part of the DS, i.e. for $\.u=A_su$ (but the
converse is not true: i.e. symmetries
of this linear system are not necessarily symmetries for the full DS),
and the linear semisimple part $B_su\pd_u$ provides another symmetry (if
$B_s\ne 0$) for the DS:
$$\{A_su,g\}=0\quad{\rm and}\quad \{B_su,f\}=0\ . \eqno(3.4)$$
\medskip
Clearly, the set $\{\psi_1,\ldots,\psi_r\}$ of the LPTI symmetries of a DS
spans a Lie algebra
\footnote{$^2$}
{Actually, multiplying a LPTI symmetry by any constant of
motion of the DS gives another symmetry, so one should more correctly
speak of a (finite dimensional) {\it module} (rather than of an infinite
dimensional algebra) of symmetries [20]. Here, we are interested in the
algebraic structure, and therefore we are considering only "independent"
(i.e. pointwise linearly independent) VFs generating symmetries.}
$\G$: it always contains the VF $\phi$ giving the dynamical flow. This
algebra may be abelian (as in Proposition 3 below) or not. An interesting
example of nonabelian symmetry algebra can be constructed starting from a
4-dimensional problem which has the "quaternionic structure" [21]: the
generators of its symmetry span the Lie algebra of the group $SU(2)$ (in
real form).
The possibility of taking in parallel or joint NF the VFs $\{\phi, \psi_1,
\ldots,\psi_r\}$ in this algebra depends on the properties of the
algebra itself, according to Theorem 2. In any case, however, being
$[\phi,\psi_a]=0$ for all $a=1,\ldots,r$, by the definition of symmetry,
the JNF is possible for $\phi$ and {\it one} at least of the $\psi_a$,
or better for all the $\psi_a$ which span an abelian subalgebra
$\H\subseteq \G$.
Let us consider finally the special case of {\it linearizable} DSs, i.e.
DSs for which all terms are nonresonant and therefore can be removed by a
(formal) coordinate transformation.
\medskip\pn
{\sl Remark 4.} According to the definition, a DS (3.1) is linearizable if
$\Ker\A^+=\{0\}$. But a simple consequence of JNF indicates also that
a condition ensuring the linearizability of a DS is that it admits a LPTI
symmetry $\psi=(Bu+\~g)\pd_u$ such that $\Ker\A^+\cap\Ker\B_s=\{0\}$.
More in general, the same is true if
$\Big(\bigcap_h\Ker\B_{h,s}\Big)\cap\Ker\A^+=\{0\}$
where $\psi_h=(B_hu+\~g_h)\pd_u\in \H$, and $\H$ is an abelian
subalgebra contained in the algebra $\G$ of the admitted symmetries, as
said before.
\medskip
We also have:
\medskip\pn
{\sl Proposition 3.} If a DS can be linearized,
then it admits $n$ independent {\it commuting} symmetries, which can be
simultaneously taken into {\it linear} form by a coordinate transformation.
If, in particular, the system has a diagonalizable $A$ with real
eigenvalues, then - once it is linearized and $A$ is diagonal -
the dilations $\zeta_i=u_i\pd_i$ (no sum over $i$) along
each direction $i$, are $n$ linear commuting symmetries for the system.
Conversely, if there is a coordinate system where the DS
admits $n$ independent linear commuting symmetries
$\s_i=B_iu\pd_u$ such that all $B_i$ are semisimple, then the DS can be
linearized.
\pn
{\sl Proof.}
In the coordinates where the DS is linear, it is easy to
construct $n$ linear commuting symmetries $B_iu\pd_u$, simply choosing
$n$ independent matrices $B_i$ commuting among themselves and with $A$
(if the matrices $I, A, A^2,\ldots,A^{n-1}$ are linearly independent,
they immediately provide the matrices $B_i$ ; even if this is not the
case, the existence of $n$ matrices $B_i$ with the required properties is
easily verified if $A$ is put in Jordan form).
The existence of the $n$ independent scalings $\zeta_i$ in the case of
diagonal $A$ is obvious. Conversely, given $n$ semisimple
commuting matrices $B_i$, they can be simultaneously diagonalized: $B_i\to
{\rm diag}\ (\b^{(i)}_1,\ldots,\b^{(i)}_n)$; now, with respect to the basis
spanned by the $n$ (independent) vectors $\b^{(i)}\equiv
(\b^{(i)}_1,\ldots,\b^{(i)}_n)$, the symmetries $\s_i=B_iu\pd_u$ become
$\s_i\to \zeta_i=u_i\pd_i$ (no sum over $i$), i.e. the independent
dilations along the directions $\b^{(i)}$. A DS admitting such
$n$ symmetries is necessarily linear.
\bigskip\bigskip\pn
{\sl Acknowledgments.}
\pn
We are grateful to prof. Giuseppe Marmo for useful discussions.
\vfill\eject
\baselineskip .40 cm
\Ref
[1] Arnold V. I. (1982) "Geometrical methods in the theory of differential
equations" (Berlin, Springer)
[2] Arnold V. I. and Il'yashenko Yu. S. (1988) "Ordinary differential
equations", in Encyclopaedia of Mathematical Sciences - vol. I,
Dynamical Systems I, (Anosov D.V. and Arnold V.I., eds.), p. 1-148
(Berlin, Springer)
[3] Bruno A. D. (1989) "Local methods in nonlinear differential equations"
(Berlin, Springer)
[4] Belitsky G. R. (1978) Russ. Math. Surveys {\bf 33}, 107;
(1979) "Normal forms, invariants, and local mappings"
(Kiev, Naukova Dumka), and (1986) Funct. Anal. Appl. {\bf 20}, 253
[5] Broer H. W. (1979) "Bifurcations of singularities in volume
preserving vector fields" (Thesis, Groningen)
[6] Dumortier F. and Roussarie R. (1980) Ann. Inst. Fourier, Grenoble {\bf
30-1}, 31
[7] Arnal D., Ben Ammar M., and Pinczon G. (1984) Lett. Math.
Phys. {\bf 8}, 467
[8] Elphick C., Tirapegui E., Brachet M. E., Coullet P., and
Iooss G. (1987), Physica {\bf D, 29}, 95
[9] Broer H. W. and Takens F. (1989) "Formally symmetric normal forms and
genericity", Dynamics Reported {\bf 2}, 39-59
[10] Cicogna G. and Gaeta G. (1990) J. Phys. A: Math. Gen. {\bf 23}, L799
[11] Cicogna G. and Gaeta G. (1994) J. Phys. A: Math. Gen. {\bf 27}, 461
[12] Courant R. and Hilbert D. (1962) "Methods of Mathematical Physics"
(New York, Interscience Publ.)
[13] Ovsjannikov L.V. (1962) "Group Properties of Differential Equations"
(Novosibirsk, USSR Acad. of Sci., English transl. 1967 by Bluman G.);
and 1982 "Group Analysis of Differential Equations" (New York,
Academic Press)
[14] Olver P.J. (1986) "Applications of Lie Groups to Differential Equations"
(Berlin, Springer)
[15] Bluman G.W., Kumei S. (1989) "Symmetries and Differential Equations"
(Berlin, Springer)
[16] Baikov V.A., Gazizov R.K., and Ibragimov N. H. (1991) J. Sov. Math.
{\bf 55}, 1450
[17] Cicogna G. and Gaeta G. (1992), J. Phys. A: Math. Gen. {\bf 25}, 1535
[18] Cicogna G. (1993) "Lie point symmetries and dynamical systems",
in "Modern Group Analysis: Advanced analytical and computational Methods in
Mathematical Physics", Proc. of the International Workshop, Acireale 1992,
ed. by Ibragimov N. H., Torrisi M., Valenti A. (Dordrecht, Kluwer),
147-153
[19] Cicogna G. and Gaeta G. (1993) Phys.Lett. A {\bf 172}, 361
[20] Gaeta G. (1994) "Nonlinear symmetries and nonlinear equations"
(Dordrecht, Kluwer), in press
[21] Cicogna G. and Gaeta G. (1985) Lett. Nuovo Cim. {\bf 44}, 65; and (1987)
J. Phys. A: Math. Gen. {\bf 20}, 79
\bye