%\magnification=1200
\parskip=10pt
\def\A{{\cal A}}
\def\B{{\cal B}}
\def\G{{\cal G}}
\def\R{{\bf R}}
\def\I{{\cal I}}
\def \d{{\rm d}}
\def\pd{\partial}
\def\sp{\sum\nolimits'}
\def \a{\alpha}
\def \b{\beta}
\def \om{\omega}
\def \ka{\kappa}
\def \z{\zeta}
\def \l{\lambda}
\def \phi{\varphi}
\def \tr {transformation}
\def \com {constant of motion}
\def \coms {constants of motion}
\def \sy {symmetry}
\def \sys {symmetries}
\def \an {analytic}
\def \co {convergen}
\def \qq {\qquad}
\def \en {\eqno}
\def \cd {\cdot}
\def \grad {\nabla}
\def \Ker {{\rm Ker}}
\def \sse {\subseteq}
\def \pn{\par\noindent}
\def \bs{\bigskip}
\def \Ref{\bs\bs\pn{\bf References}\parskip=7 pt\parindent=0 pt}
\def \section#1{\bs\bs \pn {\bf #1} \bigskip}
\def\~#1{\widetilde #1}
\def\.#1{\dot #1}
\def\^#1{\widehat #1}
\baselineskip 0.6 cm
{\nopagenumbers
~ \vskip 4 truecm
{\bf
\centerline {
Convergent Normal Forms of Symmetric Dynamical Systems}
\vskip 2 truecm
\centerline {G. Cicogna}}
\centerline{Dipartimento di Fisica, Universit\`a di Pisa,
P.za Torricelli 2, I-56126, Pisa, Italy}
{\tt \centerline{E-mail : cicogna@ipifidpt.difi.unipi.it}}
\vskip 4 truecm
\pn
{\bf Abstract.}
\pn
It is shown that the presence of Lie-point-symmetries of
(non-Hamiltonian) dynamical systems can ensure the
convergence of the coordinate transformations which take the dynamical
sytem (or vector field) into Poincar\'e-Dulac normal form.
\vfill\eject }
~ \vskip 1.5 truecm
\baselineskip 0.53 cm
\section{1. Introduction}
A well known and interesting procedure, going back to the classical work of
Poincar\'e, for investigating \an\ vector fields (VF) $X_f$
$$X_f\equiv \sum_{i=1}^n f_i(u){\pd\over{\pd u_i}}\equiv f\cd \grad \qq
(u\in \R^n)\en(1)$$
or the associated dynamical sytems (DS)
$${\d u\over{\d t}}=f(u)\qq u=u(t)\en(2)$$
in a neighbourhood of a stationary point,
is that of introducing some new coordinates in which the given VF takes
its ``simplest'' form, i.e. the normal form (NF) (in the sense of
Poincar\'e-Dulac [1-4]). These coordinate \tr s are usually performed by
means of recursive techniques: in general the normalizing \tr s (NT) are
actually purely formal \tr s, and only very special conditions can
ensure their \co ce and the (local) analyticity of the NF [1-4]. In the
investigation of these problems, a
relevant role can be played by the presence of some \sy\ property
[5,6] (see also [7]) of the VF $X_f$, i.e. by the presence of some VF $X_g$
$$X_g\equiv \sum_{i=1}^n g_i(u){\pd\over{\pd u_i}}\equiv g\cd \grad
\en(3)$$
such that
$$ [X_f,X_g]=0\en(4)$$
In terms of the DS (2), the \sy\ VF $X_g$ provides the Lie generator of a
(possibly non-linear) Lie-point-symmetry of the DS, and this can be
conveniently expressed in the form of the Lie-Poisson bracket
$$\{f,g\}_i\equiv (f\cd\grad)g_i-(g\cd\grad) f_i \ =\ 0 \qq (i=1,\ldots,n)
\en(5)$$
In this context, Bruno and Walcher [8] showed that the existence of an \an\
\sy\ for a 2-dimensional DS is enough to ensure \co ce of the NT; in [9] the
\co ce is obtained also for DS of dimension $n>2$, combining the
existence of \sys\ with other conditions involving also
the \coms\ of the DS; in [10,11] the role of \sys\ is investigated
in view of the problem of linearizing the DS.
In this paper, we will discuss some generalizations and some
results along the same lines.
\vfill\eject
\section {2. Preliminary results}
We will freely use $f$ both to denote the VF $X_f$ (1) and to refer to the DS
(2); let us introduce the notation
$$f(u)=Au+F(u)\en(6)$$
where $f$ is assumed to be \an\ in a neighbourhood of the stationary point
$u_0=0$, and its linear part $A=(\grad f)(0)$ a
semisimple (and not zero) matrix. The NF of $f$ will be written
$$\^f(u)=Au+\^F(u)\en(7)$$
(the notation \ $\^\cd$\ will be always reserved to NF; there is no
danger of confusion if $u$ is used to denote also the ``new'' coordinates),
and $\^F(u)$ contains the ``resonant terms'' with respect to $A$, i.e.
the terms such that
$$\^F(u)\in\Ker(\A)\en(8)$$
where $\A$ is the ``homological operator'' defined by
$$\A(h)=\{Au,h\}\en(9)$$
This fact can be conveniently stated in the form:
\pn
{\bf Proposition 1.} Every NF $\^f$ admits the linear \sy\ $g_A\equiv Au$.
\pn
Let us recall some well known and useful facts.
\pn
{\bf Lemma 1.} Every \sy
$$g(u)=Bu+G(u)\en(10)$$
of a NF $\^f$ is also a \sy\ of the linear part $Au$ of $\^f$. The analogous
result is true for the \coms\ of the DS (6): i.e., if a scalar function
$\mu=\mu(u)$ is such that $\^f\cd\grad\mu=0$ then also $Au\cd\grad\mu=0$.
\pn
Denoting by $\G_f$ and $\I_f$ the set of the \sys\ and respectively of
the \coms\ of $f$, we can then write
$$\G_{\^f}\sse\G_{Au} \qq{\rm and} \qq \I_{\^f}\sse\I_{Au}\en(11)$$
This is true not only for \an\ quantities, but also for quantities
expressed by means of formal power series.
\pn
{\bf Lemma 2.} Given the matrix $A$, the most general NF has the form
$$\^F(u)=\sum_j \mu_j(u) M_ju \qq{\rm with}\qq \mu_j(u)\in\I_{Au}
\qq {\rm and}\qq [M_j,A]=0 \en(12)$$
where the sum is extended to a set of linearly independent matrices
$M_j$ (the set of these matrices clearly includes $A$).
\pn
{\bf Lemma 3.} If $f$ admits a linear \sy\ $g_B=Bu$, then the NF $\^f$
also admits this \sy . If $f$ admits a (possibly formal) \sy\
$g=Bu+G(u)$ and $B$ is semisimple, then $Bu$ is a \sy\ of the NF $\^f$, or
-- in other words -- $\^F$ is a NF also with respect to $B$, i.e.
$\^F\in\Ker(\A)\cap\Ker(\B)$.
The proofs of these Lemmas are well known and can be found, e.g., in
[7,12-14]. Let us also recall the basic conditions, found by Bruno [1,2],
and called Condition $\om$ and Condition A, which ensure
the \co ce of the NT of a given DS. Denoting by $\l_1,\ldots,\l_n$ the
eigenvalues of the matrix $A$, then the first condition is
\pn
{\sl Condition $\om$}: let $\om_k=\min|(q,\l)|$ for all positive
integers $q_i$ such that $\sum_{i=1}^n q_i<2^k$ and
$(q,\l)=\sum_iq_i\l_i\ne 0$: then
$$\sum_{k=1}^\infty 2^{-k}\ln \om_k^{-1}<\infty$$
\pn
This is a very weak condition, and we explicitly
assume from now on that it is always satisfied. The other one, instead,
is a quite strong restriction on the form of the NF. To state this
condition in its simplest form, let us assume for a moment that there is
a straight line through the origin in the complex plane which contains
all the eigenvalues $\l_i$ of $A$, and that there are eigenvalues lying
on both sides of this line with respect to the origin. Then the condition
reads
\pn
{\sl Condition A}: there is a coordinate \tr\ changing $f$ to $\^f$,
where $\^f$ has the form
$$\^f=Au+\a(u)Au$$
and $\a(u)$ is some scalar-valued power series \big(with $\a(0)=0$\big).
\pn
In the case there is no line in the complex plane which satisfies the
above property, then Condition A should be modified [1] (or even
weakened: for instance, if there is a straight line through the origin
such that all the $\l_i$ lie on the same side of this line, then the
eigenvalues belong to a Poincar\'e domain [1,3] and the \co ce is
guaranteed without any other condition); but in all the applications
below, where in particular only linear NF will be ultimately
concerned, the above formulation of Condition A is enough to cover
all the cases to be considered, and we can say [1,2] that
there is a \co t NT if the above conditions are satisfied. Clearly, here
and in the following, ``\co ce'' stands for ``\co ce in some open
neighbourhood of $u_0=0$''.
\section{3. Symmetries and \co ce of the NT: a general result}
Let us finally state the first result of this paper. It can be noted that
quite strong assumptions are needed; but it is known, on the other hand, that
the \co ce of the NT is quite ``exceptional''. The examples given below
will show how, thanks to additional \sy\ properties, these assumptions
can happen to be verified. Let us remark that obviously, for any constant
$c$, then $cf$ is a (trivial) \sy\ of $f$; therefore, it is understood
that when we assume the existence of some \sy\ of $f$ we will always
refer to {\it nontrivial} \sys , i.e. to \sys\ $g\not= cf$.
\pn
{\bf Theorem 1.} Given the \an\ VF $f$, let us write its NF, according to
Lemma 2, in the form
$$\^f=Au+\a(u) Au+\sp\mu_j(u)M_ju\equiv Au+\a(u) Au+\^F_1(u)\en(13)$$
where (here and in the following) $\sum'$ is the sum extended to the
matrices $M_j\not=A$. Assume $\^F_1(u)\not=0$ (otherwise Condition A is
sufficient of ensure \co ce of the NT), and:
\parskip 0 pt\pn
{\sl a)} assume that $f$ admits an \an\ \sy\
$$g=Bu+G(u)\qq {\rm such\ that}\qq B=aA\en(14)$$
where $a$ is a (possibly vanishing) constant;
\pn
{\sl b)} assume that the equation
$$\{\^F_1,S\}=0\en(15)$$
for the unknown
$$S=S(u)=\sp\nu_j(u)M_ju \qq {\rm with} \qq \nu_j(u)\in\I_{Au}\qq {\rm and}
\qq \nu_j(0)=0\en(15')$$
has only the trivial solution
$$S=c\^F_1(u) \qq\qq (c={\rm constant}) $$
Then $f$ can be put into NF by means of a \co t NT.
\parskip 10 pt\pn
{\bf Proof}. First of all, if $a=0$ in assumption {\sl a)}, one
can consider, instead of
$g$, the \sy\ $g'=f+g$ having linear part $Au$; it is then not
restrictive to assume $a=1$, i.e. $B=A$. Once $f$ is put into NF $\^f$,
the \sy\ $g$ will become a (possibly formal) \sy\ $\~g$
$$\~g=Au+\b(u)Au+\sp\nu_j(u)M_ju\qq{\rm with}\qq\b(u),\nu_j(u)\in\I_{Au}
\en(16)$$
this is indeed the most general \sy\ of a NF, thanks to Lemma 1. The \sy\
condition $\{f,g\}=0$ in the new coordinates reads $\{\^f,\~g\}=0$;
evaluating term by term this bracket, one is left with
$$\Big\{\a Au\ ,\ \sp_j \nu_j M_ju\Big\}+\Big\{\sp_j
\mu_j M_ju\ ,\ \b Au\Big\}
+
\Big\{\sp_j\mu_j M_ju\ ,\ \sp_k\nu_k M_ku\Big\}\ =\ 0$$
or
$$\Big(\sp_j\mu_j(u)M_ju\cd\grad\b-\sp_j
\nu_j(u)M_ju\cd\grad\a\Big)Au\ +\
\Big\{\sp_j\mu_j(u)M_ju\ ,\ \sp_k\nu_k(u)M_ku\Big\}\ =\ 0 \en(17)$$
All other terms in fact vanish thanks to Proposition 1 and Lemmas 2 and
3. Now, in eq.(17), the bracket $\{\ \cd\ ,\ \cd\ \}$ produces, through
the matrix commutators $[M_j,M_k]$, only terms
proportional to $M_ju$ (and not to $Au$: this can be easily seen in a
basis in which $A$ is diagonal), therefore the terms appearing into
the $\big(\quad\big)$ and the bracket $\{\ \cd\ ,\ \cd\ \}$ are both zero.
The last bracket has just the form
$\{\^F_1,S\}$, and therefore assumption {\sl b)} gives $S=c\^F_1$, i.e.
$\nu_j(u)=c\mu_j(u)$. From the
vanishing of the first $\big(\quad\big)$ in (17), and using again
assumption {\sl b)}, one obtains similarly $\b(u)=c\a(u)$,
and then either $\~g=c\^f$, which is impossible because $g\not=cf$, or
$$\~g(u)\ =\ Au \en(18)$$
This means that the \tr\ which puts $f(u)$ into $\^f(u)$ transforms
$g(u)=Au+G(u)$ into $\~g(u)=Au$, therefore the \sy\ $g(u)$ satisfies
Condition A and there is \co t \tr\ which puts $g(u)$ into NF. Under this
\co t \tr\ $f(u)$ is transformed into NF $\^f$, as a consequence of the
last part of Lemma 3.
\pn
{\bf Remark 1.} One can see that the assumption {\sl b)} of Theorem 1 is
equivalent to the assumption that the NF $\^f$ admits only linear \sys\
$Lu$. For a practical point of view (see the Examples below), it is quite
simpler to verify the property {\sl b)}.
\pn
{\bf Remark 2.} In the particular case that $\^F_1(u)$ has the form
$$\^F_1(u)=\mu(u)Mu\en(19)$$
(with $M\not= A$), then assumption {\sl b)} is actually equivalent to the
very simple following one: there are no common (\an , formal or fractional)
\coms\ of the two linear problems
$$\.u=Au\qq{\rm and}\qq\.u=Mu\en(20)$$
Indeed, assume there is some $\ka=\ka(u)\in\I_{Au}\cap\I_{Mu}$, then
$S=\ka(u)Au\not= c\^F_1$ would satisfy $\{\^F_1,S\}=0$; notice,
incidentally, that one would also get in this case $\ka(u)\in\I_{\^f}$\ .
The converse is easily obtained by explicit calculations. This case has
been already considered in [9]; the result for 2-dimensional
DS in [8] can be viewed as a particular case of this (see [9] for
details).
\bs
The apparent difficulty in the application to concrete cases of the
above results is that, in general, one does not known -- a priori -- the
NF, and then it seems to be impossible to check if the assumptions of
Theorem 1 (or even Condition A) are verified by the NF. However, as the
foregoing Examples will show, other \sy\ properties of the VF may provide,
once again, the decisive help on this point.
\pn
{\bf Example 1.} Consider a 3-dimensional \an\ DS
$$\.u=f(u)=Au+F(u) \qq {\rm with}\qq A={\rm diag}(1,1,-2)\en(21)$$
with $u\equiv(x,y,z)\in\R^3$ and assume that $f(u)$ possesses the linear
$SO_2$ \sy\ generated by $Lu\cd\grad$ where
$$L=\pmatrix{0 &1 & 0\cr
-1 &0 &0 \cr
0 &0 &0 } \en(22)$$
i.e. $f$ is ``equivariant'' under rotations in the plane $(x,y)$. Putting
$r^2=x^2+y^2$, this implies that $F(u)$ must be written in the form
$$ F(u)=\phi_0(r^2,z)Au+\phi_1(r^2,z)Iu+\phi_2(r^2,z)Lu \en(23)$$
where $I$ is the identity matrix in $\R^3$.
If we now choose, for instance,
$$\phi_0=0,\qq \phi_1=a_1r^2z+a_2z^3,\qq \phi_2=b\phi_1\en(24)$$
where $a_1,a_2,b$ are constants $\not=0$, then the DS admits also the
non-linear \sy\
$$G(u)=r^2z(I+bL)u\en(25)$$
Notice that the assumption $a_2\not=0$ ensures that this DS is {\it not}
a NF, and that the above \sy\ is not trivial.
Then assumption {\sl a)} of Theorem 1 is satisfied. Now, the NF of the
above DS (21-24) must be of the form
$$\^f=Au+\a(r^2z)Au+\mu_1(r^2z)Iu+\mu_2(r^2z)Lu \en(26)$$
where $\a,\mu_1,\mu_2$ depend only on $\ka=r^2z$, as a consequence of Lemma 3
(i.e., the equivariance under $SO_2$ is preserved), and of Lemma 2
(the resonance condition).
We have to look for the solutions $S$ of the equation
$\{\^F_1,S\}=0$, where the unknown $S$ can be written
$$S=\nu_1(u)Iu+\nu_2(u)Lu\qq{\rm with}\qq\nu_1(u),\nu_2(u)\in\I_{Au}\en(27)$$
\big(it is easy to see that no other matrices can appear in the r.h.s. of
(26-27)\big), and where $\nu_1,\nu_2$ must be functions only of the two
functionally independent quantities
$x^2z,xyz\in\I_{Au}$. The condition $\{\^F_1,S\}=0$ gives
the first-order system of linear partial differential equations
$$\eqalign{(\mu_1 u+\mu_2Lu)\cd\grad\nu_1=\nu_1 u\cd\grad\mu_1\cr
(\mu_1 u+\mu_2Lu)\cd\grad\nu_2=\nu_1 u\cd\grad\mu_2 }\en(28)$$
Observing that
$Lu\cd\grad\mu_1=0$ and $Au\cd\grad\mu_1=0=u\cd\grad\mu_1-3z\pd_z\mu_1$
and the same for $\nu_1$, i.e. $u\cd\grad\nu_1=3z\pd_z\nu_1$, one gets
from the first of the (28) the characteristic equation
for the unknown $\z=\z(u)$ defined by $\nu_1(u)=\z(u)\mu_1(u)$
$${\d x\over y}\ =\ -\ {\d y\over x}\ =\ {\mu_2\over\mu_1}\
{\d z\over 3z} \en(29)$$
which shows that $\z$ must be a function of $r^2$ and of some other
variable of the form $v={\rm arctg}(y/x)+Z(z)$. On the other hand,
$\mu_1$ is a function of $\ka=r^2z$, and $\nu_1$ is a function
of the quantities $x^2z,xyz\in\I_A$
(possibly also of $x/y$, of course, the only requirement is that $\mu_1,\
\nu_1$ are power series in $x,y,z$); it is then easy to see that
$\z={\rm const}$, i.e. that $\nu_1=c\mu_1$. Proceeding in the same way
for the other equation in (28), one can conclude that
$$S\ =\ c\^F_1$$
and then also the assumption {\sl b)} in Theorem 1 is satisfied, and
therefore there is a \co t NT.
Notice that it was essential in the calculations for the example above
that $\mu_1$ and $\mu_2$ are both $\not=0$, and this is in fact
guaranteed by the normalizing procedure: indeed, at the lowest order,
the resonant terms $r^2zIu$ and $r^2zLu$ are orthogonal to $z^3Iu$
and $z^3Lu$ (with respect to standard scalar product [3,12,14] introduced
in the vector space of homogeneous polynomials, where the homological
operator $\A$ is defined), then at the first step
of the normalization procedure the resonant terms are not changed, i.e.
one has $\mu_1=a_1r^2z+\ldots$ and $\mu_2=a_1b\ r^2z+\ldots$;
and then -- at any further step of the iteration -- the lower order
terms are not altered.
\section{4. Symmetries and convergence of the NT: a special case}
Let us now come back to the special case that
$\^F_1(u)$ can be written in the form
$$\^F_1(u)=\mu(u) Mu\en(30)$$
In Remark 2 we have seen that assumption {\sl b)} of
Theorem 1 can be replaced by the requirement that there are no
simultaneous \coms\ of $Au$ and $Mu$. Assume now that, as in Example 1,
the DS admits not only the non-linear \sy\ $g(u)$ \big(assumption {\sl a)} of
Theorem 1\big), but also a \sy\ generated by some linear VF $Lu\cd\grad$, and
that $g(u)$ satisfies
$$\{g,Lu\}=0\en(31)$$
In the NT the \sy\ is conserved step by step [14], therefore
$g(u)$ also will be
transformed, once $f$ is in NF $\^f$, into some $\~g(u)$ which is
symmetric under $Lu$: $\{\~g,Lu\}=0$. From this remark we see that
it is sufficient to look for the common \coms\ of $Au$ and $Mu$
{\it only in the set} of those
scalar functions $\ka=\ka(u)$ which are left invariant by $Lu$; in other
words, we can conclude that if the set of these simultaneously invariant
functions contains only trivially constant numbers, i.e.
$$\I_{Au}\cap\I_{Mu}\cap\I_{Lu}=\R \en(32)$$
then no other non-linear \sys\ are allowed, and $\~g(u)$ becomes necessarily
$\~g(u)=Au$; then, using similar arguments as above, the \co ce of the NT is
guaranteed. In conclusion, we can state:
\pn
{\bf Theorem 2.} Assume that $f$ admits a \sy\ $g$ as in assumption {\sl a)}
in Theorem 1, and also a \sy\ generated by a linear VF $Lu\cd\grad$ such
that in addition (31) and (32) are satisfied. Then, if $\^F_1$ has the form
(30), the NT is \co t. The result can be trivially extended to the
case that $f$ admits an algebra of (more than one) \sys\ $L_ku\cd\grad$.
\pn
{\bf Example 2.} The same as Example 1, here with $b=0$ and $a_1,a_2\not=
0$. In the NF now $\mu_1=a_1r^2z+\ldots\not=0$, but $\mu_2$ may be zero.
If $\mu_2\not= 0$, Theorem 1 can be applied.
If $\mu_2=0$, then $\^F_1$ has the form (30);
on the other hand, it is clear that no $SO_2$-invariant \an\ functions are
simultaneously \coms\ of $\.u=Au$ and $\.u=u$, then all assumptions of
Theorem 2 are satisfied, and the NT is \co t.
The example given in [9] can be viewed as another example of Theorem 2,
in the presence of a larger \sy\ (the Lie algebra of the group $SO_3$).
\bs
To conclude, it can be interesting to point out the following peculiar
property of the present approach.
\pn
{\bf Remark 3.} All the results in this paper
are peculiar of {\it non-Hamiltonian} DS: indeed, an Hamiltonian DS never
satisfies the crucial hypothesis, i.e. assumption {\sl b)} of Theorem 1. Let
in fact $H=H(u)$, with $u\equiv(q,p)\in\R^{2m}$, be an \an\ Hamiltonian and
$$\.u=J\grad H$$
be the associated DS, where $J$ is the symplectic matrix. Writing
$H=H_0+H_R$, where $H_0$ is the quadratic part of $H$, we have clearly
$Au=J\grad H_0$ and $F(u)=J\grad H_R$, and the requirement that $F(u)$ is
in NF, $\^F(u)=J\grad\^H_R$, becomes now the requirement that $H_0$ is a
\com\ of $\^H_R$ (cf. [15,16]). Then, eq. (15) always admits nontrivial
solutions of the form
$$S=\eta(H_0)\^F_1$$
for any (regular) function $\eta$ of $H_0$.
\vfill\eject
\baselineskip 0.5 cm
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\bye