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\markboth{\hfil Invariance of Poincar\'e-Lyapunov polynomials\hfil
EJDE--1998/23}%
{EJDE--1998/23\hfil Pierre Joyal \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol.\ {\bf 1998}(1998), No.~23, pp. 1--8. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or
http://ejde.math.unt.edu
\newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113}
\vspace{\bigskipamount} \\
Invariance of Poincar\'e-Lyapunov polynomials under the group of rotations
\thanks{ {\em 1991 Mathematics Subject Classifications:} 58F14, 58F21,
58F35, 34C25.
\hfil\break\indent
{\em Key words and phrases:} focus, invariance of Poincar\'e-Lyapunov
polynomials, \hfil\break\indent weighted-homogeneity.
\hfil\break\indent
\copyright 1998 Southwest Texas State University and University of
North Texas. \hfil\break\indent
Submitted June 25, 1998. Published October 9, 1998. \hfil\break\indent
This research was partially supported by NSERC and FCAR
} }
\date{}
\author{Pierre Joyal}
\maketitle
\begin{abstract}
We show that the Poincar\'e-Lyapunov polynomials at a focus of a
family of real polynomial vector fields of degree $n$ on the plane
are invariant under the group of rotations. Furthermore,
we show that under the multiplicative group
${\mathbb C}^*=\{\rho {\rm e}^{i\psi}\}$, they are invariant up to a
positive factor. These results follow from the weighted-homogeneity
of the polynomials that we define in the text.
\end{abstract}
\newtheorem{prop}{Proposition}
\newtheorem{lemma}[prop]{Lemma}
\newtheorem{coro}[prop]{Corollary}
\newcommand{\qed}{\vrule height 1.3ex width 1.0ex depth- 0.0ex}
\section{Introduction}
Let us consider a real analytic vector field on the plane having a
non-degenerate focus at the origin, that is, the Jacobian matrix of
the vector field at the focus is not singular.
After a linear transformation, we can suppose that the Jacobian matrix
at the focus has the form
\begin{equation}\label{mat}
\left(\begin{array}{rr}
a & -b \\
b & a
\end{array}
\right),\qquad b\ne 0.
\end{equation}
Let $\Sigma$ be a local cross section
with one end point at the origin and $U\subseteq\Sigma$, a neighborhood
of the origin in $\Sigma$.
Recall that the displacement function in the neighbourhood of
the origin is the Poincar\'e map $P\colon U\to\Sigma$ minus the
Identity.
One can show that the displacement function in a neighborhood of
the origin has the following form (see [1]):
\begin{equation}\label{eq2}
r=({\rm e}^{2\pi a/b}-1)r_0+u_3r_0^3+u_5r_0^5+u_7r_0^7+\cdots\, .
\end{equation}
All the coefficients of the even powers of $r_0$ are equal to zero.
When all the coefficients vanish, the origin is a center.
Instead of calculating these coefficients to determine if an equilibrium
point is a center, Poincar\'e gave in [2] another method which
resembles the search for a Lyapunov function to establish the stability
of a focus. Let us recall this method.
Looking at (\ref{eq2}), we see that ${dr/dr_0}\ne 0$
in a punctured neighborhood of the origin, if $a\ne 0$. Suppose that
$a=0$.
If the vector field is linear, the integral curves are circles around
the
origin: $x^2+y^2=k$ ($k$ a constant), or in polar coordinates $r^2=k$.
If the vector field is not linear, it is natural to look for integral
curves
that are small perturbations of these circles. Using polar coordinates,
one tries to find integral curves of the form
\begin{equation}\label{eq3}
H(r,\theta) = r^2 + H_3(\theta)r^3 + H_4(\theta)r^4 + \cdots = k\,.
\end{equation}
If the origin is a center and if $H=k$ is an integral curve, then
$$
{dH\over dt}=\frac{\partial H}{\partial r}\dot r+
\frac{\partial H}{\partial \theta}\dot\theta = 0\,.
$$
Looking at the coefficients of the powers of $r$, this equation
generates
an infinite system of equations with the unknows $H_j(\theta)$ (see
section 2).
If the origin is not a center, then the equation above cannot be solved.
However, as we will see later on, one can formally solve the equation
$$
{dH\over dt}= P_1r^4 + P_2r^6 + P_3r^8 + \cdots\,,
$$
where $P_j$, $j=1,2,\ldots$ are constants.
The sign of the first non-zero $P_j$ controls the type of stability of
the focus. If $P_j>0$, the focus is unstable; it is stable otherwise.
In fact, it is possible to find
$H=r^2 + H_3(\theta)r^3 + \cdots + H_{2j+1}(\theta)r^{2j+1}$ such that
$$
{dH/dt\over r^{2j+2}}\Big|_{r=0}= P_j\,.
$$
$H$ is a Lyapunov function for the focus (see proposition 1 and
corollary
2).
If all the $P_j$ vanish, it is possible to solve the system and
the series in (\ref{eq3}) converges in a neighborhood of the origin
(see [2]).
There are no standard names for the constants $P_j$. Some call
them focal numbers (or quantities), others call them Lyapunov
constants. These names do not match the
definitions of Andronov {\it et al\/} [1]. According to [1], the
$j^{\rm th}$ focal value (or quantity) is the $j^{\rm th}$ derivative
of the displacement function $r$ in (\ref{eq2}).
If the first non-vanishing derivative of $r$ is of order $k=2j+1\ge 3$
($j\ge 1$), then it is called the $k^{\rm th}$ Lyapunov value.
But the $P_j$ are not in general equal to the $u_j$ in (\ref{eq2}).
Moreover, in the case of a family of vector fields, the $P_j$ are in
fact
polynomial functions of the parameters (as we will see later on).
We adopt the following definition.
\paragraph{Definition}
$P_j$ is the $j^{\rm th}$ Poincar\'e-Lyapunov constant.
In the case of a family of vector fields,
$P_j$ will be called the $j^{\rm th}$ Poincar\'e-Lyapunov polynomial
(associated with this family). \medskip
We will study these polynomials for the family of all
polynomial vector fields of degree $n$ on the plane.
We will prove that they are invariant under the group of rotations
$S^1=\{\,{\rm e}^{i\psi}\,\}$ and also invariant under the
multiplicative
group ${\mathbb C}^*=\{\,\rho {\rm e}^{i\psi}\,\}$ modulo a positive
factor.
Precisely, $\forall j\ge 1$ and for $g=\rho{\rm e}^{i\psi}\in {\mathbb
C}^*$,
\[
P_j(g(a_{rs}))=\rho^{2j}P_j(a_{rs})\,,
\]
where the $a_{rs}$ are the parameters of the family of all
polynomial vector fields of degree $n$ on the plane. In this statement,
it is important to distinguish a Poincar\'e-Lyapunov polynomial from
the corresponding Poincar\'e-Lyapunov constant (the value of this
polynomial for a certain vector field). Indeed, the statement says
that the polynomials are also weighted-homogeneous in a certain sense
that we will define in section 3.
\section{ Poincar\'e's Method}
We suppose that the family of all polynomial vector fields of degree $n$
has an equilibrium point at the origin with a Jacobian matrix of
the form (\ref{mat}) where $a=0$. We will slightly modify
Poincar\'e's procedure to obtain the main result of this article.
Dividing the family by $b$, it takes the following form in the
coordinates
$z=x+iy$ and $\bar z$:
\begin{equation}\label{eq4}
\begin{array}{lll}
\dot z &=& \displaystyle{
iz + \sum_{m=2}^n\sum_{j+k=m} a_{jk}z^j\bar z^k,}\\
\dot{\bar z} &=& \displaystyle{
-i\bar z + \sum_{m=2}^n\sum_{j+k=m} \bar a_{kj}z^j\bar z^k.}
\end{array}
\end{equation}
Setting $r=\sqrt{z\bar z}$ and $\theta=(1/2i)\ln(z/\bar z)$, we obtain:
\begin{equation}\label{eq5}
\begin{array}{lll}
\dot r &=& \displaystyle{
{1\over 2r}(\dot z\bar z + z\dot{\bar z})
= (1/2)\sum_{m=2}^n F_m({\rm e}^{i\theta})r^m }\\ [4pt]
\dot\theta &=& \displaystyle{{1\over 2r^2}(-i\dot z\bar z + iz\dot{\bar
z})
= 1 + (1/2)\sum_{m=2}^n G_m({\rm e}^{i\theta})r^{m-1},}
\end{array}
\end{equation}
where
\begin{eqnarray}\label{eq6}
F_m({\rm e}^{i\theta}) &=& a_{0m}{\rm e}^{-(m+1)i\theta}
+ \sum_{j+k=m;\ j\ne 0} (a_{jk}+\bar a_{(k+1)(j-1)}){\rm
e}^{(j-k-1)i\theta}
\nonumber \\
&\quad& + \bar a_{0m}{\rm e}^{(m+1)i\theta} \\
G_m({\rm e}^{i\theta}) &=& -ia_{0m}{\rm e}^{-(m+1)i\theta}
+\sum_{j+k=m;\ j\ne 0}(-ia_{jk}+i\bar a_{(k+1)(j-1)}){\rm
e}^{(j-k-1)i\theta}
\nonumber \\
&\quad& + i\bar a_{0m}{\rm e}^{(m+1)i\theta}.\nonumber
\end{eqnarray}
One must find a function
$$
H(r,e^{i\theta}) = r^2 + H_3(e^{i\theta})r^3 + H_4(e^{i\theta})r^4 +
\cdots
$$
such that
\begin{equation}\label{eq7}
{d H\over dt} = \frac{\partial H}{\partial r}\dot r +
\frac{\partial H}{\partial \theta} \dot\theta
= P_1r^4 + P_2r^6 + P_3r^8 + \cdots\,.
\end{equation}
We will see, as Poincar\'e did, that it is in general impossible to find
$H(r,e^{i\theta})$ such that ${dH/dt}=0$, except if the origin is a
center.
In this case, all the constants $P_j$ vanish. We have:
$$\displaylines{
{d H\over dt}
= (F_2 + H_3')r^3
+\left({3\over 2}H_3F_2 + F_3 + {1\over 2}H_3' G_2
+ H_4'\right)r^4 + \cdots \hfill\cr
+\left({n\over 2}H_nF_2 + \cdots + {3\over 2}H_3F_{n-1} + F_n
+ {1\over 2}H_3' G_{n-1}
+ \cdots + {1\over 2}H_n' G_2 + H_{n+1}'\right)r^{n+1}
\hfill\cr
+ \left({{n+1}\over 2}H_{n+1}F_2 + \cdots + {3\over 2}H_3F_n
+ {1\over 2}H_3' G_n
+ \cdots + {1\over2}H_{n+1}' G_2 +
H_{n+2}'\right)r^{n+2}\hfill\cr
+ \left({{n+2}\over 2}H_{n+2}F_2 + \cdots + {4\over 2}H_4F_n
+ {1\over 2}H_4' G_n
+ \cdots + {1\over2}H_{n+2}' G_2 +
H_{n+3}'\right)r^{n+3}\hfill\cr
+ \cdots \hfill
}
$$
\paragraph{Notation 1}
Let us denote the coefficient of $r^k$ in the previous expression
by $L_k({\rm e}^{i\theta})+H'_k$.
\begin{prop}
Let $m$ be the smallest integer such that $P_m\ne 0$. Then the system
of equations $L_k({\rm e}^{i\theta})+H'_k=0$ ($3\le k\le 2m+1$)
with the unknowns $H_k$ has a solution. $H_k$ has only powers of
${\rm e}^{i\theta}$ of the same parity as $k$. There is no $H_{2m+2}$
such
that
$L_{2m+2}({\rm e}^{i\theta})+H'_{2m+2}=0$.
\end{prop}
\paragraph{Proof} In the sequel, we will say simply powers instead of
powers of ${\rm e}^{i\theta}$. If we can find $H_k'$, then $H_k$ and
$H_k'$ ($k\ge 3$) have the same powers. From (6) we see that
$F_j$ and $G_j$ ($j\ge 2$)
have (only) powers of the parity opposite to that of $j$. Since
$H'_3=-F_2$,
$H'_3$ and $H_3$ have odd powers. Up to constants,
the terms in $L_4$ are $H_3F_2$, $F_3$ and $H'_3G_2$, where the
powers in $H_3$, $F_2$, $H'_3$ and $G_2$ are odd.
Then $L_4$ has even powers.
The coefficient of ${\rm e}^{0i\theta}$ in $L_4$ is $P_1$. If
$P_1=0$, we can find $H_4({\rm e}^{i\theta})$ such that
$L_4({\rm e}^{i\theta})+H'_4=0$; in this case $H_4$ has even
powers. If $P_1\ne 0$, it is impossible to solve the equation.
Let $m\ge 2$. We proceed by induction. Let us suppose that it is
possible
to solve the equations $L_k({\rm e}^{i\theta})+H'_k=0$ up to $k=2m$
and that the powers in $H'_k$ and $H_k$ have the same parity as $k$.
Up to constants, the terms in $L_k$ are of the form
$H_rF_s$, $F_{k-1}$ and $H'_rG_s$, where $r+s=k+1$.
If $k=2m+1$ is odd, then $F_{k-1}$ has
odd powers. Since $r+s$ is even, $s$ and $r$ have the same parity and
the powers in $H_rF_s$ and $H'_rG_s$ are odd. We conclude that
$L_{2m+1}({\rm e}^{i\theta})+H'_{2m+1}=0$ has a solution and
that $H'_{2m+1}$ and $H_{2m+1}$ have odd powers. Similar arguments
show that, when $k=2m+2$, $F_{k-1}$, $H_rF_s$ and $H'_rG_s$ have
even powers; then $L_{2m+2}({\rm e}^{i\theta})+H'_{2m+2}=0$ has a
solution
if and only if $P_m$, the coefficient of ${\rm e}^{0i\theta}$ in
$L_{2m+2}$,
is zero. If $P_m=0$, then $H'_{2m+2}$ and $H_k$ have even powers.
\hfill\qed
\begin{coro}
Let $m$ be the smallest integer such that $P_m\ne 0$. Then the function
$r^2 + H_3(\theta)r^3 + \cdots + H_{2m+1}(\theta)r^{2m+1}$, i.e., the
solution
of the system of equations $L_k({\rm e}^{i\theta})+H'_k=0$ ($3\le k\le
2m+1$),
is a Lyapunov function for the focus. If $P_m<0$, the focus is
stable. Otherwise it is unstable.
\end{coro}
To find the Poincar\'e-Lyapunov polynomials we proceed as follows.
Equating ${dH/dt}$ with the right hand side of (\ref{eq7}), we get an
infinite set of differential equations with the unknowns $H_j$
($j\ge 3$) and $P_k$ ($k\ge 1$), where $P_k$ is the coefficient of
${\rm e}^{0i\theta}$ in $L_{2k+2}$.
If $n=2k$ is even, the system is:
\begin{eqnarray}\label{eq8}
H_3' &=& -F_2 \nonumber\\
H_4' &=& P_1- {3\over2}H_3F_2 + F_3 - {1\over2}H_3' G_2 \\
& &\hskip -20pt\cdots \nonumber \\
H_{2k+1}' &=&\hskip -1pt
-{2k\over2}H_{2k}F_2 -\cdots-{3\over2}H_3F_{2k-1} - F_{2k}
- {1\over2}H_3' G_{2k-1} - \cdots - {1\over2}H_{2k}' G_2 \nonumber \\
H_{2k+2}'&=& P_k-{{2k+1}\over2}H_{2k+1}F_2
- \cdots -{3\over2}H_3F_{2k} \nonumber \\
& &\qquad\qquad\qquad\qquad\qquad\qquad
- {1\over2}H_3' G_{2k} - \cdots - {1\over2}H_{2k+1}' G_2 \nonumber \\
& &\hskip -20pt\cdots \nonumber
\end{eqnarray}
If $n=2k-1$ is odd, the last lines become:
\begin{eqnarray}\label{eq9}
H_{2k+1}' &=& - {2k\over2}H_{2k}F_2 - \cdots - {3\over2}H_3F_{2k-1}
- {1\over2}H_3' G_{2k-1} - \cdots - {1\over2}H_{2k}' G_2 \nonumber \\
H_{2k+2}'&=& P_k-{{2k+1}\over2}H_{2k+1}F_2 - \cdots
-{4\over2}H_4F_{2k-1} \\
& &\qquad\qquad\qquad\qquad\qquad\qquad
- {1\over2}H_4' G_{2k-1}-\cdots - {1\over2}H_{2k+1}' G_2 \nonumber \\
& &\hskip -20pt\cdots \nonumber
\end{eqnarray}
Poincar\'e used the sine
and the cosine functions instead of ${\rm e}^{i\theta}$.
\section{ The Main Result}
Letting $z=\alpha w$ ($\alpha = \rho {\rm e}^{i\psi}$), the vector field
(\ref{eq4}) becomes (writing just one equation):
$$
\dot w = iw + \sum_{m=2}^n\sum_{j+k=m} a_{jk}\alpha^{j-1}\bar a^kw^j\bar
w^k.
$$
Then we obtain:
\begin{lemma}
Under the action of the element $\rho e^{i\psi}$ of the group ${\mathbb
C}^*$,
$a_{rs}$ and $\bar a_{rs}$,
where $r+s=m$, are respectively changed to
$a_{rs}\rho^{m-1}{\rm e}^{(r-s-1)i\psi}$ and\newline
$\bar a_{rs}\rho^{m-1}{\rm e}^{(s-r+1)i\psi}$.
\end{lemma}
\paragraph{Definition}
Let $c\in{\mathbb C}$ be a constant. If $r+s=m$, the weight of
$ca_{rs}$ or $c\bar a_{rs}$ with respect to $\rho$ is $m-1$ .
The respective weights of $ca_{rs}$ and $c\bar a_{rs}$ with respect to
$\psi$ are $r-s-1$ and $s-r+1$.
\begin{lemma}\label{lem4}
Let $c\in{\mathbb C}$ be a constant. Each $ca_{rs}$ or $c\bar a_{rs}$ in
$F_m$
and $G_m$ (see (\ref{eq6})) have a weight with respect of $\rho$
equal to $m-1$. The weight with respect to $\psi$ of each monomial
in the coefficient of ${\rm e}^{ti\theta}$ is $t$.
\end{lemma}
\paragraph{Proof} Because $j+k=m$ ($j,k\ge 0$), $(k+1)+(j-1)=m$ ($j\ne
0$) and $0+m=m$,
equation (6) implies that
the weights with respect to $\rho$ of $ca_{jk}$, $c\bar a_{(k+1)(j-1)}$
and $c\bar a_{m0}$ in $F_m$ and $G_m$ are indeed equal to $m-1$.
The weight with respect to $\psi$ of
$ca_{jk}$ is $j-k-1$, that of $c\bar a_{(k+1)(j-1)}$ ($j\ne 0$),
$(j-1)-(k+1)+1=j-k-1$ and that of $c\bar a_{0m}$, $m-0+1=m+1$.
\hfill\qed \medskip
Since each monomial in the coefficient of ${\rm e}^{si\theta}$ has the
same weights, we can, without ambiguity, talk about of the
{\it weights of this coefficient\/}.
The following notation will help to easily determine the weights
of the coefficient of ${\rm e}^{si\theta}$ in $F_m$ and $G_m$.
\paragraph{Notation}
Let us denote the coefficient of ${\rm e}^{si\theta}$ in $F_m$ by
$c{\scriptstyle [m-1,s]}$. The coefficients of the ${\rm
e}^{si\theta}$'s in $G_m$
will be denoted in order by
$$
-ic{\scriptstyle [m-1,-m-1]}, d{\scriptstyle [m-1,-m+1]},\ldots,
d{\scriptstyle [m-1,m-1]}, ic{\scriptstyle [m-1,m+1]}\,.
$$
In the particular case of the family of polynomial vector fields of
degree 3, one gets:
\begin{eqnarray*}
\dot r &=&
{1\over2}\left(c{\scriptstyle [1,-3]}{\rm e}^{-3i\theta} +
c{\scriptstyle [1,-1]}{\rm e}^{-i\theta}
+ c{\scriptstyle [1,1]}{\rm e}^{i\theta} + c{\scriptstyle [1,3]}{\rm
e}^{3i\theta}\right)r^2
\\
& & + {1\over2}\left(c{\scriptstyle [2,-4]}{\rm e}^{-4i\theta}
+ c{\scriptstyle [2,-2]}{\rm e}^{-2i\theta} + c{\scriptstyle [2,0]} +
c{\scriptstyle [2,2]}{\rm
e}^{2i\theta}
+ c{\scriptstyle [2,4]}{\rm e}^{4i\theta}\right)r^3\\
\dot\theta &=&
1+ {1\over2}\left(-ic{\scriptstyle [1,-3]}{\rm e}^{-3i\theta}
+d{\scriptstyle [1,-1]}{\rm e}^{-i\theta}+d{\scriptstyle [1,1]}{\rm
e}^{i\theta}
+ic{\scriptstyle [1,3]}{\rm e}^{3i\theta}\right)r \\
& & +{1\over2}\left(-ic{\scriptstyle [2,-4]}{\rm e}^{-4i\theta}
+d{\scriptstyle [2,-2]}{\rm e}^{-2i\theta} +d{\scriptstyle [2,0]} +
d{\scriptstyle [2,2]}{\rm
e}^{2i\theta}
+ ic{\scriptstyle [2,4]}{\rm e}^{4i\theta}\right)r^2.
\end{eqnarray*}
\begin{lemma}
The following relations are satisfied:
$$
\bar c{\scriptstyle [m-1,s]}= c{\scriptstyle [m-1,-s]}\ \hbox{and}\
\bar d{\scriptstyle [m-1,s]} = d{\scriptstyle [m-1,-s]}.
$$
Moreover, $c{\scriptstyle [m-1,0]}$ and $d{\scriptstyle [m-1,0]}$ are
real.
\end{lemma}
\paragraph{Proof} $F_m$ and $G_{m}$ are real expressions, since the
original family of
vector fields is real. Because in (\ref{eq4}), $\dot z\bar z+z\dot{\bar
z}$
and $-i\dot z\bar z +iz\dot{\bar z}$ are sums of conjugate terms,
$F_m$ and $G_m$ are are also sums of conjugate terms. Precisely,
the conjugate of the coefficient of ${\rm e}^{si\theta}$
is the coefficient of the
conjugate of ${\rm e}^{si\theta}$. Then
$\bar c{\scriptstyle [m-1,s]}=c{\scriptstyle [m-1,-s]}$ and
$\bar d{\scriptstyle [m-1,s]}=d{\scriptstyle [m-1,-s]}$.
When $s=0$, the terms $c{\scriptstyle [m-1,0]}$ and $d{\scriptstyle
[m-1,0]}$ are
self-conjugate, and therefore real.
\hfill\qed
\paragraph{Definition}
Let $h$ be a monomial in the unknowns $c{\scriptstyle [j,s]}$ and
$d{\scriptstyle [k,t]}$.
The weights of $h$
with respect to $\rho$ and $\psi$ are the sums of the respective weights
of
its unknowns. We will say that a polynomial is weighted-homogeneous of
degree
$(k,r)$ if all its monomials have the same weights $k$ and $r$ with
respect
to $\rho$ and $\psi$ respectively.
\begin{prop}
Let $Q_t$ be the coefficient of ${\rm e}^{ti\theta}$ in $H_s$. Then
$Q_t$ is
weighted-homogeneous of degree $(s-2,t)$. $P_k$ is weighted-homogeneous
of
degree $(2k,0)$.
\end{prop}
\paragraph{Proof} Let us look at the system of equations (\ref{eq8}) or
(\ref{eq9}).
According to lemma~\ref{lem4} and the paragraph following it,
the statement is true for all the coefficients $Q_t$ in $H_3$,
since $H_3' = - F_2$. Since $H_s'=-L_s$ (see notation 1),
the result follows by induction.
\begin{coro}
$P_k$ is invariant under the group of rotations $S^1$ and is invariant
under
the group $C^*$ modulo a positive constant.
\end{coro}
\section{ Conclusion}
We have proved not only that $\forall j\ge 1$ and for
$g=\rho{\rm e}^{i\psi}\in {\mathbb C}^*$,
$P_j(g(a_{rs}))=\rho^{2j}P_j(a_{rs})$,
where $P_j$ is a Poincar\'e-Lyapunov polynomial, but
also that $P_j$ is weighted-homogeneous of degree $(2j,0)$
(according to definition 3).
This result has at least two goals.
New directions of research related to Hilbert's $16^{\rm th}$ problem
which look
promising
have been given by H. Zoladek in [3] and [4]. One of the questions
raised by
the Hilbert's $16^{\rm th}$ problem
is about the maximum number of limit cycles that exist in the family of
polynomial vector fields of degree less or equal to
$n$. A minor question, but closely related to, is to determine the
maximum
number of limit cycles near a center-focus.
Zoladek proved in [3] that the family of polynomial vector fields of
degree
less or equal to
two has at most 3 limit cycles near a center-focus.
In [4], he proved that a family of degree less or equal to
three, but without its quadratic part, has at most 5
limit cycles near a center-focus.
The proofs follow from his main result that says
the ideal generated by the Poincar\'e-Lyapunov polynomials is a linear
combination, with polynomial coefficients in
the $a_{rs}$, of the first Poincar\'e-Lyapunov polynomials. He utilizes
for
it the invariance of the Poincar\'e-Lyapunov polynomials under the group
of
rotations, but the arguments for proving the invariance, though correct,
are
rather elliptic. The present article gives a detailed proof.
One knows the importance of the Poincar\'e-Lyapunov polynomials to
determine
the stability of an equilibrium point.
One could hope to find the Poincar\'e-Lyapunov polynomials for certain
low degree polynomial vector fields. Indeed, using a computer, one could
list all the monomials of $P_j$, since they must satisfy the (two)
homogeneity condition(s). Using the explicit system (8) or (9),
one could find the coefficients of the monomials.
\paragraph{\bf Remark} The author has received from
J.P. Fran\-\c coi\-se, C. Rousseau and R. Roussarie the main arguments
of
another proof of the invariance of the Poincar\'e-Lyapunov polynomials
under the group of rotations. They do not have a result on the
homogeneity
with respect to the weights.
\begin{thebibliography}{0}
\bibitem{[1]} A. A. Andronov {\it et al}, Theory of Bifurcations of
Dynamic
Systems on a plane, {\it John Wiley \& Sons\/}, 1973.
\bibitem{[2]} H. Poincar\'e, Oeuvres de Poincar\'e,
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\bibitem{[3]} H. Zoladek, Quadratic systems with center and their
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\end{thebibliography}
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{\sc Pierre Joyal}\\\
D\'epartement d'informatique et de math\'ematique\\
Universit\'e du Qu\'ebec \`a Chicoutimi\\
555 boul. de l'Universit\'e, Chicoutimi, G7H 2B1, Canada\\
E-mail address: Pierre\_Joyal@uqac.uquebec.ca
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