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\markboth{\hfil saddle conic of quadratic planar differential systems\hfil EJDE--2000/23}
{EJDE--2000/23\hfil D. Boularas \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol.~{\bf 2000}(2000), No.~23, pp.~1--9. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)}
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A note on the saddle conic of quadratic planar differential systems
\thanks{ {\em Mathematics Subject Classifications:} 34C05, 34C20.
\hfil\break\indent
{\em Key words and phrases:} nonlinear differential systems, saddle points,
\hfil\break\indent
qualitative theory of ordinary differential systems, critical points,
invariant theory.
\hfil\break\indent
\copyright 2000 Southwest Texas State University and University of
North Texas. \hfil\break\indent
Submitted December 16, 1999. Published March 28, 2000.} }
\date{}
%
\author{D. Boularas}
\maketitle
\begin{abstract}
We give some properties of the saddle conic of quadratic differential
systems. We also deduce semi-algebraic conditions for the existence
of one, two or three saddle points (in terms of affine invariants).
\end{abstract}
\newtheorem{theorem}{Theorem}
\newtheorem{prop}{Proposition}
\newtheorem{lem}{Lemma}
\section{Motivations and introduction}
As shown in the report by Reyn \cite{reyn92}, many publications
are devoted to the qualitative analysis of the planar quadratic differential
system
\begin{eqnarray}\label{sdq}
&\frac{dx}{dt} = a_{0,0}+ a_{1,0}x+a_{0,1}y +
a_{2,0}x^2 +a_{1,1}xy+a_{0,2}y^2& \\
&\frac{dy}{dt} = b_{0,0}+ b_{1,0}x+b_{0,1}y +
b_{2,0}x^2 +b_{1,1}xy+b_{0,2}y^2 ,&\nonumber
\end{eqnarray}
where $ a_{i,j}$ and $ b_{i,j}$ are real-valued coefficients.
Note that these systems form a $12$-dimensional linear space, denoted by
${\cal A}$, and that a complete qualitative study requires determining the
partition of the phase plane into trajectories. Following
Leontovitch \& Maier \cite{LeonMa},
this partition is completely defined by the number and the nature of
critical points, the separatrix structure, and the location of closed
trajectories (the famous 16th Hilbert problem). This work deals
with the first of these three questions.
In \cite{vulpfin,vulpinf}, Baltag and Vulpe established a complete tableau of
the number and multiplicity of the critical points in the plane, including those
at infinity. Their conditions are algebraic and semi-algebraic (equalities and
inequalities) given in terms of center-affine invariants and covariants.
From the symbolic computation point of view, this approach is very interesting
since it provides a simple algorithm giving the number and the multiplicity of
critical points without solving (with radicals) algebraic equations.
The research on the nature of critical points falls into two
different categories. The first direction concerns the famous
center-focus problem. Based on Dulac's early work (1908) and on
Kapteyn's work (1912), a series of contributions has been presented
(see \cite{sib1} for the evolution of this
question). Finally, explicit conditions (in terms of invariants and
covariants) for finding systems (\ref{sdq}) with one or
two centers were obtained in \cite{driss91}.
The second direction is centered around the question of the coexistence of
critical points of different types. It was initiated by Berlinskii
\cite{ber} who established, among others, Theorem ~\ref{thber}
below. However, he did not characterize the possible situations by
algebraic or semi-algebraic conditions on the coefficients of (\ref{sdq}).
P. Curtz gave the first set of sufficient conditions expressed in terms of
coefficients of systems (\ref{sdq}) for the existence of saddle
points.
The study of the second direction problem leads us to consider the
determinant of the Jacobian of the vector field associated to the system
(\ref{sdq}),
$$q(x, y)= \left|
\begin{array}{cc}
a_{1,0}+2a_{2,0}x +a_{1,1}y & a_{0,1}+a_{1,1}x +2a_{0,2}y\\
b_{1,0}+2b_{2,0}x +b_{1,1}y & a_{0,1}+b_{1,1}x +2b_{0,2}y\\
\end{array}
\right| .$$
It is clear that a critical point $(x_0, y_0)$ is a saddle point if
and only if $q(x_0, y_0) < 0$. We call the algebraic curve $q(x,
y) = 0$ the saddle conic because it induces a partition of the phase
plane into three regions characterized by the relations
$q(x_0, y_0) < 0$, $q(x_0, y_0) >0$ and $q(x_0, y_0) = 0$
and the first one contains the saddle
points and the second one, the anti-saddle points.
In this note, we establish some algebraic and geometric properties of
the polynomial $q(x,y)$ and give affine conditions for the existence of
one, two, or three saddle points for system (\ref{sdq}).
All computations are made with Maple and the package {\bf SIB}
\cite{driss00} which
contains minimal systems of generators of center-affine and affine
covariants of systems (\ref{sdq}).
\section{Review of invariants and covariants of differential systems}
Planar quadratic differential systems with real coefficients form
a ${\mathbb R}$-vector-space of dimension $12$
(precisely, isomorphic to ${\mathbb R} ^2 \oplus {\mathbb R} ^2
\bigotimes ({\mathbb R}^2)^{\star } \oplus S_2 \bigotimes
({\mathbb R} ^2)^{\star }$ where
$({\mathbb R}^2)^{\star }$ is the dual of ${\mathbb R} ^2$ and
$ S_2$ the space of algebraic quadratic forms).
Using Einstein notation, they can be written in the condensed form
\begin{equation}\label{sdqt}
\frac{dx^j}{dt} = a^j + a^j_{\alpha }x^{\alpha } + a^j_{\alpha\beta
}x^{\alpha }x^{\beta }\quad
(j,\alpha , \beta = 1,2).
\end{equation}
where $x= (x^1,x^2)^T \in {{\mathbb R} }^2$ (the letter $T$ means
transposed) and
$a^j_{\alpha}x^{\alpha } = a^j_{1 }x^{1 }+ a^j_{2 }x^{2 }$,
$a^j_{\alpha\beta }x^{\alpha }x^{\beta } = a^j_{11}(x^{1 })^{2 } +
2a^j_{12}x^{1}x^{2} + a^j_{22 }(x^{2})^{2}$. The
Einstein notation will be adopted in the whole paper: We suppress
the symbol $\sum $ (sum) in all contractions.
In addition, let $Aff(2,{\mathbb R})$ be the group of affine
transformations
\begin{eqnarray}\label{gaf}
x \mapsto y = P^{-1}(x - p)
\end{eqnarray}
with
$$ P = \left(\begin{array}{ll}
p^1_1 & p^1_2 \\
p^2_1 & p^2_2
\end{array}\right), \quad \det(P) \neq 0
\quad \mbox{and} \quad p = (p^1,p^2)^T .$$
It acts rationally over ${\cal A }$ following the rational
representation
$$\rho : G \mapsto GL({\cal A }) $$
where $ GL({\cal A }) $ is the group of automorphisms of ${\cal A }$. Putting $\rho
(P,p) (a) = b$, this representation is defined by
the formulae:
$$ \displaylines{
b^j= q^j_i( a^i + a^i_{\alpha }p^{\alpha } + a^i_{\alpha\beta
}p^{\alpha }p^{\beta }) , \cr
b^j_{\alpha }= q^j_ip^{\beta }_{\alpha}( a^i_{\beta } + 2
a^i_{\beta \gamma }p^{\gamma }) , \cr
b^j_{\alpha \beta }= q^j_ip^{\gamma }_{\alpha}
p^{\delta }_{\beta}a^i_{\gamma\delta } ,\cr
}$$
where $Q =(q^j_i)$ is the inverse matrix of $P$.
Let $ {\cal {\mathbb R} }[a, x]$ denote the algebra of polynomials whose
indeterminates are components of a generic vector $a$ of ${\cal
A}\times {\mathbb R} ^2$:
$a^1 , a^2, a^1_1, a^1_2, \ldots , a^1_{22},a^2_{22}, x^1, x^2$.
The representation of the group $Aff(2,{\mathbb R})$ on
$GL({\cal A}\times {\mathbb R}^2)$ is the direct sum of $\rho $ and
$Aff(2,{\mathbb R})$. It is denoted $\mbox{{\cal r}}$.
\paragraph{Definition.}
A polynomial function $K \in {\cal {\mathbb R} }[a, x]$ is said to be
a $Aff(2,{\mathbb R})$-covariant of ${\cal A}$ if there exists a function
$\lambda :Aff(2,{\mathbb R}) \to {\mathbb R} $ such that
$$ \forall g \in G, \quad \left( K \circ \mbox{{\cal r}} \right)
(g) = \lambda (g) . K\, .$$
If $\lambda (g) \equiv 1 $, then the invariant is said absolute.
Otherwise, it is said relative.
An $Aff(2,{\mathbb R})$-invariant is an $Aff(2,{\mathbb R})$-covariant
which does not depend on $x$.
It can be proved \cite{dieu} that the function $\lambda $ is a character
group of $Aff(2,{\mathbb R})$ and equal to $\det (Q)^{-\kappa } $,
where the integer $\kappa $ is called the weight of the covariant
(or invariant).
The above definitions hold for any subgroup of $Aff(2,{\mathbb R})$,
in particular for the center-affine group denoted $Gl(2,{\mathbb R})$
(put in the affine group $p \equiv 0$) or the special
group denoted $Sl(2,{\mathbb R})$ ($ \det (P) = 1 $ and
$p \equiv 0)$.
The sets of $Sl(2,{\mathbb R})$-covariants or invariants and
homogeneous $Gl(2,{\mathbb R})$-covariants (called also center-affine
covariants) or $Gl(2,{\mathbb R})$-invariants
(center-affine invariants) are the same.
The algebras of $Sl(2,{\mathbb R})$-invariants and
$Sl(2,{\mathbb R})$-covariants are finitely generated.
In \cite{driss00} a package denoted {\bf SIB} is elaborated with
Maple. It contains minimal systems of generators of the algebras of
center-affine (denoted $J_1, \ldots , J_{36}$, $K_1, \ldots , K_{33}$)
and affine covariants (denoted by $Q_1, \ldots , Q_{36}$).
\section{Algebraic Properties of the Saddle Conic}
Let us introduce, for the systems ({\ref{sdqt}), the following
quantities:
$$A_{00} = \left|
\begin{array}{cc}
a^1_{1}&a^1_{2}\\
a^2_{1}&a^2_{2}
\end{array}
\right| ,\,
A_{i0} = \left|
\begin{array}{cc}
a^1_{1i}&a^1_{2}\\
a^2_{1i}&a^2_{2}
\end{array}
\right| ,\,
A_{0i} = \left|
\begin{array}{cc}
a^1_{1}&a^1_{2i}\\
a^2_{1}&a^2_{2i}
\end{array}
\right| ,\,
A_{ij} = \left|
\begin{array}{cc}
a^1_{1i}&a^1_{2j}\\
a^2_{1i}&a^2_{2j}
\end{array}
\right| .$$
The saddle conic of ({\ref{sdqt}) has the expression (we represent
here by $x$ the previous vector $(x,y)$):
\begin{eqnarray*}
q(x) &=& A_{00} + 2(A_{10}+A_{01})x^1 + 2(A_{20}+A_{02})x^2 \\
&&+4[A_{11}(x^1)^2 + (A_{12} + A_{21}) x^1x^2 + A_{22}(x^2)^2]\,.
\end{eqnarray*}
It contains all the information about the distribution of saddles and
antisaddles in the phase plane.
Following \cite{driss00}, the polynomial $q(x,y)$ is an affine {\it
absolute } covariant,
$$q(x,y) = \frac{1}{2}(Q_1^2 - Q_2)=
\frac{1}{2}[(J_1^2 - J_2)+4(J_1K_1 - K_3)+4(K_1^2 - K_7)],$$
where $ Q_1$ and $ Q_2$ are affine covariants. Let us consider its two
discriminants,
\begin{eqnarray} \label{disc1}
&4\delta _1 = \left|
\begin{array}{cc}
4A_{11} &2(A_{12}+A_{21})\\
2(A_{12}+A_{21})&4A_{22}
\end{array}
\right| , &\\
\label{disc2}
& 2\delta _2 = \left|
\begin{array}{ccc}
4A_{11} &2(A_{12}+A_{21})&A_{01}+A_{10}\\
2(A_{12}+A_{21})&4A_{22} &A_{02}+A_{20}\\
A_{01}+A_{10} &A_{02}+A_{20} &A_{00}
\end{array}
\right| . &
\end{eqnarray}
With the help of package {\bf SIB}, we obtain the affine invariants
$$\displaylines{
2\delta _1 = 2J_7 - J_8 - J_9,\cr
\delta _2 = 4J_1(J_{12} - J_{11}) - J^2_1(J_8 + J_9 - 2J_7) +
2J_2(J_9 - J_7) + 2(J_4 - J_5)(2J_3 - J_4 - J_5).
}$$
The total degree of $\delta _1$ is $4$ and that of $\delta _2 $ is
$6$.
\begin{lem}[\cite{sib2}, p. 56]
The quadratic homogeneous parts of the equations $(\ref{sdqt})$ have
a common factor if, and only if, $\delta _1 = 0$.
\end{lem}
\paragraph{Proof.} The resultant of the polynomials $a_{11}^1(x^1)^2 +
2a_{12}^1x^1x^2 \, +\, a_{22}^1(x^2)^2$ and $a_{11}^2(x^1)^2 +
2a_{12}^2x^1x^2 \, +\, a_{22}^2(x^2)^2$ is equal to $\delta _1$.
\begin{lem}
The differential system (\ref{sdqt}) can be reduced by a rotation into the form
\begin{eqnarray*} \label{sql}
\frac{dx^1}{dt}& = &a^1 + a^1_{\alpha }x^{\alpha },\\
\frac{dx^2}{dt}& = &a^2 + a^2_{\alpha }x^{\alpha } +
a^2_{\alpha\beta }x^{\alpha }x^{\beta } \nonumber
\end{eqnarray*}
if, and only if, $A_{11}(x^1)^2 + (A_{12} + A_{21}) x^1x^2 +
A_{22}(x^2)^2= K^2_1 - K_7 = 0$.
\end{lem}
\paragraph{Proof.} The necessary condition is trivial. Suppose that
$A_{11}(x^1)^2 + (A_{12} + A_{21}) x^1x^2 + A_{22}(x^2)^2 = 0$.
That means that
$$ \left| \begin{array}{cc}
a^1_{11}&a^1_{12}\\
a^2_{11}&a^2_{12}
\end{array}\right|(x^1)^2 + \left| \begin{array}{cc}
a^1_{11}&a^1_{22}\\
a^2_{11}&a^2_{22}
\end{array}\right|x^1x^2 + \left| \begin{array}{cc}
a^1_{12}&a^1_{22} \\
a^2_{12}&a^2_{22}
\end{array}\right|(x^2)^2= 0 .$$
Consequently, there exists two real constants $k_1$ and $k_2$ such that
$k_1^2+k_2^2= 1$ and $k_1a^1_{\alpha\beta }x^{\alpha
}x^{\beta }+k_2a^2_{\alpha\beta }x^{\alpha }x^{\beta }=
0$.
Then the rotation $X^1:= k_1x^1+ k_2x^2$, $X^2:= - k_2x^1+
k_1x^2$ leads the initial system to the sought form.
From this lemma it follows the proposition:
\begin{prop}
If the system (\ref{sdqt}) has four isolated critical points
(real or complex), then $K^2_1 - K_7 \neq 0$.
\end{prop}
\section{Geometric Properties of the Saddle Conic}
Suppose that(\ref{sdqt}) has four isolated critical points. By
\cite{cop}, any three of these points are never into the same
straight line. Then
it is possible to find an affine transformation of the plane, denoted
$\Phi $ such that
the points $(0,0)^T = O$, $(0,1)^T = A$,
$(1,0) = B$, and $(c,d) = D$ become critical for the transformed
system
\begin{eqnarray}\label{sdqr}
\frac{dy^i}{dt} = b^i + b^i_{\alpha }y^{\alpha } + b^i_{\alpha\beta
}y^{\alpha }y^{\beta }\quad
(i,\alpha , \beta = 1,2)
\end{eqnarray}
whose coefficients verify the relations:
\begin{eqnarray}\label{relations}
&b^{i}= 0\,,\quad b^{i}_{1}= -b^i_{11}\,, \quad b^i_2 = -
b^i_{22}\,,\quad (i = 1,2)& \\
&b^i_{11}c(c-1)+2b^i_{12}cd+b^i_{22}d(d-1) = 0,\quad (i = 1,2).&\nonumber
\end{eqnarray}
Moreover, $cd \neq 0$ and $c+d- 1 \neq 0$.
Let $B_{ij}$, $(i,j = 1,2)$ be the transformed quantities of
$A_{ij}$ and ${\tilde \delta }_{1}, {\tilde \delta }_{2}$ the
expressions of $ \delta _1$ and $ \delta _2$ where the $A_{ij}$ are replaced
by $B_{ij}$.
Since the affine invariants $ \delta _1$ and $ \delta _2$ are relative
and of weight 2, we have
$$ {\tilde \delta }_{1} = \Delta ^{-2}\delta _{1} \quad
\mbox{and} \quad {\tilde\delta }_{2} = \Delta ^{-2}\delta _{2}\,,$$
where $\Delta $ is the determinant of the linear part of $\Phi $.
\paragraph{Remark}
The signs of the affine invariants $ \delta _1$ and $ \delta _2$ do not
change under the affine transformation of the plane.
We have arrived at the interesting geometrical fact.
\begin{lem}\label{quad}
The quadrilateral whose vertices are the four isolated singular
points of the quadratic system (\ref{sdqt}) is convex (resp. not convex) if and
only if $\delta _1 = 2J_7 - J_8 - J_9 < 0$
(resp. $\delta _1 >0$).
\end{lem}
\paragraph{Proof.} Note that the
quadrilateral is not convex if, and only if, one vertex lies in the
triangle formed by other vertices. For
systems (\ref{sdqr} -\ref{relations}), the vertices $O,\,
A,\, B$ being fixed, the quadrilateral is convex if and only if
$(c+d-1)cd > 0$.
Taking into account the relations (\ref{relations}) we obtain
$$ \widetilde {\delta }_1
= -2\frac{(c+d-1)(b^1_{11}b^2_{22}-b^2_{11}b^1_{22})^2}{c d} \,. $$
Moreover, $B_{12}= b^1_{11}b^2_{22}-b^2_{11}b^1_{22} \neq 0$,
because $K^2_1 - K_7 \neq 0$. This completes the proof.
\smallskip
A saddle point is an elementary critical point whose corresponding linearized
system admits real eigenvalues of opposite signs. Its geometrical
index is equal to $-1$. All other elementary critical points (nodes,
center and foci) of geometrical index $+1$ are called
anti-saddles.
To know whether a given critical point $x_0$ is a saddle or not we
have to compute the determinant of the linearized part around the
considered point, i.e., $q(x_0)$: $x_0$ is a saddle if and only if
$q(x_0)< 0$. In the case of four isolated critical points we get the
following result which was established the first time by Berlinski
\cite{ber}.
\begin{theorem}\label{thber}
Suppose that there are four real critical points. If the
quadrilateral with vertices at the points is convex then two opposite
critical points are saddles and the other two are antisaddles. But
if the quadrilateral is not convex then either the three exterior
vertices are saddles and the interior antisaddle or the exterior
vertices are antisaddles and the interior vertex a saddle.
\end{theorem}
\begin{figure}[ht]
\begin{center}
\epsffile{fig1.ps}
\end{center}
\caption{Case with two anti-saddle points}
\end{figure}
\begin{figure}[ht]
\begin{center}
\epsfxsize=\hsize
\epsffile{fig2.ps }
\end{center}
\caption{Case with one and with three saddle points}
\end{figure}
\paragraph{Proof \cite{cop}.} After substitution $x_0$ by critical points $O$, $A$,
$B$ and $D$ in (\ref{sdqr} - \ref{relations}), we obtain:
$$ \displaylines{
q(0) = B_{12}, \quad q(A) = -\frac{(c+d-1)B_{12}}{d}, \cr
q(B) = - \frac{(c+d-1)B_{12}}{c}, \quad q(D) = (c+d-1)B_{12}\,.\cr
}$$
Consequently,
$$q(0)q(A)q(B)q(D)= \frac{(c+d-1)^3B_{12}^4}{cd}\,.$$
If ${\tilde \delta }_1 < 0$, the quadrilateral OABD is
convex and $q(0)q(A)q(B)q(D) > 0$. There are three
possibilities: zero, two, or four saddles. We shall show that the first
and third cases cannot hold.
If $c + d - 1 < 0$, $q(0)$ and $q(D)$ have opposite
sign. Then, there exists at least one saddle point and one
anti-saddle point.
If $c + d - 1 > 0$ and taking into account the inequality
$(c+d-1)/(cd) > 0$, we have necessarily $cd > 0$. Because
$c + d - 1 > 0$, this implies that $c > 0$ and $d >0$. Thus,
the quantities $q(0)$ and $q(A)$ are of opposite signs.
If ${\tilde \delta }_1 >0$, then the quadrilateral OABD is not
convex and $q(0)q(A)q(B)q(D)< 0$. This implies that there
exists either one or three saddle points.
\\[2mm]
Using the Poincar\'e's index of vectors fields
around critical points another simplified
proof of this theorem was proposed in \cite{ses}.\\
Actually, the second discriminant of the saddle conic may distinguish
between the cases of one and three saddle points.
\begin{theorem}
Suppose that the differential system (\ref{sdqt}) admits four real
isolated critical points. Then
\begin{itemize}
\item (\ref{sdqt}) has one saddle point if, and only if, $\delta _1 > 0$ and
$\delta _2 < 0$,
\item (\ref{sdqt}) has two saddle points if, and only if, $\delta _1 < 0$,
\item (\ref{sdqt}) has three saddle points if, and only if, $\delta _1 > 0$ and
$\delta _2 > 0$.
\end{itemize}
\end{theorem}
\paragraph{Proof.} For the systems (\ref{sdqr}), (\ref{relations}),
we have
$$\widetilde {\delta }_2
= \frac{(c+d-1)(c+d)(c-1)(d-1)(b^1_{11}b^2_{22}-b^2_{11}b^1_{22})^3}
{c^2 d^2} \,. $$
Suppose that ${\tilde \delta }_1 >0$; i.e.,
$(c+d-1)/(cd) < 0$ and $0$ is a
saddle point: if $q(D)$ is of negative sign, then $c + d - 1 > 0$ and
$c$ and $d$ are of opposite sign. Thus, one of the points $A$ or $B$
is of saddle type. Without loss of generality, we can suppose that $A$
is a saddle point. Then $c <0$, $d > 1$ and
$c+d > 1 >0$. Necessarily, $\widetilde {\delta }_2 > 0$.
Suppose that $q(D)$ is of positive sign, then $c + d - 1 < 0$ and
$c$ and $d$ have the same sign. If $c$ and $d$ are of negative sign,
then there are three antisaddle points and $\widetilde {\delta }_2 <0$.
If $c$ and $d$ are of positive sign, then $A$ and $B$ are of
saddle type and $0< c <1$, $0< d <1$. There are three saddle points
and $\widetilde {\delta }_2 >0$.
This result is partially obtained in \cite{cur1,cur2}.
\paragraph{Acknowledgments} The author wishes to express his sincere
thanks to the anonymous referees for carefully reading the manuscript.
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}\end{thebibliography}
\noindent{\sc Driss Boularas }\\
LACO, D\'epartement de Math\'ematiques \\
Facult\'e des Sciences, Universit\'e de Limoges,\\
123, Avenue A. Thomas, 87060, Limoges, France \\
email: boularas@alpha1.unilim.fr
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