\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{graphicx}
\usepackage{epic}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 314, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2015/314\hfil Global phase portraits]
{Global phase portraits of quadratic systems with an ellipse and a
straight line as invariant algebraic curves}
\author[J. Llibre, J. Yu \hfil EJDE-2015/314\hfilneg]
{Jaume Llibre, Jiang Yu}
\address{Jaume Llibre \newline
Departament de Matem\`atiques, Universitat Aut\`onoma
de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain}
\email{jllibre@mat.uab.cat}
\address{Jiang Yu (corresponding author) \newline
Department of Mathematics, Shanghai Jiao Tong University,
Shanghai 200240, China}
\email{jiangyu@sjtu.edu.cn}
\thanks{Submitted September 4, 2015. Published December 24, 2015.}
\subjclass[2010]{34C05}
\keywords{Quadratic system; first integral; global phase portraits;
\hfill\break\indent invariant ellipse; invariant straight line}
\begin{abstract}
In this article we study a class of integrable quadratic systems
and classify all its phase portraits. More precisely, we
characterize the class of all quadratic polynomial differential
systems in the plane having an ellipse and a straight line as
invariant algebraic curves. We show that this class is integrable
and we provide all the different topological phase portraits that
this class exhibits in the Poincar\'{e} disc.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks
\section{Introduction and statement of main results}
A \emph{planar polynomial differential system} is a differential
system of the form
\begin{equation}\label{eq0}
\begin{gathered}
\dot x= P(x,y), \\
\dot y= Q(x,y),
\end{gathered}
\end{equation}
where $P$ and $Q$ are real polynomials. We say that the polynomial
differential system \eqref{eq0} has degree $n$, if $n$ is the
maximum of the degrees of the polynomials $P$ and $Q$. Usually a
\emph{polynomial differential system of degree $2$} is denoted simply
as a \emph{quadratic system}. The dot in \eqref{eq0} denotes
derivative with respect to the independent variable $t$.
Let $U$ be a dense and open subset of $\mathbb{R}^2$. A non-locally
constant function $H: U\to \mathbb{R}$ is a \emph{first integral} of the
differential system \eqref{eq0} if $H$ is constant on the orbits of
\eqref{eq0} contained in $U$, i.e.
\[
\frac{dH}{dt}= \frac{\partial H}{\partial x}(x,y) P(x,y)+\frac{\partial H}{\partial y}(x,y)
Q(x,y)=0
\]
in the points $(x,y)\in U$. We say that a quadratic system is \emph{
integrable} if it has a first integral $H: U\to \mathbb{R}$.
Quadratic systems have been studied intensively, and more than one
thousand papers have been published about these polynomial
differential equations of degree $2$, see for instance the
references quoted in the books of Ye \cite{Ye, Ye2} and Reyn
\cite{Reyn1}. But the problem of classifying all the integrable
quadratic system remains open.
For a quadratic system the notion of integrability reduces to the
existence of a first integral, so the following natural question
arises:
\begin{quote}
Given a quadratic system, how to recognize if it has a
first integral? or
Given a class of quadratic systems
depending on parameters, how to determine the values of the
parameters for which the system has a first integral?
\end{quote}
At this moment these questions do not have a good answer.
Many classes of integrable quadratic systems have been studied, and
for them all the possible global topological phase portraits have
been classified. One of the first of these classes studied was the
classification of the quadratic centers and their first integrals
which started with the works of Dulac \cite{Du}, Kapteyn \cite{K1,
K2}, Bautin \cite{Ba}, Lunkevich and Sibirskii \cite{LS}, Schlomiuk
\cite{Sc}, \.{Z}o\l\c{a}dek \cite{Zo}, Wei and Ye \cite{YY}, Art\'{e}s,
Llibre and Vulpe \cite{ALV}. The class of the homogeneous
quadratic systems, see Lyagina \cite{LY}, Markus \cite{M}, Korol
\cite{KOR}, Sibirskii and Vulpe \cite{SV}, Newton \cite{NW}, Date
\cite{DA} and Vdovina \cite{VD},\ Another class is the one formed
by the Hamiltonian quadratic systems, see Art\'{e}s and Llibre
\cite{AL}, Kalin and Vulpe \cite{KV} and Art\'{e}s, Llibre and Vulpe
\cite{ALV}.
In this article we study a new class of integrable quadratic
systems and classify all its phase portraits. More precisely we
analyze the class of all quadratic polynomial differential systems
having an ellipse and a straight line as invariant algebraic curves.
Our first result is to provide a normal form for all quadratic
polynomial differential systems having an ellipse and a straight
line as invariant algebraic curves.
\begin{theorem}\label{t1}
A planar polynomial differential system of degree 2 having an
ellipse and a straight line as invariant algebraic curves, after an
affine change of coordinates, can be written as
\begin{equation}\label{pe1}
\begin{aligned}
\dot x&=-cy(x-r), \\
\dot y&=C(x^2+y^2-1)+cx(x-r),
\end{aligned}
\end{equation}
where $c, C \in \mathbb{R}$.
\end{theorem}
This theorem is proved in section \ref{s2}.
In the next result we present the first integrals of the polynomial
differential system of degree $2$ having an ellipse and a straight
line as invariant algebraic curves.
\begin{theorem}\label{t2}
The quadratic polynomial differential systems \eqref{pe1} have the
following first integrals:
\begin{itemize}
\item[(a)] $H= x^2+y^2$ if $C=0$ and $c\ne 0$;
\item[(b)] $H=x$ if $C\ne 0$ and $c=0$;
\item[(c)] $H= (x-r)^{2C/c} ( x^2 + y^2-1)$ if $Cc\ne 0$.
\end{itemize}
Moreover, the quadratic polynomial differential systems \eqref{pe1}
have no limit cycles.
\end{theorem}
This theorem is proved in section \ref{s2}.
\begin{figure}[htb]
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(120,170)(1,2)
\put(0,0){\includegraphics[width=120mm]{fig1}} % tu1.eps
\put(3.6,141){$C=0$, $c\neq 0$}
\put(10.5,137){$r>0$}
\put(35,141){$C=0$, $c\neq 0$}
\put(40,137){$r=0$}
\put(66,141){$C\neq 0$, $c=0$}
\put(74,137){$L_9$}
\put(103,141){$R_1$}
\put(13,105){$R_2$}
\put(41,105){$R_3,R_5$}
\put(75,105){$R_4$}
\put(98,105){$R_6,R_7,L_4$}
\put(13,72){$R_8$}
\put(44.5,72){$L_1$}
\put(72.5,72){$L_2,L_3$}
\put(105,72){$L_5$}
\put(13,37){$L_6$}
\put(44.5,37){$L_7$}
\put(76,37){$L_8$}
\put(107,37){$p_1$}
\put(46,2){$P_2$}
\put(76,2){$P_3$}
\end{picture}
\end{center}
\caption{Phase portraits of systems \eqref{pe1}} \label{fig1}
\end{figure}
In the next theorem we present the topological classification of all
the phase portraits of planar polynomial differential system of
degree $2$ having an ellipse and a straight line as invariant
algebraic curves in the Poincar\'{e} disc. For a definition of the
Poincar\'{e} compactification and Poincar\'{e} disc see section \ref{s3},
and for a definition of a topological equivalent phase portraits of
a polynomial differential system in the Poincar\'{e} disc see sections
\ref{s3} and \ref{s4}.
\begin{theorem}\label{t3}
Given a planar polynomial differential system of degree $2$ having
an ellipse and a straight line as invariant algebraic curves its
phase portrait is topological equivalent to one of the $18$ phase
portraits of Figure \ref{fig1}.
\end{theorem}
This theorem is proved in section \ref{s5}.
\section{Proofs of Theorems \ref{t1} and \ref{t2}} \label{s2}
Suppose that a polynomial differential system in the plane has an
invariant ellipse and an invariant straight line. Then, first we do
an affine transformation that changes the ellipse to a circle and of
course the straight line to another straight line, second we
translate the center of the circle to the origin of coordinates,
third we rescale the coordinates in order that the circle has radius
one, and finally we rotate the coordinates around the origin until
the straight line takes the form $x-r=0$ with $r\ge 0$. Hence, we
can assume that the systems having an ellipse and a straight line as
invariant algebraic curves, without loss of generality, these curves
are
\begin{equation*}%\label{in}
f_1(x,y)=x^2+y^2-1=0 \quad
\text{and}\quad f_2(x,y)=x-r=0,\quad r\ge 0.
\end{equation*}
We shall need the following result which is a consequence of
\cite[Corollary 6]{LRS}, which characterizes all rational
differential systems having two curves $f_1=0$ and $f_2=0$ as
invariant algebraic curves. Since this result plays a main role in
this work and its proof given in \cite[Theorem 2.1]{LMR} is
shorter, for completeness we present it here.
\begin{theorem}\label{t0}
Let $f_1$ and $f_2$ be polynomials in $\mathbb{R}[x,y]$ such that
the Jacobian $\{f_1,f_2\} \not\equiv 0$. Then any planar polynomial
differential system which admits $f_1=0$ and $f_2=0$ as invariant
algebraic curves can be written as
\begin{equation} \label{mm}
\dot{x}=\varphi_1\{x,f_2\}+\varphi_2\{f_1,x\}, \quad
\dot{y}=\varphi_1\{y,f_2\}+\varphi_2\{f_1,y\},
\end{equation}
where $\varphi_1=\lambda_1 f_1$ and $\varphi_2=\lambda_2 f_2$, with
$\lambda_1$ and $\lambda_2$ being arbitrary polynomial functions.
\label{teogeral}
\end{theorem}
\begin{proof}
Consider the vector fields
\begin{equation*}
\{\ast,f_2\}=\det \begin{pmatrix}
\frac{\partial\ast}{\partial x} & \frac{\partial\ast}{\partial y}\\[4pt]
\frac{\partial f_{2}}{\partial x} & \frac{\partial f_{2}}{\partial y}\\
\end{pmatrix}
\quad \text{and} \quad \{f_1,\ast\}=\det \begin{pmatrix}
\frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y}\\[4pt]
\frac{\partial\ast}{\partial x} & \frac{\partial\ast}{\partial y}\\
\end{pmatrix}.
\end{equation*}
Using this notation and denoting by $X$ the vector field associated
to system \eqref{mm}, we have
\begin{equation}\label{mm2}
X(\ast)=\varphi_1\{\ast,f_2\}+\varphi_2\{f_1,\ast\}.
\end{equation}
In this way
\begin{equation*}
X(f_1)=\varphi_1\{f_1,f_2\}+\varphi_2\{f_1,f_1\}=\lambda_1 f_1
\{f_1,f_2\}=Kf_1.
\end{equation*}
Hence $f_1=0$ is an invariant algebraic curve of the polynomial
vector field $X$ associated to system \eqref{mm} with cofactor
$K=\lambda_1 \{f_1,f_2\}$. Analogously we can show that $f_2=0$ is
also an invariant algebraic curve of $X$.
Now we prove that the vector field $X$ is the most general
polynomial vector field which admits $f_1=0$ and $f_2=0$ as
invariant algebraic curves. Indeed let $Y=(Y_1(x,y),Y_2(x,y))$ be an
arbitrary polynomial vector field having $f_1=0$ and $f_2=0$ as
invariant algebraic curves. Then taking
\begin{equation*}
\varphi_1=\frac{Y(f_1)}{\{f_1,f_2\}} \quad \text{and} \quad
\varphi_2=\frac{Y(f_2)}{\{f_1,f_2\}}
\end{equation*}
and substituting the expressions of $\varphi_1$ and $\varphi_2$ in
the expression \eqref{mm2} of the vector field $X$ we obtain for an
arbitrary polynomial $F$ that
\begin{equation*}
X(F)=Y(f_1)\frac{\{F,f_2\}}{\{f_1,f_2\}}+Y(f_2)\frac{\{f_1,F\}}{\{f_1,f_2\}}.
\end{equation*}
Substituting
\begin{equation*}
Y(f_1)=Y_1\frac{\partial f_1}{\partial x}+Y_2\frac{\partial
f_1}{\partial y} \quad \text{and} \quad Y(f_2)=Y_1\frac{\partial
f_2}{\partial x}+Y_2\frac{\partial f_2}{\partial y}
\end{equation*}
in $X(F)$ we have that $X(F)=Y(F)$. Therefore the theorem is proved,
because of the arbitrariness of the function $F$.
\end{proof}
Using this theorem we have the following proof.
\begin{proof}[Proof of Theorem \ref{t1}]
Noting that
$$
\{x,f_2\}=0,\ \{y,f_2\}=-1,\ \{f_1,x\}=-2y,\ \{f_1,y\}=2x,
$$
and applying Theorem \ref{t0} we can write systems \eqref{eq0} of
degree less than or equalt $2$ having an ellipse and a straight
line as invariant
algebraic curves as
\begin{align*}%\label{pe2}
\dot x&=-2\lambda_2 y(x-r), \\
\dot y&=-\lambda_1(x^2+y^2-1)+2\lambda_2x(x-r),
\end{align*}
where $\lambda_1,\lambda_2$ are arbitrary constants. Then we have system
\eqref{pe1}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{t2}]
Statements (a) and (b) follow easily.
It is immediate that the function $H$ given in statement (c) on the
orbits of system \eqref{eq1} satisfies
\[
\frac{dH}{dt}=\frac{\partial H}{\partial x}(-c y(x-r))+ \frac{\partial H}{\partial
y}(C(x^2+y^2-1)+c x(x-r))=0.
\]
So $H$ is a first integral of system \eqref{eq1}, and this proves
statement (c).
Since both first integrals are defined in the whole plane except
perhaps on the invariant straight line $x=r$, the system has no
limit cycles. This completes the proof of the theorem.
\end{proof}
\section{Poincar\'e compactification} \label{s3}
Let
\[
\mathcal{X}= P(x,y)\frac{\partial}{\partial x}+ Q(x,y)\frac{\partial}{\partial y}
\]
be the planar polynomial vector field of degree $n$ associated to
the polynomial differential system \eqref{eq0} of degree $n$. The
\emph{Poincar\'e compactified vector field $p(\mathcal{X})$ associated to
$\mathcal{X}$} is an analytic vector field on $\mathbb{S}^2$ constructed as follows
(see, for instance \cite {GO}, or Chapter 5 of \cite {DLA}).
Let $\mathbb{S}^2=\{y=(y_1,y_2,y_3) \in \mathbb{R}^3: y_1^2+y_2^2+y_3^2=1\}$ (the
\emph{Poincar\'e sphere}) and $T_y \mathbb{S}^2$ be the tangent plane to
$\mathbb{S}^2$ at point $y$. We identify the plane $\mathbb{R}^2$ where we have
our polynomial vector field $\mathcal{X}$ with the tangent plane
$T_{(0,0,1)}\mathbb{S}^2$. Consider the central projection $f:
T_{(0,0,1)}\mathbb{S}^2 \to \mathbb{S}^2$. This map defines two copies
of $\mathcal{X}$, one in the northern hemisphere and the other in the
southern hemisphere. Denote by $\mathcal{X}'$ the vector field $Df \circ \mathcal{X}$
defined on $\mathbb{S}^2$ except on its equator $\mathbb{S}^1=\{y \in
\mathbb{S}^2:y_3=0\}$. Clearly the equator $\mathbb{S}^1$ is identified to the
\emph{infinity} of $\mathbb{R}^2$. In order to extend $\mathcal{X}'$ to a vector field
on $\mathbb{S}^2$ (including $\mathbb{S}^1$) it is necessary that $\mathcal{X}$ satisfies
suitable conditions. In the case that $\mathcal{X}$ is a planar polynomial
vector field of degree $n$ then $p(\mathcal{X})$ is the only analytic
extension of $y_3^{n-1} \mathcal{X}'$ to $\mathbb{S}^2.$ On $\mathbb{S}^2 \backslash
\mathbb{S}^1$ there are two symmetric copies of $\mathcal{X}$, and knowing the
behaviour of $p(\mathcal{X})$ around $\mathbb{S}^1$, we know the behaviour of $\mathcal{X}$
at infinity.
The projection of the closed northern hemisphere of $\mathbb{S}^2$ on
$y_3=0$ with the mapping $(y_1,y_2,y_3) \mapsto (y_1,y_2)$ is called the
\emph{Poincar\'e disc,} and it is denoted by $\mathbb{D}^2$. The Poincar\'e
compactification has the property that $\mathbb{S}^1$ is invariant under
the flow of $p(\mathcal{X})$.
We say that two polynomial vector fields $\mathcal{X}$ and $\mathcal{Y}$ on $\mathbb{R}^2$ are
\emph{topologically equivalent} if there exists a homeomorphism on
$\mathbb{S}^2$ preserving the infinity $\mathbb{S}^1$ carrying orbits of the
flow induced by $p(\mathcal{X})$ into orbits of the flow induced by $p(\mathcal{Y})$,
preserving or reversing simultaneously the sense of all orbits.
As $\mathbb{S}^2$ is a differentiable manifold, for computing the
expression for $p(\mathcal{X})$, we can consider the six local charts $U_i =
\{y \in \mathbb{S}^2: y_i>0\}$, and $V_i=\{y \in \mathbb{S}^2: y_i<0\}$ where
$i=1,2,3;$ and the diffeomorphisms $F_i: U_i \to \mathbb{R}^2$ and
$G_i: V_i \to \mathbb{R}^2$ for $i=1,2,3$ are the inverses of the
central projections from the planes tangent at the points
$(1,0,0),(-1,0,0),(0,1,0),(0,-1,0),(0,0,1)$ and $(0,0,-1)$
respectively. If we denote by $(u,v)$ the value of $F_i(y)$ or
$G_i(y)$ for any $i=1,2,3$ (so $(u,v)$ represents different things
according to the local charts under consideration), then some easy
computations give for $p(\mathcal{X})$ the following expressions:
\begin{gather}
v^n \Delta (u,v) \Big( Q \big( \frac{1}{v} , \frac{u}{v} \big)
- u P \big( \frac{1}{v}, \frac{u}{v} \big),
-v P \big(\frac{1}{v}, \frac{u}{v} \big) \Big) \quad \text{in } U_1, \label{l4.1} \\
v^n \Delta (u,v) \Big( P \big(\frac{u}{v} , \frac{1}{v} \big)
- u Q \big(\frac{u}{v}, \frac{1}{v} \big),
-v Q \big(\frac{u}{v},\frac{1}{v} \big)\Big) \quad \text{in } U_2, \label{l4.2} \\
\Delta(u,v)( P(u,v), Q(u,v))\quad \text{in } U_3,\nonumber
\end{gather}
where $\Delta(u,v)=(u^2+v^2+1)^{-(n-1)/2}$. The expression
for $V_i$ is the same as that for $U_i$ except for a multiplicative
factor $(-1)^{n-1}$. In these coordinates for $i=1,2$, $v=0$ always
denotes the points of $\mathbb{S}^1.$ \emph{In what follows we omit the
factor $\Delta(u,v)$ by rescaling the vector field $p(\mathcal{X})$.} Thus we
obtain a polynomial vector field in each local chart.
\section{Separatrices and canonical regions}\label{s4}
Let $p(\mathcal{X})$ be the Poincar\'{e} compactification in the Poincar\'{e} disc
$\mathbb{D}$ of the polynomial differential system \eqref{eq0} defined in
$\mathbb{R}^{2}$, and let $\Phi$ be its analytic flow. Following
Markus \cite{M2} and Neumann \cite{Ne} we denote by $(U,\Phi)$ the
flow of a differential system on an invariant set $U\subset \mathbb{D}$
under the flow $\Phi$. Two flows $(U,\Phi)$ and $(V,\Psi)$ are \emph{
topologically equivalent} if and only if there exists a
homeomorphism $h: U\to V$ which sends orbits of the flow $\Phi$ into
orbits of the flow $\Psi$ either preserving or reversing the
orientation of all the orbits.
The flow $(U,\Phi)$ is said to be \textit{parallel} if it
is topologically equivalent to one of the following flows:
\begin{itemize}
\item[(i)] The flow defined in $\mathbb{R}^{2}$ by the differential system
$\dot{x}=1$, $\dot{y}=0$, called \textit{strip flow}.
\item[(ii)] The flow defined in $\mathbb{R}^{2}\setminus\left\{
(0,0)\right\} $ by the differential system in polar coordinates
$\dot {r}=0$, $\dot{\theta}=1$, called \textit{annular flow}.
\item[(iii)] The flow defined in $\mathbb{R}^{2}\setminus\left\{
(0,0)\right\} $ by the differential system in polar coordinates
$\dot {r}=r$, $\dot{\theta}=0$, called \textit{spiral or radial
flow}.
\end{itemize}
It is known that the separatrices of the vector field $p(\mathcal{X})$ in the
Poincar\'{e} disc $\mathbb{D}$ are:
\begin{itemize}
\item[(I)] all the orbits of $p(\mathcal{X})$ which are in the boundary
$\mathbb{S}^1$ of the Poincar\'{e} disc (i.e. at the infinity of $\mathbb{R}^2$),
\item[(II)] all the finite singular points of $p(\mathcal{X})$,
\item[(III)] all the limit cycles of $p(\mathcal{X})$, and
\item[(IV)] all the separatrices of the hyperbolic sectors of the
finite and infinite singular points of $p(\mathcal{X})$.
\end{itemize}
Moreover such vector fields $p(\mathcal{X})$, coming from polynomial vector
fields \eqref{eq0} of $\mathbb{R}^2$ having finitely many singular points
finite and infinite, have finitely many separatrices. For more
details see for instance \cite{LLNZ}.
Let $\mathcal{S}$ be the union of the separatrices of the flow $(\mathbb{D},\Phi)$
defined by $p(\mathcal{X})$ in the Poincar\'{e} disc $\mathbb{D}$. It is easy to check
that $\mathcal{S}$ is an invariant closed set. If $N$ is a connected
component of $\mathbb{D}\setminus \mathcal{S}$, then $N$ is also an invariant set
under the flow $\Phi$ of $p(\mathcal{X})$, and the flow
$\left(N,\Phi|_{N}\right)$ is called a \textit{canonical region} of
the flow $(\mathbb{D},\Phi)$ .
\begin{proposition}\label{MN1}
If the number of separatrices of the flow $(\mathbb{D},\Phi)$ is finite,
then every canonical region of the flow $(\mathbb{D},\Phi)$ is parallel.
\end{proposition}
For a proof of this proposition see \cite{Ne} or \cite{LLNZ}.
The \textit{separatrix configuration} $\mathcal{S}_c$ of a flow $(\mathbb{D},\Phi)$
is the union of all the separatrices $\mathcal{S}$ of the flow together
with an orbit belonging to each canonical region. The separatrix
configuration $\mathcal{S}_c$ of the flow $\left( \mathbb{D},\Phi\right)$ is said
to be topologically equivalent to the separatrix configuration
$\mathcal{S}^{\ast}_c$ of the flow $\left( \mathbb{D},\Phi^{\ast}\right)$ if there
exists an orientation preserving homeomorphism from $\mathbb{D}$ to $\mathbb{D}$
which transforms orbits of $\mathcal{S}_c$ into orbits of $\mathcal{S}^{\ast}_c$,
and orbits of $\mathcal{S}$ into orbits of $\mathcal{S}^{\ast}$.
\begin{theorem}
[Markus--Neumann--Peixoto]\label{TMN}
Let $\left(\mathbb{D},\Phi\right)$ and
$\left(\mathbb{D},\Phi^{\ast}\right)$ be two compactified Poincar\'{e} flows
with finitely many separatrices coming from two polynomial vector
fields \eqref{eq0}. Then they are topologically equivalent if and
only if their separatrix configurations are topologically
equivalent.
\end{theorem}
For a proof of this result we refer the reader to \cite{M2, Ne, Pe}.
It follows from the previous theorem that in order to classify the
phase portraits in the Poincar\'{e} disc of a planar polynomial
differential system having finitely many separatrices finite and
infinite, it is enough to describe their separatrix configuration.
This is what we have done in Figure \ref{fig1}, where we also have added the
invariant straight line $x=r$ with $r\ge 0$ and the invariant circle
$x^2+y^2=1$.
\section{Phase portraits}\label{s5}
It is clear that the phase portrait of the quadratic polynomial
differential system \eqref{pe1} with $C=0$, if formed by all the
invariant circles centered at the origin of coordinates, intersected
with the invariant straight line $x=r$ filled of equilibria,
providing the two first phase portraits of Figure \ref{fig1}.
In what follows we shall study the phase portraits of system
\eqref{pe1} with $C\ne 0$.
Doing the rescaling of the time $\tau= C t$, and renaming $c/C$
again by $c$, we have the quadratic system
\begin{equation}\label{eq1}
\begin{gathered}
\dot x=-c y(x-r), \\
\dot y= x^2+y^2-1+c x(x-r),
\end{gathered}
\end{equation}
with $c\in \mathbb{R}$ and $r\ge 0$.
\begin{remark} \rm
System \eqref{eq1} is reversible because it
does not change under the transformation $(x,y,t)\to (x,-y,-t)$.
Hence we know that the phase portrait of system
\eqref{eq1} is symmetric with respect to the $x-$axis.
\end{remark}
The way for studying the phase portraits of systems \eqref{eq1} is
the following. First we shall characterize all the finite equilibria
of those systems together with their local phase portraits. After we
do the same for the infinite equilibria, and finally using this
information on the equilibria and the existence of the invariant
straight line $x=r$ with $r\ge 0$, and of the invariant ellipse
$x^2+y^2=1$ we shall provide the classification of all the phase
portraits of systems \eqref{eq1}.
\subsection{The finite singular points}
The finite singular points of system \eqref{eq1} are characterized
in the next result.
\begin{proposition}\label{p1}
System \eqref{eq1} has the following finite singular points:
\begin{itemize}
\item[(a)] if $c=0$ all the points of circle $x^2+y^2=1$;
\item[(b)] if $c\notin \{-1,0\}$ the singular points are
\begin{equation}\label{sp}
\begin{gathered}
M_\pm= \big(r,\pm \sqrt{1-r^2}\big) \quad \text{if $0\le r<1$}, \\
M= (1,0) \quad \text{if $r=1$}, \\
N_\pm= (x^*_\pm,0)=\Big( \frac{cr\pm \sqrt{\Delta}}{2(c+1)},0
\Big) \quad \text{if $\Delta>0$}, \\
N= (x^*,0)=\Big(\frac{cr}{2(c+1)},0 \Big) \quad \text{if
$\Delta=0$},
\end{gathered}
\end{equation}
where $\Delta= c^2 r^2+4(c+1)$;
\item[(d)] if $c= -1$ the singular points are
\begin{equation}\label{spm}
\begin{gathered}
(0, \pm 1) \quad \text{if $r=0$}, \\
\big(\frac{1}{r},0\big) \text{ and } (r, \pm\sqrt{1-r^2})
\quad \text{if $0< r<1$}, \\
\big(\frac{1}{r},0\big) \quad \text{if $r\ge 1$}.
\end{gathered}
\end{equation}
\end{itemize}
\end{proposition}
\begin{proof}
The proof follows easily studying the real solutions of the system
$c y(x-r)=0$, $x^2+y^2-1+c x(x-r)=0$.
\end{proof}
\begin{figure}[htb]
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(97,50)(0,0)
\dashline{1}(5,35)(97,35)
\dashline{1}(5,5)(97,5)
\dashline{1}(45,5)(45,35)
\dashline{1}(65,5)(65,48)
\dashline{1}(85,5)(85,50)
\put(97,4){$\rightarrow$}
\put(84.1,48){$\uparrow$}
\qbezier(5,7)(24,9)(35,27)
\qbezier(35,27)(45,42)(55,27)
\qbezier(55,27)(62,16)(65,5)
\put(82,49){$r$}
\put(33,43){$R_8$}
\put(61,43){$L_8$}
\put(73,43){$R_7$}
\put(93,43){$R_6$}
\put(20,36){$L_7$}
\put(43,36){$P_2$}
\put(53,36){$L_6$}
\put(73,36){$L_5$}
\put(93,36){$L_4$}
\put(66,32){$P_1$}
\put(86,32){$1$}
\put(59,26){$R_3$}
\put(11,18){$R_5$}
\put(23.5,18){$L_3$}
\put(36,18){$R_4$}
\put(55.5,18){$L_2$}
\put(65.5,18){$L_1$}
\put(73,18){$R_2$}
\put(93,18){$R_1$}
\put(0,8){$\Delta=0$}
\put(43,1){$-2$}
\put(63,1){$P_3$}
\put(84,1){$0$}
\put(97,1){$c$}
\end{picture}
\end{center}
\caption{Bifurcation diagram}
\label{fig2}
\end{figure}
We write the curve $\Delta=0$ in the strip $\{ (c,r): c\in \mathbb{R},\,
0\le r \le 1\}$ as
\begin{equation}\label{c3}
c_\pm(r)=\frac{-2\pm 2\sqrt{1-r^2}}{r^2}.
\end{equation}
Obviously $c_-(r)\leq-2\leq c_+(r)$.
Now we define the regions
\begin{align*}
R_1=&\{(c,r) : 0\le r<1,\, c>0\},\\
R_2=&\{(c,r):0\le r<1,\, -10\},\\
R_7=&\{(c,r):11,\, c=-1\},\\
L_9=&\{(c,r):r\ge 0,\, c=0 \}.
\end{align*}
and the points $P_1(c,r)=(-1,1)$, $P_2(c,r)=(-2,1)$ and $P_3(c,r)=
(-1,0)$. See Figure \ref{fig2}.
For definitions of elliptic and hyperbolic sectors, cusp, and
hyperbolic, semi-hyperbolic and nilpotent singular points see
\cite{DLA}.
\begin{proposition}\label{Tr01}
System \eqref{eq1} has the following finite singular points if its
parameters $(c,r)$ are in
\begin{itemize}
\item[($R_1$)] two hyperbolic saddles $M_\pm$ and two centers $N_\pm$.
\item[($R_2$)] four hyperbolic singular points: $M_+$ is an unstable node,
$M_-$ is a stable node, and $N_\pm$ are saddles.
\item[($R_3$)] three hyperbolic singular points: $M_+$ is an unstable node,
$M_-$ is a stable node, and $N_+$ is saddle; and a center $N_-$.
\item[($R_4$)] two hyperbolic singular points: $M_+$ is an unstable node,
and $M_-$ is a stable node.
\item[($R_5$)] three hyperbolic singular points: $M_+$ is an unstable node,
$M_-$ is a stable node, and $N_-$ is saddle; and a center $N_+$.
\item[($R_{6,7}$)] one hyperbolic saddle $N_+$ and a center $N_-$.
\item[($R_8$)] two centers $N_\pm$.
\item[($L_1$)] three hyperbolic singular points: $M_+$ is an unstable node,
$M_-$ is a stable node, and $N$ is a saddle.
\item[($L_{2,3}$)] two hyperbolic singular points: $M_+$ is an unstable
node and $M_-$ is a stable node, and a nilpotent cusp $N$.
\item[($L_4$)] $M=(1,0)$ is a nilpotent saddle and $N=(-1/(c+1),0)$ is
a center.
\item[($L_5$)] $M=(1,0)$ is a nilpotent singular point formed by one
elliptic sector and one hyperbolic sector, and $N=(-1/(c+1),0)$ is a
hyperbolic saddle.
\item[($L_{6,7}$)] $M=(1,0)$ is a nilpotent singular point formed by one
elliptic sector and one hyperbolic sector, and $N=(-1/(c+1),0)$ is a
center.
\item[($L_8$)] $N(1/r,0)$ is a center.
\item[($L_9$)] all the points of circle $x^2+y^2=1$ are singular
points.
\item[($P_1$)] $M=(1,0)$ is a nilpotent singular point formed by one
elliptic sector and one hyperbolic sector.
\item[($P_2$)] $M=(1,0)$ is a degenerated singular point formed by the
union of two elliptic sectors.
\item[($P_3$)] two hyperbolic singular points: $M_+$ is an unstable node
and $M_-$ is a stable node.
\end{itemize}
\end{proposition}
\begin{proof}
On the curve $L_9$ we have that $c$ is zero, so the straight lines
$x=\text{constant}$ are invariant by system \eqref{eq1}, see the
phase portrait $L_9$ in Figure \ref{fig1}.
In the following we always assume $c\ne 0$. We distinguish two cases
in the study of the finite singular points of system \eqref{eq1}.
\smallskip
\noindent\textbf{Case 1:} Singular points on the invariant straight
line $x=r$. Clearly system \eqref{eq1} has no singular point on
$x=r$ when $r>1$, a unique singular point $M$ when $r=1$, and the
singular points $M_\pm$ for $0\leq r<1$, see \eqref{sp}.
\smallskip
\noindent\textbf{Subcase 1.1:} $0\leq r<1$. The eigenvalues of the
Jacobian matrix of system \eqref{eq1} at $M_+$ are $-c\sqrt{1-r^2}$
and $2\sqrt{1-r^2}$, and at $M_-$ are $c\sqrt{1-r^2}$ and
$-2\sqrt{1-r^2}$. Therefore, $M_\pm$ are hyperbolic saddles if
$c>0$; and $M_+$ is an unstable hyperbolic node and $M_-$ is a
stable hyperbolic node if $c<0$, see for more details Theorem 2.15
of \cite{DLA} where are described the local phase portraits of the
hyperbolic singular points.
\smallskip
\noindent\textbf{Subcase 1.2:} $r=1$. The Jacobian matrix of system
\eqref{eq1} at $M(1,0)$ is
$$
J_M= \begin{pmatrix}
0 & 0\\
c+2 & 0
\end{pmatrix}.
$$
\smallskip
\noindent\textbf{Subcase 1.2.1:} $c\ne -2$. So $M$ is a nilpotent
singular point. Using \cite[Theorem 3.5]{DLA} for studying the
local phase portraits of the nilpotent singular points we get that
$M$ a nilpotent saddle if $c>0$, and if $c<0$ and different from
$-2$ is reunion of one elliptic sector with one hyperbolic sector.
\smallskip
\noindent\textbf{Subcase 1.2.2:} $c=-2$. Using the polar blowing-up
centered at $M$, i.e. $x=\rho\cos\theta+1$ and $y=\rho\sin\theta$,
system \eqref{eq1} becomes
\begin{equation}\label{bl}
\begin{gathered}
\dot\rho=\rho^2\sin\theta,\\
\dot\theta=-\rho\cos\theta.
\end{gathered}
\end{equation}
The singular points of \eqref{bl} on $\{\rho=0\}$ are located at
$\theta=\pm\pi/2$. Then $(0,\pi/2)$ is an unstable hyperbolic node
and $(0,-\pi/2)$ is a stable hyperbolic node. Doing a blowing down
we obtain that $M$ is formed by the union of two elliptic sectors.
See picture $P_2$ in Figure \ref{fig1}.
\smallskip
\noindent\textbf{Case 2:} Singular points on the straight line $y=0$.
\smallskip
\noindent\textbf{Subcase 2.1:} $\Delta>0$ and $c\ne -1$. Then system
\eqref{eq1} has two singular points $N_\pm= (x^*_\pm,0)$, see
\eqref{sp}. The Jacobian matrix of system \eqref{eq1} at the points
$N_\pm$ is
\begin{equation}\label{jac}
J= \begin{pmatrix}
0& -c(x_\pm^*-r)\\
\pm \sqrt{\Delta} & 0
\end{pmatrix}.
\end{equation}
It is easy to check that
$$
(x_+^*-r)(x_-^*-r)=\frac{r^2-1}{c+1}, \quad
(x_+^*-r)+(x_-^*-r)=-\frac{(c+2)r}{c+1}.
$$
\smallskip
\noindent\textbf{Subcase 2.1.1:} $0\leq r<1$. Then we have
$x_-^*-1$, and $x_\pm^*>r$ when $-2\leq c<-1$, and
$x_\pm^*0$, and saddles when $-11$. Then $x_\pm^*-1$;
$x_+^*0$; $N_-$ is a saddle and $N_+$ a center when $-10$ or $c<-1$ and $c\not=-2$, but
$N$ is a center of system \eqref{eq1} because this system is
reversible with respect to $x$--axis.
\smallskip
\noindent\textbf{Subcase 2.1.3.2:} $c= -2$.
\smallskip
\noindent\textbf{Subcase 2.2:} $\Delta=0$ and $c\ne -1$, we have from
\eqref{c3} that $c=c_\pm(r)$, and from \eqref{jac} the singular
point $N=(x^*,0)$ is nilpotent. Taking $(x,y)=(X+x^*,Y)$, after
$(X,Y)=(x,y)$, and rescaling the independent variable $t$ by
$\tau=rc(c+2)t/(2(c+1))$, we obtain
\begin{gather*} %\label{g1}
\dot x= y-\frac{2(c+1)}{r(c+2)}xy, \\
\dot y= \frac{2(c+1)}{rc(c+2)}\left((c+1)x^2+y^2\right).
\end{gather*}
By \cite[Theorem 3.5]{DLA} the origin of the previous system is a
cusp.
\smallskip
\noindent\textbf{Subcase 2.3:} $c=-1$. On $y=0$ there is the unique
singular point $N(1/r,0)$ when $r>0$.
\smallskip
\noindent\textbf{Subcase 2.3.1:} $r\notin \{0,1\}$. The eigenvalues of
the Jacobian matrix of system \eqref{eq1} at $N$ are
$\pm\sqrt{1-r^2}$, which implies that $N$ is obviously a linear
center when $r>1$ (and consequently a center because of the
reversibility of the system), and $N$ is a saddle when $0< r<1$.
\smallskip
\noindent\textbf{Subcase 2.3.2:} $r=1$. The unique singular point of
the system is $(1,0)$, the Jacobian matrix on it is
$\begin{pmatrix}
0 & 0\\
1 & 0
\end{pmatrix}$.
So it is a nilpotent singular point. By \cite[Theorem 3.5]{DLA}
its local phase portrait is formed by one hyperbolic and one
elliptic sector (see picture $P_1$ in Figure \ref{fig1}).
\smallskip
\noindent\textbf{Subcase 2.3.3:} $r=0$. No singular points on $y=0$.
Finally, taking into account all this information on the finite
singular points, we can organize it, as it appears in the statements
of the proposition.
\end{proof}
\subsection{The infinite singular points}
\begin{proposition}\label{p2}
The following two statements hold.
\begin{itemize}
\item[(a)] If $c\ne -1$ system \eqref{eq1} has a pair of infinite
singular points, which are saddles if $c<-1$, and nodes if $c>-1$.
\item[(b)] If $c=-1$ the infinity of system \eqref{eq1} is filled of
singular points.
\end{itemize}
\end{proposition}
\begin{proof}
Considering the infinite singular of \eqref{eq1}, we take
\begin{equation}\label{chau}
x=\frac{1}{v},\quad y=\frac{u}{v}.
\end{equation}
and the time rescaling $t=v\tau$. Then system \eqref{eq1} in the
local chart \eqref{chau} is
\begin{equation}\label{infu}
\begin{aligned}
\dot u&= (u^2+1)(1+c)-v^2-cr(u^2+1)v,\\
\dot v&=cu(1-rv)v.
\end{aligned}
\end{equation}
If $c\ne -1$, there is no singular point of system \eqref{infu} on
$v=0$. Taking
\begin{equation}\label{chav}
x=\frac{u}{v},\quad\ y=\frac{1}{v}.
\end{equation}
and the time rescaling $t=v\tau$. Then system \eqref{eq1} in the
local chart \eqref{chav} is
\begin{equation}\label{infv}
\begin{aligned}
\dot u&= uv^2+cr(1+u^2)v-u(1+u^2)(1+c),\\
\dot v&=v(v^2-1+cr uv-(c+1)u^2).
\end{aligned}
\end{equation}
If $c\ne -1$, the origin is a singular point of \eqref{infv}. It is
easy to get that the eigenvalues of the Jacobian matrix at the
origin are $-1$ and $-(c+1)$, which implies that system \eqref{eq1}
has a pair of infinite saddles if $c<-1$, and a pair of node if
$c>-1$.
If $c= -1$, it is obtained from \eqref{infu} and \eqref{infv} that
the infinity $v=0$ of the Poincar\'e disc is filled with singular
points. Furthermore, we reduce \eqref{infu} into
\begin{equation}\label{infc}
\begin{aligned}
\dot u&= -z+r(u^2+1),\\
\dot z&=-u(1-rz).
\end{aligned}
\end{equation}
If $r=0$, the origin of system \eqref{infc} is a saddle, that is,
there is a pair of infinite singular point of system \eqref{infc}.
\end{proof}
According to Theorem \ref{t2}, Proposition \ref{Tr01} and Proposition
\ref{p2}, and using the invariant straight line $x=0$ with $r\ge 0$
and the invariant circle $x^2+y^2=1$, we obtain the global phase
portraits of system \eqref{eq1} in Poincar\'e disc described in
Figure \ref{fig1}.
\subsection*{Acknowledgements}
The first author is partially supported by a MINECO/ \\ FEDER grant
MTM2008-03437 and MTM2013-40998-P, an AGAUR grant number 2014
SGR568, an ICREA Academia, the grants FP7-PEOPLE-2012-IRSES 318999
and 316338, the grant UNAB13-4E-1604, and from the recruitment
program of high-end foreign experts of China.
The second author is partially supported by the state key program of
NNSF of China grant number 11431008, NSF of Shanghai grant number
15ZR1423700 and NSF of Jiangsu, BK 20131285.
\begin{thebibliography}{99}
\bibitem{AL} J. C. Art\'esm J. Llibre;
\emph{Quadratic Hamiltonian vector fields}, J. Differential Equations
\textbf{107} (1994), 80--95.
\bibitem{ALV} J. C. Art\'{e}s, J. Llibre, N. Vulpe;
\emph{Complete geometric invariant study of two classes of quadratic
systems}, Electronic J. of Differential Equations \textbf{2012}, No. 09
(2012), 1--35.
\bibitem{Ba} N. N. Bautin;
\emph{On the number of limit cycles which appear with the variation
of coefficients from an equilibrium position of focus or center
type}, Mat. Sbornik \textbf{30} (1952), 181--196, Amer. Math. Soc.
Transl. Vol. 100 (1954), 1--19.
\bibitem{DA} T. Date;
\emph{Classification and analysis of two--dimensional homogeneous
quadratic differential equations systems}, J. of Differential
Equations \textbf{32} (1979), 311--334.
\bibitem{Du} H. Dulac;
\emph{D\'{e}termination et integration d'une certaine classe d'\'{e}quations
diff\'{e}rentielle ayant par point singulier un centre}, Bull. Sci.
Math. S\'{e}r. (2) \textbf{32} (1908), 230--252.
\bibitem{DLA} F. Dumortier, J. Llibre, J.C. Art\'es;
\emph{Qualitative theory of planar differential systems},
Universitext, Springer--Verlag, 2006.
\bibitem{GO} E. A. Gonz\'alez;
\emph{Generic properties of polynomial vector fields at infinity},
Trans. Amer. Math. Soc. \textbf{143} (1969), 201--222.
\bibitem{KV} Yu.F. Kalin, N.I. Vulpe;
\emph{Affine--invariant conditions for the topological discrimination
of quadratic Hamiltonian differential systems}, Differential
Equations \textbf{34} (1998), no. 3, 297--301.
\bibitem{K1} W. Kapteyn;
\emph{On the midpoints of integral curves of differential equations
of the first degree}, Nederl. Akad. Wetensch. Verslag. Afd. Natuurk.
Konikl. Nederland (1911), 1446--1457 (Dutch).
\bibitem{K2} W. Kapteyn;
\emph{New investigations on the midpoints of integrals of
differential equations of the first degree}, Nederl. Akad. Wetensch.
Verslag Afd. Natuurk. \textbf{20} (1912), 1354--1365; \textbf{21}, 27--33
(Dutch).
\bibitem{KOR} N. A. Korol;
\emph{The integral curves of a certain differential equation},
(in Russian), Minsk. Gos. Ped. Inst. Minsk (1973), 47-51.
\bibitem{LLNZ} W. Li, J. Llibre, M. Nicolau, X. Zhang;
\emph{On the differentiability of first integrals of two dimensional
flows}, Proc. Amer. Math. Soc. \textbf{130} (2002), 2079--2088.
\bibitem{LMR} J. Llibre, M. Messias, A. Reinol;
\emph{Global phase portraits of quadratic and cubic systems with two
nonconcentric circles as invariant algebraic curves}, appear in
Dynamical systems: An International J.
\bibitem{LRS} J. Llibre, R. Ram\'{\i}rez, N. Sadovskaia;
\emph{Inverse problems in ordinary differential equations:
Applications to mechanics}, J. of Dynamics and Differential
Equations \textbf{26} (2014), 529--581.
\bibitem{LS} V. A. Lunkevich, K. S. Sibirskii;
\emph{Integrals of a general quadratic differential system in cases
of a center}, Differential Equations \textbf{18} (1982), 563--568.
\bibitem{LY} L. S. Lyagina;
\emph{The integral curves of the equation
$y'=(ax^2+bxy+cy^2)/(dx^2+exy+fy^2)$} (in Russian),
Usp. Mat. Nauk, \textbf{6-2(42)} (1951), 171--183.
\bibitem{M} L. Markus;
\emph{Quadratic differential equations and non--associative
algebras}, Annals of Mathematics Studies, Vol \textbf{45}, Princeton
University Press, 1960, pp 185--213.
\bibitem{M2} L. Markus;
\emph{Global structure of ordinary differential equations in the
plane:} Trans. Amer. Math Soc. \textbf{76} (1954), 127--148.
\bibitem{Ne} D. A. Neumann;
\emph{Classification of continuous flows on 2--manifolds}, Proc.
Amer. Math. Soc. \textbf{48} (1975), 73--81.
\bibitem{NW} T. A. Newton;
\emph{Two dimensional homogeneous quadratic differential systems},
SIAM Review \textbf{20} (1978), 120--138.
\bibitem{Pe} M. M. Peixoto;
\emph{Dynamical Systems. Proccedings of a Symposium held at the
University of Bahia}, 389--420, Acad. Press, New York, 1973.
\bibitem{Reyn1} J. W. Reyn;
\emph{Phase portraits of planar quadratic systems}, Mathematics and
Its Applications (Springer), \textbf{583}, Springer, New York, 2007.
\bibitem{Sc} D. Schlomiuk;
\emph{Algebraic particular integrals, integrability and the problem
of the center}, Trans. Amer. Math. Soc. \textbf{338} (1993), 799--841.
\bibitem{SV} K. S. Sibirskii, N. I. Vulpe;
\emph{Geometric classification of quadratic differential systems},
Differential Equations \textbf{13} (1977), 548--556.
\bibitem{VD} E. V. Vdovina;
\emph{Classification of singular points of the
equation $y'=(a_0x^2+a_1xy+a_2y^2)/(b_0x^2+b_1xy+b_2y^2)$ by Forster's
method} (in Russian), Differential Equations \textbf{20} (1984),
1809--1813.
\bibitem{Ye} Y. Ye, et al.;
\emph{Theory of Limit Cycles}, Transl. Math. Monographs \textbf{66},
Amer. Math. Soc., Providence, 1984.
\bibitem{Ye2} Y. Ye;
\emph{Qualitative Theory of Polynomial Differential Systems},
Shanghai Scientific \& Technical Publishers, Shanghai, 1995 (in
Chinese).
\bibitem{YY} Y. Wei, Y. Ye;
\emph{On the conditions of a center and general integrals of
quadratic differential systems}, Acta Math. Sin. (Engl. Ser.) \textbf{
17} (2001), 229--236.
\bibitem{Zo} H. \.{Z}o\l\c{a}dek;
\emph{Quadratic systems with center and their perturbations}, J.
Differential Equations \textbf{109} (1994), 223--273.
\end{thebibliography}
\end{document}