\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{graphicx,epic}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 141, pp. 1--19.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/141\hfil Global phase portraits]
{Global phase portraits for quadratic systems with a hyperbola and a
straight line as invariant algebraic curves}

\author[J. Llibre, J. Yu \hfil EJDE-2018/141\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}

\dedicatory{Communicated by Peter Bates}

\thanks{Submitted May 25, 2016. Published July 11, 2018.}
\subjclass[2010]{34C05}
\keywords{Quadratic system, first integral, global phase portraits,
\hfill\break\indent invariant hperbola, invariant straight line}

\begin{abstract}
 In this article we consider a class of quadratic polynomial differential
 systems in the plane having a hyperbola and a straight line as
 invariant algebraic curves, and we classify all its phase portraits.
 Moreover these systems are integrable and we provide their first integrals.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction and statement of main results}

In this article we consider the planar quadratic differential system
\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 such that the maximum of the degree
of $P$ and $Q$ is $2$. The dot in system \eqref{eq0} denotes
derivative with respect to the independent variable $t$. We introduce
some definitions.

Let $f$ is a nonconstant polynomial in the variable $x$ and $y$.
The algebraic curve $f(x,y)=0$ is {\it an invariant curve} of system \eqref{eq0},
if there exists some polynomial K(x,y) such that
$$
\mathcal X(f)=P\frac{\partial f}{\partial x}+Q\frac{\partial f}{\partial y}=Kf,
$$
and $K(x,y)$ is called the cofactor of the invariant curve $f(x,y)=0$.

Let $H(x,y)$ be a function defined in a dense and open subset $U$ of $\mathbb{R}^2$.
The function $H(x,y)$ is a {\it first integral} of system \eqref{eq0}
if $H$ is constant on the solutions of system \eqref{eq0} contained in $U$,
i.e.
\[
\mathcal X(H)\big|_U=P\frac{\partial H}{\partial x}(x,y)+ Q\frac{\partial H}{\partial y}\big|_U=0.
\]
And a quadratic system is {\it integrable} in $U$ if it has a first integral
$H$ in $U$.

Up to now several hundred of papers have been published studying differential
 aspects of quadratic systems, as their integrability, their limit cycles,
their global dynamical behavior, and $\cdots$, see for instance the
references quoted in the books of Reyn
\cite{Reyn1} and Ye \cite{Ye, Ye2}. But it remains many open problems of
these systems. For example, the problem of the maximum number and distribution
of limit cycles, or the problem of classifying all the integrable quadratic
systems, remain open.

Darboux \cite{Dar} introduced the relation between the existence of invariant
algebraic curves on a polynomial differential system and its integrability,
see for more details in \cite{CL,DLA}.

Dulac \cite{Du} started the studying of the
classification of the quadratic centers and their first integrals,
see also \cite{ALV,Ba,K1, K2,LS,Sc,YY,Zo}.
Art\'es and Llibre \cite{AL}
 studied the Hamiltonian quadratic systems, see also \cite{KV,ALV}.
Markus \cite{M} studied the class of homogeneous quadratic systems,
see also \cite{DA,KOR,LY,NW,SV,VD}.

In this paper we concern about {\it 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}?
In \cite{LYu} and \cite{LMR} the authors proved the integrability of
the class of quadratic systems having an ellipse and a straight line
as invariant algebraic curves, or two non-concentric circles as
invariant algebraic curves, respectively. Additionally the authors provided
all the different topological phase portraits that
 these classes exhibit in the Poincar\'e disc.


In this paper we want to study a new class of integrable quadratic
systems, the ones having a hyperbola and a straight line as invariant
algebraic curves. We prove their integrability and classify all their
phase portraits.

Our first result is to provide a normal form for a class of quadratic
polynomial differential systems having a hyperbola and a straight
line as invariant algebraic curves.

\begin{theorem}\label{thm1}
A class of planar polynomial differential system of degree $2$ having a
hyperbola and a straight line as invariant algebraic curves after an
affine change of coordinates can be written as either
\begin{equation}\label{pe23}
\begin{gathered}
\dot x=c \frac{a^2-b^2}{a^2}y(x-\delta), \\
\dot y=C\Big(\frac{x^2}{a^2-b^2}-\frac{a^2-b^2}{a^2}y^2-1\Big)
 +\frac{c}{a^2-b^2}x(x-\delta),
\end{gathered}
\end{equation}
or
\begin{equation}\label{pe2}
\begin{gathered}
\dot x=-cx(x-r), \\
\dot y=C(2xy-1)+cy(x-r),
\end{gathered}
\end{equation}
where $a,b, c, C \in \mathbb R$ with $a\ne 0, a\ne b$ and $\delta=\{0,1\}$.
\end{theorem}

Theorem \ref{thm1} is proved in section \ref{s2}.
In the next result we present the first integrals of the polynomial
differential system of degree $2$ having a hyperbola and a straight
line as invariant algebraic curves.

\begin{theorem}\label{thm2}
The quadratic polynomial differential systems \eqref{pe23} have the following
first integrals:
\begin{itemize}
\item[(a)] $H=x$ if $c=0$;
\item[(b)] $H=(x-\delta)^{(-2\frac{a^4+b^4}{(a^2-b^2)^2}\frac{C}{c})}
(\frac{x^2}{a^2-b^2}-\frac{a^2-b^2}{a^2}y^2-1)$ if $c\ne 0$;
\end{itemize}
and systems \eqref{pe2} have the following first integrals:
\begin{itemize}
\item[(c)] $H=x$ if $c=0$;
\item[(d)] $H= (x-r)^{2C/c} ( 2xy-1)$ if $c\ne 0$.
\end{itemize}
Moreover, the quadratic polynomial differential systems \eqref{pe23} and \eqref{pe2}
have no limit cycles.
\end{theorem}

This theorem  is proved in section \ref{s2}.
In the next theorem we present the topological classification of all
the phase portraits of planar polynomial differential system of
degree $2$ having a hyperbola and a straight line as invariant
algebraic curves in the Poincar\'e disc.

\begin{theorem}\label{thm3}
Given a planar polynomial differential system \eqref{pe23} and \eqref{pe2} of
degree $2$ having a hyperbola and a straight line as invariant algebraic curves its
phase portrait is topological equivalent to one of the $38$ phase
portraits of Figures \ref{fig1}, \ref{LC1}, \ref{RF1}.
\end{theorem}

The above  theorem is proved in sections \ref{s4} and \ref{s5}.

\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(120,106)(0,0)
\put(12,67){$L_{01}$} 
\put(43,67){$L_{02}$}
\put(74,67){$L_{03}$}
\put(105,67){$L_{04}$}
\put(12,33){$\omega_1$} 
\put(43,33){$\omega_2$}
\put(74,33){$\omega_3$}
\put(105,33){$L_{05}$}
\put(43,0){$P_{0}$}
\put(70,0){$C=0$}
\put(0,3){\includegraphics[width=12cm]{fig1}} % L01.eps
\end{picture}
\end{center}
\caption{Phase portraits for system \eqref{pe2}.}
\label{fig1}
\end{figure}

\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(120,140)(0,0)
\put(9,106){$L_{11},L_{12}$}
\put(43,106){$L_{21}$}
\put(73,106){$L_{22}$}
\put(99,106){$L_{31},L_{32}$}
\put(12,71){$L_{33}$}
\put(43,71){$L_{34}$}
\put(73,71){$L_{41}$}
\put(103,71){$L_{42}$}
\put(12,35){$L_{43}$}
\put(43,35){$P_{1}$}
\put(73,35){$P_{2}$}
\put(103,35){$L_{0-}$}
\put(12,0){$L_{0+}$}
\put(43,0){$L_{44}$}
\put(73,0){$L_{45}$}
\put(101,0){$C=0$}
\put(0,3){\includegraphics[width=12cm]{fig2}} %  LC1.eps
\end{picture}
\end{center}
\caption{Phase portraits I for system \eqref{pe23}.}
\label{LC1}
\end{figure}

\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(120,108)(0,0)
\put(10,70){$R_1,R_2$}
\put(40,70){$R_{01},\Omega_6$} 
\put(71,70){$R_3,\Omega_5$} 
\put(101,70){$R_4,\Omega_4$}
\put(13,35){$R_5$}
\put(43,35){$R_6$}
\put(74,35){$R_7$}
\put(102,35){$R_8, \Omega_1$}
\put(13,0){$R_9$}
\put(41,0){$R_{02},\Omega_2$}
\put(71,0){$R_{03},\Omega_3$}
\put(104,0){$R_{10}$}
\put(0,3){\includegraphics[width=12cm]{fig3}} % RF1.eps
\end{picture}
\end{center}
\caption{\small The phase portraits II of system \eqref{pe23}.}
\label{RF1}
\end{figure}

\section{Quadratic polynomial differential systems with hyperbola}\label{s2}

In this section we consider that system \eqref{eq0} has an invariant
hyperbola and an invariant straight line. Then by an affine transformation
we can change the hyperbola to the following norm form and to any straight line
\begin{equation}\label{hy}
\mathcal H:\ f_1(x,y)=x^2-y^2-1=0,\ \text{and}\ \mathcal L: \
f_2(x,y)=ax+by-\delta=0,
\end{equation}
where $\delta=1$ or $0$. Without loss of generality, let $a\geq 0$ for $\mathcal L$.
 According to the properties of the hyperbola, we classify the straight
line in the following four cases:
(i) $a=\pm b$;
(ii) $0<a^2<b^2$,
(iii) $a^2>b^2$, (iv) $a=0$.

For $a\neq 0, a^2-b^2\neq 0$, doing the transformation
$$
\begin{pmatrix}
x\\y
\end{pmatrix}
=\begin{pmatrix}
\frac{a}{a^2-b^2} & -\frac{b}{a}\\
-\frac{b}{a^2-b^2} & 1
\end{pmatrix}
\begin{pmatrix}
u\\v
\end{pmatrix},
$$
the curves \eqref{hy} change into
\begin{equation}\label{ic23}
\mathcal H:\ f_1(x,y)=\frac{x^2}{a^2-b^2}-\frac{a^2-b^2}{a^2}y^2-1=0,
\ \text{and}\ \mathcal L: \ f_2(x,y)=x-\delta=0,
\end{equation}
for cases (ii) and (iii), where we rename $(u,v)$ by $(x,y)$. Hence,
in the last two cases, without loss of generality, we have that $a>0,b\geq0$.
For case (i), doing the transformation
$$
\begin{pmatrix}
x\\y
\end{pmatrix}
=\begin{pmatrix}
\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}\\
-\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}
\end{pmatrix}
\begin{pmatrix}
u\\v
\end{pmatrix},
$$
the invariant hyperbola and the
invariant straight line can be written as
\begin{equation}\label{ic1}
\mathcal H:\ f_1(x,y)=2xy-1=0,\ \text{and}\ \mathcal L: \
f_2(x,y)=x-r=0,
\end{equation}
where $r=\delta/(\sqrt{2}b)$. Finally, case (iv) pass to case (ii)
by the transformation $(x,y)\to (y,x)$.

Next we provide a normal form for the quadratic
polynomial differential systems having a hyperbola and a straight
line as invariant algebraic curves $\mathcal H$ in Theorem \ref{thm1}.
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.

\begin{theorem}\label{thm0}
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*}
\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.
\end{theorem}

Using this theorem we will prove Theorem \ref{thm1}.

\begin{proof}[Proof of Theorem \ref{thm1}]
First for the cases (ii) and (iii) noting that
$$
\{x,f_2\}=0,\quad \{y,f_2\}=-1,\quad
\{f_1,x\}=2\frac{a^2-b^2}{a^2}y,\quad \{f_1,y\}=\frac{2x}{a^2-b^2},
$$
and applying Theorem \ref{thm0} we can write systems \eqref{eq0} of
degree $\leq 2$ having the hyperbola and the straight line given
in \eqref{ic23} as invariant algebraic curves into the form
\begin{gather*}
\dot x=2\lambda_2 \frac{a^2-b^2}{a^2}y(x-\delta), \\
\dot y=-\lambda_1\Big(\frac{x^2}{a^2-b^2}-\frac{a^2-b^2}{a^2}y^2-1\Big)
+\frac{2\lambda_2}{a^2-b^2}x(x-\delta),
\end{gather*}
where $\lambda_1,\lambda_2$ are arbitrary constants. Then we have system
\eqref{pe23}.

Second for the case (i), noting that
$$
\{x,f_2\}=0,\quad \{y,f_2\}=-1,\quad \{f_1,x\}=-2x,\quad \{f_1,y\}=2y,
$$
and applying Theorem \ref{thm0} we can write systems \eqref{eq0} of
degree $\leq 2$ having the hyperbola and the straight line given in
\eqref{ic1} as invariant algebraic curves into the form
\begin{gather*}
\dot x=-2\lambda_2x(x-r), \\
\dot y=-\lambda_1(2xy-1)+2\lambda_2y(x-r),
\end{gather*}
where $\lambda_1,\lambda_2$ are arbitrary constants, obtaining system \eqref{pe2}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2}]
Statements (a) and (c) follow easily.
It is immediate that the function $H$ given in statement (b) or (d) on the
orbits of system \eqref{pe23} or \eqref{pe2} satisfies
\[
\frac{dH}{dt}=\frac{\partial H}{\partial x}\dot x+ \frac{\partial H}{\partial
y}\dot y=0.
\]
So $H$ is a first integral of system \eqref{pe23} or \eqref{pe2}, and this proves
statement (b) and (d).

Since both first integrals are defined in the whole plane except
perhaps on the invariant straight line $x=\delta$, or $x=r$, the systems has no
limit cycles. This completes the proof of the theorem.
\end{proof}

If $C=0$ in systems \eqref{pe23} and \eqref{pe2}, then it is easy to verify
that they are equivalent to a linear differential system with a saddle and
the straight line $x=\delta$ or $x=r$ filled of singular points, respectively.
Then the phase portraits of systems \eqref{pe23} and \eqref{pe2} are shown
 in last picture of Figures \ref{fig1} and \ref{LC1} with the title $C=0$,
respectively. Assume $C\neq 0$. Doing the rescaling of the time $\tau= C t$,
and renaming $\rho=c/C$ system \eqref{pe23} becomes
\begin{equation}\label{eq23}
\begin{gathered}
\dot x=\rho \frac{a^2-b^2}{a^2}y(x-\delta), \\
\dot y=\frac{x^2}{a^2-b^2}-\frac{a^2-b^2}{a^2}y^2-1
+\frac{\rho}{a^2-b^2}x(x-\delta),
\end{gathered}
\end{equation}
and the quadratic system corresponding to \eqref{pe2} writes as
\begin{equation}\label{eq1}
\begin{gathered}
\dot x=-\rho x(x-r), \\
\dot y= 2xy-1+\rho y(x-r),
\end{gathered}
\end{equation}
with $\rho\in \mathbb{R}$ and $r\ge 0$.

\begin{remark}\label{rm} \rm
In system \eqref{eq1} we only consider the case $r\ge 0$. If $r<0$,
then it can be changed into the case of $r\ge0$ by the transformation
$(x,y,t)\to (-x,-y,-t)$.

System \eqref{eq23} 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{eq23} is symmetric with respect to the $x-$axis.
\end{remark}

In the following section we shall prove our main Theorem \ref{thm3}
for systems \eqref{eq23} and \eqref{eq1}.


\section{Phase portraits of system \eqref{eq1}}\label{s4}

In this section we consider the case of $a^2-b^2=0$, and take the normal
form as system \eqref{eq1}.

\subsection{Finite singular points}

The finite singular points of system \eqref{eq1} are characterized
in the following result.

\begin{proposition}\label{prop1}
System \eqref{eq1} has the following finite singular points.
\begin{itemize}
\item[(a)] If $\rho=0$ all the points of the hyperbola $2xy-1=0$.

\item[(b)] If $\rho\neq 0$ and $r=0$ there is no singular point.

\item[(c)] If $\rho>0$ and $r\neq 0$ the singular points are
$B_1(0,-1/(\rho r))$ and $B_2(r,1/(2r))$, and are saddles.

\item[(d)] If $\rho<0$ and $r\neq 0$ the singular points are
$B_1(0,-1/(\rho r))$ and $B_2(r,1/(2r))$, the first is a saddle and
the second a node.
\end{itemize}
\end{proposition}

\begin{proof}
It follows easily from \eqref{eq1} that statements $(a)$ and $(b)$ hold.
Noting that the Jacobian matrices of system \eqref{eq1} at the points
$B_1$ and $B_2$ are
$$
\begin{pmatrix}
\rho r & 0\\
-\frac{\rho+2}{2r} & -\rho r
\end{pmatrix}, \quad
\begin{pmatrix}
-\rho r& 0\\
\frac{\rho+2}{2r} & 2r
\end{pmatrix}
$$
respectively, it follows the proof of statements $(c)$ and $(d)$.
\end{proof}


\subsection{Infinite singular points}

\begin{proposition}\label{prop0}
System \eqref{eq1} has the following infinite singular points.
\begin{itemize}
\item[(a)] If $\rho=-1$ the infinity of system \eqref{eq1} is filled
of singular points.

\item[(b)] If $\rho\neq -1$, system \eqref{eq1} has two pairs of infinite
 singular points. There exits a pair of infinite nodes for $\rho<-1$ and $\rho>0$, while a pair of infinite saddles for $-1<\rho<0$. The other pair of infinite singular points are the union of a parabolic sector and a hyperbolic sector if $\rho<-1$ and $r\geq 0$, if $\rho>0$ and $r=0$, while they are a pair of infinite singular points each one formed by the union of a parabolic sector and an elliptic sector if $-1<\rho<0$ and $r\geq 0$, or if $\rho>0$ and $r>0$.
\end{itemize}
\end{proposition}

\begin{proof}
Doing the change of variables we take
\begin{equation}\label{chau}
x=\frac{1}{v},\quad\quad y=\frac{u}{v},
\end{equation}
and the time rescaling $t=v\tau$, system \eqref{eq1} in the
coordinates $(u,v)$ is
\begin{equation}\label{infa}
\begin{gathered}
\dot u= -v^2-2r\rho uv+2(\rho+1)u,\\
\dot v=\rho v(1-rv).
\end{gathered}
\end{equation}
Obviously system \eqref{infa} has a unique singular point $(0,0)$,
which is an unstable node for $\rho>0$,
a saddle if $-1<\rho<0$, and a stable node for $\rho<-1$.

Doing the change of variables
\begin{equation}\label{chav}
x=\frac{u}{v},\quad\quad y=\frac{1}{v},
\end{equation}
and the time rescaling $t=v\tau$, system \eqref{eq1} becomes
\begin{equation}\label{infb}
\begin{gathered}
\dot u= -u(2(\rho+1)u-2r\rho v-v^2),\\
\dot v=v(v^2-(\rho+2)u+r\rho v).
\end{gathered}
\end{equation}
Hence $(0,0)$ is a degenerated singular point.
Using the blowing-up technique we obtain that it is formed by a pair
of parabolic sectors and an elliptic sector if $-1<\rho<0$ and $r\geq 0$,
or $\rho>0$ and $r>0$. And it is formed by a pair of parabolic sectors
and a hyperbolic sectors if $\rho<-1$ and $r\geq 0$, or $\rho>0$ and $r=0$.

As an example we study the case $-1<\rho<0$ and $r> 0$. Considering the
degenerated singular point $(0,0)$ of \eqref{infb}, using the polar blowing-up,
$$
u=\gamma\cos\theta,\quad v=\gamma\sin\theta,
$$
we have
\begin{equation}\label{blow}
\begin{gathered}
\dot \gamma = ((\cos^2\theta+1)(r\sin\theta-\cos\theta)\rho
 -2\cos\theta)\gamma^2+\sin^2\theta \gamma^3,\\
\dot \theta =-\rho\cos\theta \sin\theta(r\sin\theta-\cos\theta)\gamma.
\end{gathered}
\end{equation}
System \eqref{blow} has simple zeroes $\theta=0, \pi/2, \pi, 3\pi/2$
and $\pm\theta^*$ on $\gamma=0$, where $\theta^*$ satisfies
$r\sin\theta-\cos\theta=0$.
It is easy to verify that $\theta=0,\pi/2$ are stable nodes,
$\theta=\pi,3\pi/2$ are unstable nodes and $\pm\theta^*$ are saddles.
Hence doing blow-down we get the phase portrait in a neighborhood
of the origin of system \eqref{blow}, shown in Figure \ref{blowf}.
Furthermore, taking into account the time scaling transformation $t=v\tau$
and that the infinite singular point of \eqref{infa} is a saddle,
we obtain the phase portraits near the boundary of the Poincar\'e disk
in Figure \ref{blowf}.

\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(120,36)(0,0)
\put(29,3){blow-down}
\put(0,0){\includegraphics[width=12cm]{fig4}}  % blow.eps
\end{picture}
\end{center}
\caption{Polar blow-down of the singular points of system \eqref{blow}.}
\label{blowf}
\end{figure}
\end{proof}

\begin{proof}[Proof of the phase portraits in Figure \ref{fig1}
for suystem \eqref{eq1} in Theorem \ref{thm3}] \quad\\
We define the following regions in the $(\rho, r)-$ plane:
\begin{gather*}
\omega_{1}=\{(\rho, r) : \rho<-1,\ r>0\},\\
\omega_{2}=\{(\rho, r) : -1<\rho<0,\ r>0\},\\
\omega_{3}=\{(\rho, r) : 0<\rho,\ r>0\},
\end{gather*}
 the straight lines:
\begin{gather*}
L_{01}=\{(\rho, r) : \rho<-1,\ r=0\}, \\
L_{02}=\{(\rho, r) : -1<\rho<0,\ r=0\}, \\
L_{03}=\{(\rho, r) : 0<\rho,\ r=0\}, \\
L_{04}=\{(\rho, r) : \rho=-1,\ r>0\}, \\
L_{05}=\{(\rho, r) : \rho=0,\ r\geq 0\},
\end{gather*}
and the point $P_{0}=(-1,0)$. In view of Propositions \ref{prop1} and \ref{prop0},
we show the bifurcation diagram of system \eqref{eq1} with respect to the
parameters $\rho$ and $r$ in Figure \ref{b0}.

\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{1mm} 
\begin{picture}(75,40)(0,0)
\put(0,10){\vector(1,0){75}}
\put(25,0){\line(0,1){40}}
\put(50,0){\vector(0,1){40}}
\put(52,40){$r$}
\put(19,26){$L_{04}$}
\put(44,26){$L_{05}$}
\put(10,20){$\omega_1$} 
\put(36,20){$\omega_2$}
\put(62,20){$\omega_3$}
\put(24,9){$\bullet$}
\put(9,6){$L_{01}$} 
\put(21,6){$P_{0}$}
\put(36,6){$L_{02}$}
\put(47,6){$0$}
\put(65,6){$L_{03}$}
\put(73,6){$\rho$} 
\end{picture}
\end{center}
\caption{Bifurcation diagram of system \eqref{eq1}.}
\label{b0}
\end{figure}

From Theorem \ref{thm2}, Propositions \ref{prop1} and \ref{prop0},
using the invariant straight lines $x=0$ and $x=r$ with $r\ge 0$,
and the invariant hyperbola $2xy=1$, we obtain the global phase
portraits of system \eqref{eq1} in the Poincar\'e disc described in
Figure \ref{fig1}. This completes the proof of Theorem \ref{thm3}.
\end{proof}

\section{Phase portraits for system \eqref{eq23}}\label{s5}

In this section we study system \eqref{eq23} for $a^2-b^2\neq 0$ and $a\ne 0$.

\subsection{Finite singular points}

The finite singular points of system \eqref{eq23} are characterized
in the following result.

\begin{proposition}\label{prop2}
System \eqref{eq23} has the following finite singular points.
\begin{itemize}
\item[(a)] If $\rho=0$ all the points of the hperbola
$\frac{x^2}{a^2-b^2}-\frac{a^2-b^2}{a^2}y^2=1$.

\item[(b)] For $\delta=1$, if $\rho\notin \{-1,0\}$ the singular points are
\begin{equation}\label{sp}
\begin{gathered}
M_\pm = (1,y^*_\pm)= \Big(1,\pm \frac{a\sqrt{1-(a^2-b^2)}}{a^2-b^2}\Big)
\quad \text{if } a^2-b^2\leq1, \\
N_\pm = (x^*_\pm,0)=\Big( \frac{\rho\pm \sqrt{\Delta}}{2(\rho+1)},0\Big) \quad
\text{if }\Delta\geq0,
\end{gathered}
\end{equation}
where $\Delta= \rho^2+4(\rho+1)(a^2-b^2)$.

If $\rho= -1$, system \eqref{eq23} has the singular point $N^c=(a^2-b^2, 0)$,
and the two singular points $M_\pm$ if $a^2-b^2<1$, or the unique singular
point $M_\pm=N^c=(1,0)$ if $a^2-b^2=1$.

\item[(c)] For $\delta=0$, if $\rho\notin \{-1,0\}$ the singular points are
\begin{equation}\label{sp1}
\begin{gathered}
M^0_\pm = (0,y^0_\pm)= \Big(0,\pm \frac{a}{\sqrt{b^2-a^2}}\Big) \quad
 \text{if } a^2-b^2<0, \\
N^0_\pm = (x^0_\pm,0)= \Big( \pm \sqrt{\frac{a^2-b^2}{\rho+1}},0\Big) \quad
 \text{if } \frac{a^2-b^2}{\rho+1}>0 .
\end{gathered}
\end{equation}
If $\rho= -1$ the singular points are $M^0_\pm$.
\end{itemize}
\end{proposition}

The proof of the above proposition follows easily studying the real solutions
of system \eqref{eq23}.

Let us denote $\eta=a^2-b^2$, we write the curve $\Delta=0$ of Proposition
 \ref{prop2} in the plane of $(\rho, \eta)$ as
\begin{equation}\label{c3}
\eta(\rho)=-\frac{\rho^2}{4(\rho+1)},
\end{equation}
which is the hyperbola with the two branches $\eta_\pm$ corresponding
to $\rho<-1$ and $\rho>-1$, respectively.

Now we define the following regions when $\delta=1$:
\begin{gather*}
R_{01}=\{(\rho, a^2-b^2): \rho<-1,\ a^2-b^2>\eta_+(\rho)\},\\
R_{02}=\{(\rho, a^2-b^2): -1<\rho<0,\ a^2-b^2<\eta_-(\rho)\},\\
R_{03}=\{(\rho, a^2-b^2): 0<\rho,\ a^2-b^2<\eta_-(\rho)\},\\
R_1=\{(\rho, a^2-b^2): \rho<-2,\ 1<a^2-b^2<\eta_+(\rho)\},\\
R_2=\{(\rho, a^2-b^2): -2<\rho<-1,\ 1<a^2-b^2<\eta_+(\rho)\},\\
R_3=\{(\rho, a^2-b^2): -1<\rho<0,\ a^2-b^2>1\},\\
R_4=\{(\rho, a^2-b^2): \rho>0,\ a^2-b^2>1\},\\
R_5=\{(\rho, a^2-b^2): \rho<-1,\ 0<a^2-b^2<1\},\\
R_6=\{(\rho, a^2-b^2): -1<\rho<0,\ 0<a^2-b^2<1\},\\
R_7=\{(\rho, a^2-b^2): 0<\rho,\ 0<a^2-b^2<1\},\\
R_8=\{(\rho, a^2-b^2): \rho<-1,\ a^2-b^2<0\},\\
R_9=\{(\rho, a^2-b^2): -1<\rho<0,\ \eta_-(\rho)<a^2-b^2<0\},\\
R_{10}=\{(\rho, a^2-b^2): \rho>0,\ \eta_-(\rho)<a^2-b^2<0\},
\end{gather*}
the curves:
\begin{align*}
L_0&=\{(\rho, a^2-b^2) : \rho=0\},\\
L_{11}&=\{(\rho, a^2-b^2):\rho<-2,\ a^2-b^2=\eta_+(\rho)\}, \\
L_{12}&=\{(\rho, a^2-b^2): -2<\rho<-1,\ a^2-b^2=\eta_+(\rho)\}, \\
L_{21}&=\{(\rho, a^2-b^2): -1<\rho<0,\ a^2-b^2=\eta_-(\rho)\}, \\
L_{22}&=\{(\rho, a^2-b^2): 0<\rho,\ a^2-b^2=\eta_-(\rho)\},\\
L_{31}&=\{(\rho, a^2-b^2): \rho<-2,\ a^2-b^2=1\},\\
L_{32}&=\{(\rho, a^2-b^2): -2<\rho<-1,\ a^2-b^2=1\},\\
L_{33}&=\{(\rho, a^2-b^2): -1<\rho<0,\ a^2-b^2=1\},\\
L_{34}&=\{(\rho, a^2-b^2): 0<\rho,\ a^2-b^2=1\},\\
L_{41}&=\{(\rho, a^2-b^2): \rho=-1, \ a^2-b^2<0\},\\
L_{42}&=\{(\rho, a^2-b^2): \rho=-1, \ 0<a^2-b^2<1\},\\
L_{43}&=\{(\rho, a^2-b^2): \rho=-1, \ 1<a^2-b^2\},\\
L_{0+}&=\{(\rho, a^2-b^2): \rho=0, \ 0<a^2-b^2\},\\
L_{0-}&=\{(\rho, a^2-b^2): \rho=0, \ a^2-b^2<0\},
\end{align*}
and the points $P_1=(-1,1)$ and $P_2=(-2,1)$, see Figure \ref{b1}.

\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{1mm} 
\begin{picture}(100,81)(0,0)
\put(0,30){\vector(1,0){100}}
\put(60,0){\vector(0,1){80}}
\dashline{1}(0,50)(100,50) 
\dashline{1}(40,0)(40,80) 
\qbezier(0,80)(29,20)(38,80)
\qbezier(42,0)(51,60)(100,0)
\put(2,78){$\eta_+$}
\put(62,78){$\eta$}
\put(4,72){$L_{11}$}
\put(31,72){$L_{12}$}
\put(41,72){$L_{43}$}
\put(61,72){$L_{0+}$}
\put(20,62){$R_{01}$}
\put(48,62){$R_3$}
\put(78,62){$R_4$}
\put(3,55){$R_1$}
\put(34,55){$R_2$}
\put(23,49.2){$\bullet$}
\put(39,49.2){$\bullet$}
\put(3,47){$L_{31}$}
\put(22,46){$P_2$}
\put(30,47){$L_{32}$}
\put(38,46){$P_1$}
\put(48,47){$L_{33}$}
\put(78,47){$L_{34}$}
\put(41,38){$L_{42}$}
\put(22,37){$R_5$}
\put(48,37){$R_6$}
\put(78,37){$R_7$}
\put(61,27){$0$}
\put(98,27){$\rho$}
\put(43,22){$R_9$}
\put(88,22){$R_{10}$}
\put(22,17){$R_8$}
\put(51,17){$R_{02}$}
\put(68,17){$R_{03}$}
\put(34,6){$L_{41}$}
\put(43.5,6){$L_{21}$}
\put(61,6){$L_{0-}$}
\put(87.5,6){$L_{22}$}
\put(93,1){$\eta_{-}$}
\end{picture}
\end{center}
\caption{Bifurcation diagram of system \eqref{eq23} when
$\delta=1$.}
\label{b1}
\end{figure}

We also define the following regions when $\delta=0$:
\begin{align*}
\Omega_1&=\{(\rho, a^2-b^2):\rho<-1,\ a^2-b^2<0\},\\
\Omega_2&=\{(\rho, a^2-b^2):-1<\rho<0,\ a^2-b^2<0\},\\
\Omega_3&=\{(\rho, a^2-b^2):0<\rho,\ a^2-b^2<0\},\\
\Omega_4&=\{(\rho, a^2-b^2):0<\rho,\ a^2-b^2>0\},\\
\Omega_5&=\{(\rho, a^2-b^2):-1<\rho<0,\ a^2-b^2>0\},\\
\Omega_6&=\{(\rho, a^2-b^2):\rho<-1,\ a^2-b^2>0\},
\end{align*}
and the straight lines $L_{0+}$, $L_{0-}$ and
\begin{gather*}
L_{44}=\{(\rho, a^2-b^2)\ : \rho=-1, \ a^2-b^2<0\},\\
L_{45}=\{(\rho, a^2-b^2)\ : \rho=-1, \ 0<a^2-b^2\},
\end{gather*}
shown in Figure \ref{b10}.

\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{1mm} 
\begin{picture}(75,43)(0,0)
\put(0,20){\vector(1,0){75}}
\dashline{1}(25,0)(25,40)
\put(50,0){\vector(0,1){40}}
\put(52,40){$\eta$}
\put(26,33){$L_{45}$}
\put(51,33){$L_{0+}$}
\put(11,27){$\Omega_6$}
\put(36,27){$\Omega_5$}
\put(61,27){$\Omega_4$}
\put(73,16){$\rho$} 
\put(11,13){$\Omega_1$}
\put(36,13){$\Omega_2$}
\put(61,13){$\Omega_3$}
\put(26,4){$L_{44}$}
\put(51,4){$L_{0-}$}
\end{picture}
\end{center}
\caption{Bifurcation diagram of sysetem \eqref{eq23} when $\delta=0$.}
\label{b10}
\end{figure}

\begin{proposition} \label{Tr01}
System \eqref{eq23} has the following finite singular points if its
parameters $(\rho, a^2-b^2)$ are in
\begin{itemize}
\item[] $(R_1\cup R_2)$ a saddle $N_+$ and a center $N_-$, or a center
$N_+$ and a saddle $N_-$.

\item[] $(R_5)$ four singular points: a stable node $M_+$,
 an unstable node $M_-$, and two saddles $N_\pm$.

\item[] $(R_6)$ four singular points: a stable node $M_+$,
 an unstable node $M_-$, and a saddle $N_+$ and a center $N_-$.

\item[] $(R_7)$ four singular points: two saddles $M_\pm$,
 a center $N_+$ and a saddle $N_-$.

\item[] $(R_9)$ four singular points: a stable node $M_-$, an unstable node $M_+$, a saddle $N_+$ and a center $N_-$.

\item[] $(R_{10})$ four singular points: three saddles $M_\pm$ and $N_-$, and a center $N_+$.

\item[] $(\Omega_1\cup R_8)$ four singular points: an unstable node $M^0_+$ or $M_+$,
 a stable node $M^0_-$ or $M_-$, and saddles $N^0_\pm$ or $N_\pm$.

\item[] $(\Omega_2\cup R_{02}\cup L_{44})$ two singular points: an unstable node $M^0_+$ or $M_+$,
and a stable node $M^0_-$ or $M_-$.

\item[] $(\Omega_3\cup R_{03})$ two singular points: $M^0_\pm$ or $M_\pm$ are saddles.

\item[] $(\Omega_4\cup R_4)$ two singular points: saddles $N^0_\pm$ or $N_\pm$.

\item[] $(\Omega_5\cup R_3)$ two singular points: centers $N^0_\pm$ or $N_\pm$.

\item[] $(\Omega_6\cup R_{01}\cup L_{45})$] no singular points.

\item[] $(L_0+\cup L_{0-})$ all singular points on the hyperbola $\mathcal H$.

\item[] $(L_{11}\cup L_{12})$ a nilpotent cusp $N$.

\item[] $(L_{21})$ three singular points: an unstable
node $M_+$, a stable node $M_-$, and a nilpotent cusp $N$.

\item[] $(L_{22})$ three singular points: two saddles $M_\pm$, and a
 nilpotent cusp $N$.

\item[] $(L_{31})$ two singular points: a nilpotent singular point
$M_\pm=N_-=(1,0)$ union of one elliptic sector with one hyperbolic sector,
and a hyperbolic saddle $N_+=(-1/(\rho+1),0)$.

\item[] $(L_{32})$ two singular points: a nilpotent singular point
$M_\pm=N_+=(1,0)$ union of one elliptic sector with one hyperbolic sector,
and a hyperbolic saddle $N_-=(-1/(\rho+1),0)$.

\item[] $(L_{33})$ two singular points: a nilpotent singular point
$M_\pm=N_+=(1,0)$ union of one elliptic sector with one hyperbolic sector, and
a center $N_-=(-1/(\rho+1),0)$.

\item[] $(L_{34})$ two singular points: a nilpotent saddle $M_\pm=N_+=(1,0)$, and
a hyperbolic saddle $N_-=(-1/(\rho+1),0)$.


\item[] $(L_{41})$ three singular points: a saddle $N^c$, an unstable hyperbolic node
$M_+$, and a stable hyperbolic node $M_-$.

\item[] $(L_{42})$ three singular points: a saddle $N^c$, an unstable hyperbolic
 node $M_-$, and a stable hyperbolic node $M_+$.

\item[] $(L_{43})$ a center $N^c$.

\item[] $(P_1)$ $M_\pm=N_\pm=(1,0)$ is a nilpotent singular point formed by one
elliptic sector, one hyperbolic sector and two parabolic sectors.

\item[] $(P_2)$ $M_\pm=N_\pm=(1,0)$ is a degenerated singular point formed by two
parabolic sectors and two hyperbolic sectors.
\end{itemize}
\end{proposition}

\begin{proof}
On $L_{0+}$ and $L_{0-}$ we have $\rho=0$. Hence the straight lines
$x=\text{constant}$ are invariant of system \eqref{eq23}, and the hyperbola
\eqref{hy} is filled with singular points, see the
phase portraits for $L_{0+}$ and $L_{0-}$ in Fig. \ref{LC1}.

In the following we always assume $\rho \ne 0$.
\begin{itemize}
\item[(A)]  Assume $\delta=1$ and distinguish two cases
in the study of the finite singular points of system \eqref{eq23}.
\end{itemize}

\noindent{\it Case} 1: On the invariant straight
line $x=1$. There are two
singular points $M_\pm$ of system \eqref{eq23} when $a^2-b^2<1$;
They coincide into a unique singular point $M(1,0)$ when $a^2-b^2=1$,
 and no singular point when $a^2-b^2>1$, see \eqref{sp}.

\noindent{\it Subcase} 1.1: $a^2-b^2<1$. The
Jacobian matrix of system \eqref{eq23} at $M_\pm$ are
\[
\begin{pmatrix}
\frac{\rho(a^2- b^2)y^*_\pm}{a^2}& 0\\
0 & -\frac{2(a^2- b^2)y^*_\pm}{a^2}
\end{pmatrix}.
\]
Therefore $M_\pm$ are saddles if $\rho>0$, and $M_+$ is a stable hyperbolic
node and $M_-$ is an unstable hyperbolic node if $\rho<0$ and $0<a^2-b^2<1$,
while $M_+$ is an unstable hyperbolic node and $M_-$ is a
stable hyperbolic node if $\rho<0$ and $a^2-b^2<0$,
see for more details \cite[Theorem 2.15]{DLA} where are described the
local phase portraits of the hyperbolic singular points.


\noindent{\it Subcase} 1.2: $a^2-b^2=1$. The Jacobian matrix of system
\eqref{eq23} at $M$ is
$$
J_M= \begin{pmatrix}
0 & 0\\
\rho+2 & 0
\end{pmatrix}.
$$
When $\rho\ne -2$, $M$ 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
$M$ is a nilpotent saddle if $\rho>0$, and if $\rho<0$ and different from
$-2$ is union of one elliptic sector with one hyperbolic sector.

When $\rho=-2$, $N_\pm$ and $M$ meet with each other into a degenerated
singular point $M$. Using the polar blowing--up
centered at $M$, i.e. $x=r\cos\theta+1$ and $y=r\sin\theta$,
system \eqref{eq23} writes
\begin{equation}\label{bl}
\begin{gathered}
\dot r=-r^2\sin\theta(\cos^2(\theta)(a^2+1)+1),\\
\dot\theta=-r\cos\theta(\cos^2(\theta)(a^2+1)-1).
\end{gathered}
\end{equation}
The singular points of system \eqref{bl} on $\{r=0\}$ are located at
$\theta=\pm\pi/2$, $\theta_k$, where $\theta_k$ satisfies
$$
(a^2+1)\cos^2\theta_k=1, \quad k=1,2,3,4,
$$
and $-\pi<\theta_3=\theta_1-\pi<\theta_4=\theta_2-\pi<\theta_1<\theta_2
=\pi-\theta_1<\pi$. All the singularities on $S^1\times\{0\}$ are hyperbolic.
Then $(0,\pi/2)$ is a stable node
$(0,-\pi/2)$ an unstable node and $\theta_k$ saddles. Doing a blowing down
we obtain that $M$ is formed by the union of two hyperbolic sectors and
four parabolic sectors, see the phase portrait $P_2$ in Figure \ref{LC1}.

\noindent{\it Case} 2: Singular points on the straight line $y=0$.
When $a^2-b^2<0$ the singular points $N_\pm$ or $N$ are located between
the two branches of the hyperbola. When $a^2-b^2>0$ it is easy to check
that there are two singular points $N_\pm$ on $y=0$ if and only if
$(\rho,a^2-b^2)\in R_1\cup R_2\cup\cdots\cup R_7\cup L_{31}\cup\cdots\cup L_{34}$.
The singular points $N_\pm$ coincide with $N$ on $y=0$ if and only if
$(\rho,a^2-b^2)\in L_{11}\cup L_{12}\cup L_{21}\cup L_{22}$, and with
$N^c$ if and only if $\rho=-1$. It is important for the phase portraits the
location of the singular points $N_\pm$ or $N$ and of the two branches of
the hyperbola, see Figures \ref{LC1}, \ref{RF1} and \ref{b1}.


\noindent{\it Subcase} 2.1: The distribution of the singular points on $y=0$.
If $a^2-b^2>0$, it is easy to check that
\begin{equation}\label{ch}
\begin{gathered}
(x_+^*\pm\sqrt{a^2-b^2})(x_-^*\pm\sqrt{a^2-b^2})
=\frac{\rho\sqrt{a^2-b^2}}{\rho+1}(\sqrt{a^2-b^2}\mp1),\\
(x_+^*-1)(x_-^*-1)=\frac{1-(a^2-b^2)}{\rho+1}.
\end{gathered}
\end{equation}
which implies that in $R_1\cup R_2$, that is when $a^2-b^2>1$ and
$\rho<-1$, the singular points $N_\pm$ are located at the same side of
the hyperbola $\mathcal H$ and of the line $\mathcal L$ given in \eqref{ic23}.
 Furthermore,  from \eqref{sp} we have
\begin{gather*}
x^*_->\frac{\rho}{2(\rho+1)}>1, \quad \text{for }  -2<\rho<-1, \\
x^*_-<\frac{\rho-\sqrt{\rho^2+4(\rho+1)}}{2(\rho+1)}=1, \quad \text{for }  \rho<-2,
\end{gather*}
Similarly, in $R_3$, that is when $a^2-b^2>1$ and $\-1<\rho<0$,
the singular points $N_\pm$ are located at the two sides of the hyperbola
$\mathcal H$ and of the line $\mathcal L$, respectively.
In $R_4$, that is when $a^2-b^2>1$ and $\rho>0$, the singular points
$N_\pm$ are located at the same side of the hyperbola $\mathcal H$,
while in the two sides of the line $\mathcal L$, respectively.

In fact we have that $-\sqrt{a^2-b^2}<x_+^*<x_-^*<1$ in $R_1$,
$1<\sqrt{a^2-b^2}<x_+^*<x_-^*$ in $R_2$,
$x_-^*<-\sqrt{a^2-b^2}<\sqrt{a^2-b^2}<x_+^*$ in $R_3$, and
$-\sqrt{a^2-b^2}<x_-^*<1<x_+^*<\sqrt{a^2-b^2}$ in $R_4$.

In the same way we can obtain that $-\sqrt{a^2-b^2}<x_+^*<\sqrt{a^2-b^2}<1<x_-^*$
in $R_5$, $x_-^*<-\sqrt{a^2-b^2}<x_+^*<\sqrt{a^2-b^2}<1$ in $R_6$, and
$-\sqrt{a^2-b^2}<x_-^*<\sqrt{a^2-b^2}<x_+^*<1$ in $R_7$.

For $a^2-b^2<0$ it follows from \eqref{sp} and \eqref{ch} that $x_+^*<0<1<x_-^*$
in $R_8$, $x_-^*<x_+^*<0$ in $R_9$, and $0<x_-^*<x_+^*<1$ in $R_{10}$.

On the curves $L_{11}\cup L_{12}\cup L_{21}\cup L_{22}$, if $\Delta=0$, then
the singular points $N_\pm$ meet into a unique singular point $N$ on $y=0$,
i.e. $x^*_\pm=x^*$. In view of \eqref{c3} it follows that $0<x^*<1$ in
$L_{11}\cup L_{22}$, $x^*>\sqrt{a^2-b^2}$ in $L_{12}$, and $x^*<0$ in $L_{21}$.

If $a^2-b^2=1$ then $0<x_+^*=-1/(\rho+1)<x_-^*=1$ in $L_{31}$,
$1=x_+^*<x_-^*=-1/(\rho+1)$ in $L_{32}$, $x_-^*=-1/(\rho+1)<-1<x_+^*=1$
in $L_{33}$, and $-1<x_-^*=-1/(\rho+1)<x_+^*=1$ in $L_{34}$.


\noindent{\it Subcase} 2.2: Classification of the singular points.
If $a^2-b^2=1$ there are two singular points on $y=0$, $N_-$ and $M=N_+$
for $\rho>-2$, while $N_+$ and $M=N_-$ for $\rho<-2$.
The singular point $M$ is also on the invariant straight line $x=1$,
which has been studied in Subcase 1.2.
The Jacobian matrix of system \eqref{eq23} at the other singular point
$N_+$ or $N_-$ is
\begin{equation*}
J= \left(\begin{array}{cc}
0& -\frac{\rho(\rho+2)}{a^2(\rho+1)}\\
-(\rho+2) & 0
\end{array}\right).
\end{equation*}
Using the fact that system \eqref{eq23}
is reversible with respect to the $x$--axis from Remark \ref{rm},
we can obtain that $N_+$ is a saddle in $L_{31}$, $N_-$ a saddle
in $L_{32}\cup L_{34}$, and $N_-$ a center in $L_{33}$.

If $\rho\ne -1$ and $\Delta>0$, then system
\eqref{eq23} has two singular points $N_\pm= (x^*_\pm,0)$ on $y=0$, see
\eqref{sp}. The Jacobian matrix of system \eqref{eq23} at the points
$N_\pm$ is
\begin{equation}\label{jac}
J= \begin{pmatrix}
0& \frac{\rho(a^2-b^2)}{a^2}(x_\pm^*-1)\\
\frac{\pm \sqrt{\Delta}}{a^2-b^2} & 0
\end{pmatrix}.
\end{equation}
It is easy from Remark \ref{rm} to prove that $N_+$ is a saddle and $N_-$
a center in $R_1\cup R_6\cup R_9$,
$N_+$ is a center and $N_-$ a saddle in $R_2\cup R_7\cup R_{10}$, $N_\pm$
are centers in $R_3$, and $N_\pm$ are saddles in $R_4\cup R_5\cup R_8$.

If $\Delta=0$ we have from
\eqref{c3} that $\eta=\eta_\pm(\rho)$, 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
$\tau=\rho^3(\rho+2)t/(8a^2(\rho+1)^2)$, we obtain
\begin{gather*}
\dot x= y-\frac{2(\rho+1)}{\rho+2}xy, \\
\dot y= -32\frac{(\rho+1)^4a^2}{\rho^5(\rho+2)}x^2+\frac{2(\rho+1)}{\rho(\rho+2)}y^2.
\end{gather*}
By \cite[Theorem 3.5]{DLA} the origin of the previous system is a
cusp in $L_{11}\cup L_{12}\cup L_{21}\cup L_{22}$.


If $\rho=-1$ system \eqref{eq23} has a singular point $N^c$ on $y=0$.
The Jacobian matrix of system \eqref{eq23} at the point $N^c$ is
\begin{equation*}
J= \begin{pmatrix}
0& -\frac{(a^2-b^2)(a^2-b^2-1)}{a^2}\\
\frac{1}{a^2-b^2} & 0
\end{pmatrix},
\end{equation*}
which implies that $N^c$ is a saddle in $L_{41}\cup L_{42}$, and a center in
$L_{43}$. If $a^2-b^2=1$ system \eqref{eq23} has a unique singular point
$M=N^c$ at $P_1$, which is union of one elliptic sector with one
hyperbolic sector as in the proof in Subcase 1.2.
\begin{itemize}
\item[(B)] Assume $\delta=0$.
\end{itemize}
There are two singular points $M^0_\pm$ on $x=0$
and two singular points $N^0_\pm$ on $y=0$ when $a^2-b^2<0$ and $\rho<-1$,
two singular points $M^0_\pm$ on $x=0$ when $a^2-b^2<0$ and $-1\leq$,
two singular points $N^0_\pm$ on $y=0$ outside of the two branches
of hyperbola when $a^2-b^2>0$ and $-1<\rho<0$, and two singular points
$N^0_\pm$ on $y=0$ between the two branches of hyperbola when $a^2-b^2>0$
and $\rho>0$. There is no singular point when $a^2-b^2>0$ and
$\rho\leq -1$. See \eqref{sp1} in Proposition \ref{prop2} and Figure \ref{RF1}.

The Jacobian matrices of system \eqref{eq23} at the points
$M^0_\pm$ and $N^0_\pm$ are
$$
\begin{pmatrix}
\frac{\rho(a^2- b^2)y^0_\pm}{a^2}& 0\\
0 & -\frac{2(a^2- b^2)y^0_\pm}{a^2}
\end{pmatrix}, \quad
\begin{pmatrix}
0& \frac{\rho(a^2- b^2)}{a^2}x^0_\pm\\
\frac{2(\rho+1)}{a^2- b^2}x^0_\pm & 0
\end{pmatrix},
$$
respectively. Similarly it is easy to obtain that $M_\pm$ are saddles
in $\Omega_3$, and $M_+$ is an unstable node, and $M_-$ a stable node
in $\Omega_1\cup\Omega_2\cup L_{44}$, $N_\pm$ are saddles in
$\Omega_1\cup\Omega_4$, $N_\pm$ are centers in $\Omega_5$, and there are no
singular points in $\Omega_6$ and $L_{45}$.
\end{proof}

\subsection{Infinite singular points}

\begin{proposition}\label{p2}
The following two statements hold.
\begin{itemize}
\item[(a)] If $\rho\ne -1$ system \eqref{eq23} has two pairs one of infinite
 saddles and the other of nodes if $\rho<0$, and two pairs of nodes if $\rho>0$.

\item[(b)] If $\rho=-1$ the infinity of system \eqref{eq23} is filled of
singular points.
\end{itemize}
\end{proposition}

\begin{proof}
Notice $a^2-b^2\neq 0$ for system \eqref{eq23}.
Doing the Poincar\'e transformation \eqref{chau}
and the time rescaling $t=v\tau$, system \eqref{eq23} in the
local chart \eqref{chau} is
\begin{equation}\label{infu}
\begin{gathered}
\dot u= -v^2+\Big(\frac{ (a^2-b^2)u^2}{a^2}-\frac{1}{a^2-b^2}\Big)
 \Big(\delta\rho v- (\rho+1)\Big),\\
\dot v=\rho \frac{a^2-b^2}{a^2}u(\delta v-1)v.
\end{gathered}
\end{equation}
First considering the infinite singular of system \eqref{eq23} with $\delta=1$.
If $\rho\ne -1$ there is two singular points $P_N(\pm\frac{a}{a^2-b^2},0)$ of
system \eqref{infu} on $v=0$. The eigenvalues of the Jacobian matrice at
 $P_N$ are $\mp\frac{2(\rho+1)}{a}$ and $\mp\frac{\rho}{a}$, which implies
that system \eqref{eq23} has two pairs of nodes if $\rho<-1$ or $0<\rho$,
and two pairs of infinite saddles if $-1<\rho<0$.


Furthermore, taking the Poincar\'e transformation \eqref{chav}
and the time rescaling $t=v\tau$, system \eqref{eq23} in the
local chart \eqref{chav} is
\begin{equation}\label{infv}
\begin{gathered}
\dot u= uv^2+\Big(\frac{u^2}{a^2-b^2}-\frac{a^2-b^2}{a^2}\Big)
 (\delta\rho v-(\rho+1)u),\\
\dot v=v^3+\frac{\delta\rho uv^2}{a^2-b^2}
 +\Big(\frac{a^2-b^2}{a^2}-\frac{(\rho+1)u^2}{a^2-b^2}\Big)v.
\end{gathered}
\end{equation}
We first consider system \eqref{eq23} with $\delta=1$. If $\rho\ne -1$
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 $\frac{a^2-b^2}{a^2}(\rho+1)$ and $\frac{a^2-b^2}{a^2}$,
which implies that system \eqref{eq23}
has a pair of infinite saddles if $\rho<-1$, and a pair of nodes if
$\rho>-1$.

If $\rho= -1$ from \eqref{infu} and \eqref{infv}
the infinity $v=0$ of the Poincar\'e disc is filled with singular
points. Furthermore we claim that the orbits from the infinity will go to
the singular points $M_\pm$. Now we prove the claim. In fact, when $\rho= -1$
we reduce the common factor $v$ of the vector field of system \eqref{infu},
then we writes it as
\begin{equation}\label{infc}
\begin{gathered}
\dot u= -v-\Big(\frac{ (a^2-b^2)u^2}{a^2}-\frac{1}{a^2-b^2}\Big),\\
\dot v=- \frac{a^2-b^2}{a^2}u(v-1).
\end{gathered}
\end{equation}
We obtain the following first integral of system \eqref{infc}
$$
H(u,v):=(v-1)^{-2}\Big(u^2+\frac{2a^2}{a^2-b^2}v
 -\frac{a^2(1+a^2-b^2)}{(a^2-b^2)^2}\Big)=(v-1)^{-2}h(u,v).
$$
If $a^2-b^2<1$ system \eqref{infc} has a pair of singular points $(u,v)=M_\pm$,
which are in fact, the singular points of system \eqref{eq23} in the finite plane,
see \eqref{sp}. Noting that $M_\pm$ are located on the invariant straight
line $v-1=0$, and $h(u,v)|_{M_\pm}=0$, we know that $M_\pm$ have to be
located on the curve $H(u,v)=c$ for any $c\in \mathbb{R}$, especially on the curve
going through $v=0$, see (a) in Figure \ref{infinity-1}.

We complete the proof of our claim. The phase portraits of system \eqref{eq23}
in $L_{41}\cup L_{42}$ are shown in Figure \ref{LC1}. In $L_{43}$ system
\eqref{eq23} has no singular point on $v-1=0$, so any curve $H(u,v)=c$
does not intersect with the line $v-1=0$.

Next we study the infinite singular points of system \eqref{eq23} with $\delta=0$.
Taking $\delta=0$ in system \eqref{infu} and \eqref{infv}, we obtain two singular
points $P_N$ of system \eqref{infu}, and the singular point $(0,0)$ of system
\eqref{infv} if $\rho\neq -1$. In a similar way as the above, we can get
that their singular points have the same properties as the systems with $\delta=1$.
But if $\rho= -1$ system \eqref{eq23} in the local chart \eqref{chau} is
\begin{equation}\label{infc0}
\begin{gathered}
\dot u= -v^2,\\
\dot v=\frac{a^2-b^2}{a^2}uv.
\end{gathered}
\end{equation}
So the infinity $v=0$ of the Poincar\'e disc is filled with singular
points. System \eqref{infc0} is equivalent to a linear differential
system with a saddle if $a^2-b^2<0$, or with a center if $a^2-b^2>0$,
and in both case with a straight line filled by singularities, see the
phase portraits (b) and (c) in Figure \ref{infinity-1}.
\end{proof}

\begin{figure}[ht]
\begin{center}

\setlength{\unitlength}{1mm}
\begin{picture}(120,50)(0,0)
\put(12,47){$u$} 
\put(60,48){$u$} 
\put(102,49){$u$}
\put(20.5,39.5){$M_+$} 
\put(36,22.5){$v$}
\put(76,23.5){$v$}
\put(118,24.5){$v$}
\put(20.5,13){$M_-$}
\put(18,1){(a)}
\put(56,1){(b)}
\put(98,1){(c)}
\put(0,6){\includegraphics[width=12 cm]{fig8}} % infinity-1.eps
\end{picture}
\end{center}
\caption{Infinite singular points of system \eqref{eq23} for $\rho=-1$.}
\label{infinity-1}
\end{figure}


\begin{proof}[Proof of the phase portraits in Figures \ref{LC1} and \ref{RF1}
for system \eqref{eq23} in Theorem \ref{thm3}] \quad\\
By Theorem \ref{thm2}, Propositions \ref{prop0}, \ref{Tr01} and
\ref{p2}, and using the invariant straight line $x=\delta$ with $\delta=0,1$
and the invariant hyperbola $\mathcal H$ in \eqref{ic23}, we obtain the global phase
portraits of system \eqref{eq23} in Poincar\'e disc described in
Figure \ref{LC1} and \ref{RF1}.
\end{proof}


\subsection*{Acknowledgements}

J. Llibre was supported by grant MTM2013-40998-P from MINECO, by
grant  2014 SGR568 from AGAUR, and by grants FP7-PEOPLE-2012-IRSES 318999
and 316338 from the recruitment program of high-end foreign experts of China.
J. Yu was supported by grants 11431008 and 11771282 from the
 NNSF of China, and by  grant 15ZR1423700 from the NSF of Shanghai.

\begin{thebibliography}{99}

\bibitem{AL} J. C. Art\'e, J. Llibre;
\emph{Quadratic Hamiltonian vector fields}, J. Differential Equations
\textbf{107} (1994), 80--95.

\bibitem{ALV} J. C. Art\'es, 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{CL} C. Christopher, C. Li;
 \emph{Limit cycles of differential equations}, CRM Barcelona, Birkhauser Verlag,
Basel, 2007.


\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{Dar} G. Darboux;
\emph{M\'emoire sur les \'equations diff\'erentielles alg\'ebriques du premier
 ordre et du premier degr\'e (m\'elanges)}, Bull. Sci. Math.
\textbf{124A} (1878),60--96,123--144,151--200.


\bibitem{Du} H. Dulac;
\emph{D\'etermination et integration d'une certaine classe d'\'equations
diff\'erentielle ayant par point singulier un centre}, Bull. Sci.
Math. S\'er. (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{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{LMR} J. Llibre, M. Messias, A. Reinol;
\emph{Global phase portraits of quadratic and cubic systems with two
nonconcentric circles as invariant algebraic curves}, to 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{LYu} J. Llibre, J. Yu;
\emph{Global phase portraits of quadratic systems with an ellipse and a
straight line as invariant algebraic curves},
Elect. J. Differential Equations, \textbf{314} (2015), 1--14.

\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{NW} T. A. Newton;
\emph{Two dimensional homogeneous quadratic differential systems},
SIAM Review \textbf{20} (1978), 120--138.


\bibitem{Reyn1} J. W. Reyn;
\emph{Phase portraits of planar quadratic systems}, Mathematics and
Its Applications, \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}