\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 75, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/75\hfil Non-existence of limit cycles]
{Non-existence of limit cycles for planar vector fields}

\author[J. Gin\'e \hfil EJDE-2014/75\hfilneg]
{Jaume Gin\'e}  % in alphabetical order

\address{Jaume Gin\'e \newline
Departament de Matem\`atica, Universitat de Lleida,
Avda. Jaume II, 69; 25001 Lleida, Catalonia, Spain}
\email{gine@matematica.udl.cat}

\thanks{Submitted December 10, 2012. Published March 18, 2014.}
\subjclass[2000]{34C05, 34C23, 37G15}
\keywords{Planar analytic vector fields; analytic differential systems;
 limit cycles}

\begin{abstract}
 This article presents sufficient conditions for the non-existence of  
 limit cycles for  planar vector fields. Classical methods for the nonexistence
 of limit cycles are connected with the theory developed here.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}

Two fundamental problems of the qualitative theory of planar differential 
equations are the center problem, and the determination of the number of 
limit cycles and their location in the phase space, see for instance 
\cite{ALGM,G2,P}. We recall that a limit cycle is an isolated periodic
solution of a differential equation, see \cite{Y}. The notion of limit 
cycle of a planar vector field was defined by Poincar\'e \cite{Po} at 
the end of 19th century. It was not until some decades later that van 
der Pol \cite{V}, Li\'enard \cite{L} and Andronov \cite{ALGM} proved that 
the periodic orbit of a self-sustained oscillation occurring in a vacuum 
tube circuit was in fact a limit cycle. This fact had been established 
by Poincar\'e himself 20 years before, see \cite{GP}. Later on the limit 
cycles have been studied extensively by mathematicians and physicists focusing 
on properties such as  nonexistence, existence, and uniqueness.

The classical method for proving the nonexistence of limit cycles in a simply
connected region is the Bendixson--Dulac method, see for instance \cite{Y} 
where variants can be found. The method of Dulac functions also gives upper 
bounds for the number of closed trajectories in a multiply connected region, 
see also \cite{Y}.

The problem of existence is the subject of the Poincar\'e--Bendixson theorem. 
The uniqueness problem is, in general, more difficult. Some criteria are 
known but the sufficient conditions of the known methods are very restrictive, 
see \cite{Y} and references therein. The Poincar\'e return map defined in a 
transversal section to the planar flow is one of the best methods for studying 
the nonexistence, existence and uniqueness of limit cycles, but in general 
such analysis is not very easy.

When the planar differential system has more than one limit cycle, the problem 
of their distribution on the plane appears. In fact all these problems are 
collected in the 16th Hilbert problem about the maximum number and distribution 
of limit cycles for a polynomial vector field of degree $n$, see \cite{H}.


In this work we consider ${\mathcal{C}}^1$ two-dimensional autonomous systems 
of differential equations
\begin{equation}
\dot{x}=P(x,y), \quad \dot{y}=Q(x,y)\,,
\label{01}
\end{equation}
defined on an open subset $U$ of $\mathbb{R}^2$ and their corresponding vector field 
$\mathcal{X}=P \partial/ \partial x+Q\partial/ \partial y$ on $U$. 
A non-constant ${\mathcal{C}}^1$  function $V:U \to \mathbb{R}$ is an 
\emph{inverse integrating factor} for $\mathcal X$ if it satisfies 
$\mathcal{X} V= V \operatorname{div} \mathcal{X}$ on $U$; that is, if
\begin{equation}\label{02}
P \frac{\partial V}{\partial x}+ Q \frac{\partial V}{\partial y}=
V \Big( \frac{\partial P}{\partial x}+ \frac{\partial Q}{\partial y} \Big),
\end{equation}
holds on $U$. In such a case the function $1/V$ is an integrating factor for 
the system of differential equations on
$U \setminus \Sigma$, where $\Sigma = \{ (x,y) \in U : V(x,y)=0 \}= V^{-1}(0)$. 
Since  $V^{-1}(0)$ is made up by solutions of the differential system \eqref{01}
the set $V^{-1}(0)$ is invariant under the local flow induced by $\mathcal{X}$ 
and as we will see immediately it contains any limit cycle contained in $U$ 
of $\mathcal{X}$. The knowledge of an inverse integrating factor defined in
 $U$ allows the computation of a first integral
in $W = U \setminus \{ V=0 \}$ by the  line integral
\[
H(x,y) = \int_{(x_0,y_0)}^{(x,y)} \frac{P(x,y) dy  - Q(x,y) dx}{V(x,y)},
\]
where $(x_0,y_0) \in W$ is any point. The inverse integrating factor has 
also been used to study the center problem, see \cite{CGG,CGGL,G}.

In \cite{GLV} the following result was established.

\begin{theorem}\label{t1}
Let $(P,Q)$ be a $\mathcal{C}^1$ vector field defined in the open subset 
$U$ of $\mathbb{R}^2$. Let $V=V(x,y)$ be a $\mathcal{C}^1$ solution of the linear
partial differential equation \eqref{02}.
If $\gamma$ is a limit cycle of the vector field $(P,Q)$ in the
simply connected domain of definition of $V$, then $\gamma$ is contained in $\Sigma
= \{ (x,y) \in U : V(x,y)=0 \}$.
\end{theorem}

The proof of this result is straightforward. The existence of the inverse 
integrating factor $V$ defined in a simply connected region $U$ implies 
that  the vector field $(P/V,Q/V)$ is Hamiltonian in $U \setminus \Sigma$. 
Since the flow of a Hamiltonian vector field preserves area and in a 
neighborhood of a limit cycle a flow does not preserve the area, 
the theorem follows. In fact the same result is obtained for any polycycle which 
is a limit set of the vector field $(P,Q)$, see \cite{EP,GG} and the references 
therein. 
Moreover it is known that there always exists a smooth inverse 
integrating factor in a neighborhood of a limit cycle
of a planar analytic vector fields, see \cite{EP}.
From this result it is clear that the inverse integrating factor 
has an important role in the qualitative study of differential equations 
and in particular in the study of the limit cycles, see \cite{GG} and references 
therein. From Theorem \ref{t1} we can give the following corollary.

\begin{corollary}  \label{coro2}
Let $(P,Q)$ be a $\mathcal{C}^1$ vector field defined in the open subset 
$U$ of $\mathbb{R}^2$. Let $V=V(x,y)$ be a $\mathcal{C}^1$ solution of the linear
partial differential equation \eqref{02}. Let
 $\Sigma = \{ (x,y) \in U : V(x,y)=0 \}$.
   The divergence $\partial (P/V) / \partial x+ \partial (Q/V)/ \partial y$ 
is identically zero in the simply connected domain of definition of 
$V \setminus \Sigma$ and then any limit cycle (in fact a closed trajectory) 
which lies entirely in the domain of definition of $V$ must be contained in $\Sigma$.
\end{corollary}

In this article we are interested in the study of
nonexistence of limit cycles using the inverse integrating factor. 
From Theorem \ref{t1} we can study the existence and nonexistence of limit 
cycles of planar vector fields from the explicit knowledge of an inverse 
integrating factor.
For instance, in \cite{DA} it is shown that system \eqref{01} with a Darboux
inverse integrating factor of the form $V = \exp(R)$ with rational $R$ cannot 
have limit cycles. It is clear that this result is, in fact, a straightforward 
consequence of Theorem \ref{t1}. First we recall the classical theorems about 
nonexistence of limit cycles.

\begin{theorem}[Bendixson] \label{B}
If the divergence $\partial P / \partial x+ \partial Q/ \partial y$ of system \eqref{01} has constant sign in a simply connected region $U$, and is not
identically zero on any subregion of $U$, then system \eqref{01} does not
 possess any limit cycle (in fact a closed trajectory) which lies entirely in $U$.
\end{theorem}

The proof is by contradiction assuming the existence of a limit cycle and 
applying Green's formula.
In fact mainly depends on the following fact.

\begin{proposition} \label{p0} 
In the interior $D$ of any closed trajectory of system \eqref{01}
of a simply connected region we have
\begin{equation}\label{es}
\int\int_D \big( \frac{\partial P}{\partial x}+ \frac{\partial Q}{\partial y} \big) 
\,dx\,dy =0.
\end{equation}
\end{proposition}

The  classical Bendixson theorem and the inverse integrating factor are related 
by the following proposition.

\begin{proposition} \label{p1} 
Let $(P,Q)$ be a $\mathcal{C}^1$ vector field defined in the open subset 
$U$ of $\mathbb{R}^2$. Let $V=V(x,y)$ be a $\mathcal{C}^1$ solution of the linear
partial differential equation \eqref{02}.
Let $\Sigma = \{ (x,y) \in U : V(x,y)=0 \}$.
If the divergence $\partial P / \partial x+ \partial Q/ \partial y$ 
of system \eqref{01} has constant sign in the simply connected domain of
definition of $V \setminus \Sigma$, then any limit cycle 
(in fact a closed trajectory) which lies entirely in the simply connected 
domain of definition of $V$ must be contained in $\Sigma$.
\end{proposition}

\begin{proof}
  Assume that $\gamma$ is a closed trajectory of system \eqref{01} which lies
entirely in the simply connected domain of definition of $V$ then integrating 
along $\gamma$ the equality \eqref{02} we obtain
\[
\oint_{\gamma} \frac{P \frac{\partial V}{\partial x}
+ Q \frac{\partial V}{\partial y}}{V}  dt = \oint_{\gamma}  
\frac{d \ln|V|}{dt} dt = - \oint_{\gamma} 
\big( \frac{\partial P}{\partial x}+ \frac{\partial Q}{\partial y} \big) dt.
\]
Obviously the left hand side of the above formula is equal to zero. 
On the other hand, the integrand on the right hand side of the equality has 
constant sign in the simply connected domain of definition of $V$. 
Hence its integral value should be different from zero, a contradiction with 
the existence of the closed trajectory except if the trajectory $\gamma$ is 
contained in $\Sigma = \{ (x,y) \in U : V(x,y)=0 \}$ because in that case 
the integrand of right hand side is not well-defined.
\end{proof}

\begin{theorem}[Bendixson-Dulac] \label{BD}
If there exists a continuously differentiable function $B(x,y)$ in a 
simply connected region $U$ such that 
$\partial (BP) / \partial x+ \partial (BQ) / \partial y$ 
has constant sign and is not identically zero in any subregion, 
then system \eqref{01} does not possess any limit cycle
(in fact a closed trajectory) which lies entirely in $U$.
\end{theorem}

The proof of Theorem \ref{BD} follows from the proof of Theorem \ref{B} 
using $BP$ and $BQ$ instead of $P$ and $Q$ respectively. 
The function $B(x,y)$ is called \emph{Dulac function}, and the method 
of proving nonexistence of closed trajectories is called the method of 
Dulac functions.  The relation of the Bendixson-Dulac theorem and the 
inverse integrating factor is established through the following proposition.

\begin{proposition}\label{p2} Let $(P,Q)$ be a $\mathcal{C}^1$ vector 
field defined in the open subset $U$ of $\mathbb{R}^2$. Let $V=V(x,y)$ be a 
$\mathcal{C}^1$ solution of the linear
partial differential equation \eqref{02}. Let $B(x,y)$ a
continuously differentiable function in $U$
such that $\partial (BP) / \partial x+ \partial (BQ) / \partial y$ 
has constant sign, then any limit cycle (in fact a closed trajectory) 
which lies entirely in the simply connected domain of definition of $V$ 
must be contained in $\Sigma = \{ (x,y) \in U : V(x,y)=0 \}$.
\end{proposition}

\begin{proof}
In the proof we use the result that if $V$ is an inverse integrating factor 
of the vector field $(P,Q)$ then the vector $(BP,BQ)$ has the inverse 
integrating factor $BV$, see \cite{CGG}. Therefore we can write the 
following equality
\[
\oint_{\gamma} \frac{BP \frac{\partial BV}{\partial x}
+ BQ \frac{\partial BV}{\partial y}}{BV}  dt 
= \oint_{\gamma}  \frac{d \ln|BV|}{dt} dt 
= - \oint_{\gamma} \big( \frac{\partial BP}{\partial x}
+ \frac{\partial BQ}{\partial y} \big) dt,
\]
and analogous arguments than in the proof of Proposition \ref{p1} are valid.
\end{proof}

In fact Theorem \ref{B} and \ref{BD} can be extended to multiply connected regions, 
see \cite{Y}. The next important result in order to study the nonexistence 
and existence of limit cycles was obtained in \cite{Ch}.

\begin{theorem}[Cherkas] \label{Ch}
Suppose that in a simply connected domain $U \subset \mathbb{R}^2$, 
there exists a function $\Psi(x,y)$ of class $\mathcal{C}^1$ and a number 
$k>0$ such that
\[
k  \Psi \operatorname{div} \mathcal{X}+ \mathcal{X} \Psi >0,
\]
then the domain $U$ contains no limit cycles of system \eqref{01}.
\end{theorem}

\section{Statement of the main results}

The following result provides a necessary for \eqref{01}
to have an inverse integrating factor of a the form $V(f(x,y))$.

\begin{theorem}\label{theo1}
The vector field $\mathcal{X}=(P,Q)$ defined in an open subset $U$ of $\mathbb{R}^2$
admits an inverse integrating factor of the form $V=V(z)$ where
$z=f(x,y)$ if and only if
\begin{equation}\label{er1}
\frac{{\rm div} \mathcal{X}}{(P f_x + Q f_y)}=\alpha(z),
\end{equation}
where $\alpha(z)$ is a function exclusively of $z$ and in such case the 
inverse integrating factor is of the form
\begin{equation}\label{er2}
V= \exp\big( \int^z \alpha(s)\, ds\big).
\end{equation}
Moreover if $\gamma$ is a limit cycle of the vector field $(P,Q)$ in the
simply connected domain of definition of \eqref{er2}, then $\gamma$
is contained in $\Sigma = \{ (x,y) \in U : e^{\,\int^z \alpha(s) ds}=0 \}$.
\end{theorem}

The following corollary to Theorem \ref{theo1} imposes  conditions for \eqref{01}
not to have limit cycles.

\begin{corollary} \label{cor}
 The following statements hold:
\begin{itemize}
\item[(i)] Taking $z=f(x,y)=x$ if $\alpha(z)$ is a function $x$
(or taking $z=f(x,y)=y$ if $\alpha(z)$ is a function $y$), then system \eqref{01}
which has the associated vector field
$\mathcal{X}=(P,Q)$ defined in the open subset $U$ of $\mathbb{R}^2$ has no limit 
cycles in the domain of definition of \eqref{er2}.

\item[(ii)] Taking $z=f(x,y)=xy$ if $\alpha(z)$ is a function $xy$, 
then system system \eqref{01} which has the associated vector field
$\mathcal{X}=(P,Q)$ defined in the open subset $U$ of $\mathbb{R}^2$ has no 
limit cycles in the domain of definition of \eqref{er2}.
\end{itemize}
\end{corollary}

In \cite{LOV}, these two statements of corollary \ref{cor} are given but they 
are not stated correctly. In both cases the authors of \cite{LOV} state that, 
under the described conditions, system \eqref{01} does not have limit cycles.
This is wrong.
The correct conclusion is that system \eqref{01} does not have limit cycles
in the simply connected domain of definition of \eqref{er2}.
In general this domain of definition is not all $\mathbb{R}^2$ as the 
following example shows.
Suppose that $\alpha(s)= 1/(s \ln s)$ which implies that 
$V= \exp\big(\int^x \alpha(s) ds\big)= \ln x$. Therefore applying Theorem 
\ref{t1} if $\gamma$ is a limit cycle of the vector field $(P,Q)$ in the
domain of definition of $V$, which is in this case $x>0$, then $\gamma$ 
is contained in $\Sigma = \{ (x,y) \in U : V(x,y)=0 \}$. 
Therefore the vector field $(P,Q)$ has no limit cycles for $x>0$ 
but it can have limit cycles in the domain $x\le 0$. 
Another example is given later in the proof of Proposition \ref{prop} 
where $\alpha(s)= -1/s$ and $V=1/x$. In fact the statement of 
Theorem \ref{t1} is not correctly stated in \cite{LOV}.

\section{Proof of Theorem \ref{theo1} and Corollary \ref{cor}}

\begin{proof}[Proof  of Theorem \ref{theo1}]
 We assume that $V=V(z)$ where $z=f(x,y)$, then applying the chain rule,  
equality \eqref{02} is transformed into
\begin{equation}\label{03}
P \frac{d V}{dz} \frac{\partial f}{\partial x} + Q \frac{d V}{dz} 
\frac{\partial f}{\partial y}=
V \big( \frac{\partial P}{\partial x}+ \frac{\partial Q}{\partial y} \big).
\end{equation}
We can isolate $dV/dz$ from equation \eqref{03} and we have
\begin{equation}\label{04}
\frac{\frac{dV}{dz}}{V} 
=\frac{\big( \frac{\partial P}{\partial x}+ \frac{\partial Q}{\partial y} \big)}
{P f_x+Qf_y},
\end{equation}
where $f_x$ and $f_y$ are the partial derivatives of $f$ with respect to $x$ 
and $y$. The left hand-side of equation
 \eqref{04} is a function of $z$, hence the right hand-side must be also a
function of $z$ and we obtain
\begin{equation}\label{05}
\frac{\frac{dV}{dz}}{V} 
=\frac{\big( \frac{\partial P}{\partial x}+ \frac{\partial Q}{\partial y} \big)}
{P f_x+Qf_y}=\alpha(z).
\end{equation}
If equality \eqref{05} is satisfied we get that the inverse integrating factor
takes the form
\[
V= e^{\int^z \frac{\big( \frac{\partial P}{\partial x}
+ \frac{\partial Q}{\partial y} \big)}{P f_x+Qf_y}}= e^{\int^z \alpha(s) ds}.
\]
Now applying Theorem \ref{01} the proof of Theorem \ref{theo1} follows.
\end{proof}

\begin{proof}[Proof of Corollary \ref{cor}]
(i) In the case $z=f(x,y)=x$ and $\alpha(z)$ is a function $x$ we have that 
equation \eqref{05} takes the form
$\frac{d}{dx}(\log V) =(\frac{\partial P}{\partial x}
+ \frac{\partial Q}{\partial y})/P=\alpha(x)$.
Hence $V=e^{\int^x \alpha(s) ds}$ and applying Theorem \ref{01} the result follows. 
The proof is analogous for the case $z=f(x,y)=y$ and when $\alpha(z)$ is 
a function $y$.


(ii) In the case $z=f(x,y)=xy$ and $\alpha(z)$ is a function $xy$ we have
 that equation \eqref{05} takes the form
$\frac{d}{dz}(\log V) =(\frac{\partial P}{\partial x}
+ \frac{\partial Q}{\partial y})/(Py+Qx)=\alpha(z)$.
Hence $V=e^{\int^z \alpha(z) dz}$ and applying Theorem \ref{01} 
we obtain the result.
\end{proof}

\section{Examples}

In this section we give two examples where we apply the results 
developed in this work.

\begin{proposition}\label{prop}
Consider the differential system
\begin{equation}\label{ss1}
\dot{x}=-x-4xy-5y^2, \quad \dot{y}=3x +2y +y^2.
\end{equation}
System \eqref{ss1} has no limit cycles in the domain
$\mathbb{R}^2\setminus\{(x,y)\ | \ x+y^2=0\}$.
\end{proposition}

\begin{proof}
Assume that system \eqref{ss1} has an inverse integrating factor which is
a function of $x+y^2$.
Therefore if we take the particular case $z=x+y^2$ in equation \eqref{03}
 we obtain
\begin{equation}\label{ex1}
(-x-4xy-5y^2) \frac{d V}{dz} + (3x +2y +y^2) \frac{d V}{dz} (2y) =
V (1-2y).
\end{equation}
Isolating $dV/dz$ from this equation and we have
\begin{equation}\label{ex2}
\frac{\frac{dV}{dz}}{V} =\frac{1-2y}{(-1 + 2 y) (x + y^2)}
=-\frac{1}{x+y^2}=-\frac{1}{z}.
\end{equation}
Integrating the differential equation \eqref{ex2} we obtain
\[
V=e^{-\int^z \frac{1}{s} ds}=\frac{1}{z}= \frac{1}{x+y^2}.
\]
Hence, applying Theorem \ref{01} the result follows. In fact if system \eqref{ss1} has a limit cycle, this limit cycle must cut the curve $x+y^2=0$.
\end{proof}

To prove the next result we need to recall the following proposition 
given in \cite{CGG} whose proof is obvious from the definition of inverse 
integrating factor.

\begin{proposition}\label{null}
Let $(P_1,Q_1)$ and $(P_2,Q_2)$ be two $C^{1}$ vector fields defined in 
an open subset $U \subset \mathbb{R}^{2}$, which have the same 
inverse integrating factor $V(x,y)$; i.e.,
\begin{gather*}
P_1 \frac{\partial V}{\partial x} + Q_1 \frac{\partial V}{\partial y} 
- \big( \frac{\partial P_1}{\partial x} + \frac{\partial Q_1}{\partial y} \big) V = 0,
\\
P_2 \frac{\partial V}{\partial x} + Q_2 \frac{\partial V}{\partial y} 
- \big( \frac{\partial P_2}{\partial x} + \frac{\partial Q_2}{\partial y} \big) V = 0,
\end{gather*}
then the vector field $(P_1 + \lambda P_2, Q_1 + \lambda Q_2)$ has also 
the function $V(x,y)$ as inverse integrating factor, for arbitrary values of 
the parameter $\lambda$.
\end{proposition}

Using this proposition we can establish the following result.

\begin{proposition}\label{prop2}
Consider the differential system
\begin{equation}\label{ss2}
\dot{x}=x+8xy-5y^2, \quad \dot{y}=6x -2y-2y^2.
\end{equation}
System \eqref{ss2} has no limit cycles in all $\mathbb{R}^2$.
\end{proposition}

\begin{proof} 
To obtain the result we decompose the system into two homogeneous systems.
\[
P_1=x,\quad Q_1= 6x-2y, \quad P_2= 8xy-5y^2, \quad Q_2=-2y^2.
\]
Now we try to compute the inverse integrating factor for each homogeneous system. 
It is well-known have that the inverse integrating factor of any homogeneous 
system is given by $V= y P_n-x Q_n$, see \cite{ChaGaGin}. 
Therefore the vector field $(P_1,Q_1)$ has the inverse integrating factor 
$V_1=x (2 x - y)$ and the vector field $(P_2,Q_2)$ has the inverse integrating 
factor $V_2=y^2 (2 x - y)$. Moreover the respective first integrals associated 
to each integrating factor are $H_1=x^2 (2 x - y)$ and $H_2=y^4 (2 x - y)$ 
respectively.


We cannot directly apply Proposition \ref{null} because the two vector fields 
have a different inverse integrating factor. Now we apply another well-known 
result which says that if $V$ is an inverse integrating factor and if $H$ 
is a first integral then $VH$ is also an inverse integrating factor. 
Hence we look for an inverse integrating factor
of system \eqref{ss2} that satisfies
$V=V_1 \mathcal{F}_1(H_1)= V_2 \mathcal{F}_2(H_2)$,
where $\mathcal{F}_1(H_1)$ and $\mathcal{F}_2(H_2)$ are arbitrary functions of
 $H_1$ and $H_2$ respectively. In our case we have
\begin{equation}\label{fin0}
V= x (2 x - y)  \mathcal{F}_1(x^2 (2 x - y)) = y^2 (2 x - y)  
\mathcal{F}_2(y^4 (2 x - y)),
\end{equation}
which simplifying implies
\begin{equation}\label{fin1}
\mathcal{F}_1(x^2 (2 x - y))= \frac{y^2}{x}  \mathcal{F}_2(y^4 (2 x - y)).
\end{equation}
Now we take $u=y^2/x$ and equation \eqref{fin1} takes the form
\begin{equation}\label{fin2}
\mathcal{F}_1(\beta)= u  \mathcal{F}_2(u^2 \beta),
\end{equation}
where $\beta=x^2 (2 x - y)$. For equation \eqref{fin2} be satisfied
we need to take
\[
\mathcal{F}_2(u^2 \beta)= \frac{1}{\sqrt{u^2 \beta}}
=\frac{1}{u\sqrt{\beta}}, \quad\text{with } u>0,
\]
which implies $\mathcal{F}_1(\beta) = 1 / \sqrt{\beta}$ and substituting in
 \eqref{fin0} we have
\begin{equation}\label{fin3}
V= x (2 x - y)  \frac{1}{x \sqrt{2 x - y}}=\sqrt{2 x - y}, \quad \text{with } 
2x-y>0,
\end{equation}
Now we can apply Proposition \ref{null} because the two vector fields $(P_1,Q_1)$ 
and $(P_2,Q_2)$ have the same inverse integrating factor and consequently \eqref{fin3} is also the inverse integrating factor of system \eqref{ss2}.
Moreover, for the case $y-2x >0$,  it is easy to see that $V=\sqrt{y-2 x}$ 
is also an inverse integrating factor of system \eqref{ss2}. Now it is clear
that we could have applied Theorem \ref{theo1} with $z=2x+y$ in order to 
find the inverse integrating factor. Finally we can apply Theorem \ref{t1} 
and we obtain that in the domain $2x-y \ge 0$ there are not limit cycles 
and  in the domain $y-2x \ge 0$. Moreover the line $2x-y=0$ 
is an invariant algebraic curve of system \eqref{fin3}, and consequently
no limit cycle can cross this line. Therefore system \eqref{fin3} has 
no limit cycles in all $\mathbb{R}^2$.
\end{proof}

The method developed in the previous example can also applied to the example 
of Proposition \ref{prop}. In that case the common inverse integrating factor 
is $V=\sqrt{x+y}$ which ensures the nonexistence of limit cycles in the domain 
$x+y \ge 0$.

\subsection*{Acknowledgements}
The author is partially supported by the
MICINN/FEDER grant number MTM2011--22877 and by an AGAUR (Generalitat de
Catalunya) grant number 2009SGR--381.


\begin{thebibliography}{99}

\bibitem{ALGM}  A. A. Andronov, E. A. Leontovich, I. I. Gordon, A. G. Maier;
 \emph{Theory of Bifurcations of Dynamic Systems on a Plane},
John Wiley and Sons, New York, 1973.

\bibitem{ChaGaGin}  J. Chavarriga, I. A. Garc\'{\i}a, J. Gin\'e;
\emph{On integrability of differential equations defined by the sum
of homogeneous vector fields with degenerate infinity.} Internat. J.
Bifur. Chaos Appl. Sci. Engrg. \textbf{11} (2001), 711--722.

\bibitem{CGG}  J. Chavarriga, H. Giacomini, J. Gin\'{e};
\emph{The null divergence factor}, Publ. Mat. \textbf{41} (1997), no. 1, 41--56.

\bibitem{CGGL} J. Chavarriga, H. Giacomini, J. Gin\'{e}, J. Llibre;
\emph{On the integrability of two-dimensional flows},
J. Differential Equations \textbf{157} (1999), 163--182.

\bibitem{Ch}  L. A. Cherkas;
\emph{Dulac function for polynomial autonomous systems on a plane},
Differ. Equ. \textbf{33} (1997), 692--701.

\bibitem{DA}  M.V. Dolov, A.A. Alekseev;
 \emph{On the absence of limit cycles in dynamical systems with an integrating 
factor of a special type}, Differentsial'nye Uravneniya \textbf{30} 
(1994) 947--954 (translation in Differential Equations \textbf{30} (1994), 876--883).

\bibitem{EP} A. Enciso, D. Peralta-Salas;
 \emph{Existence and vanishing set of inverse integrating factors for analytic 
vector fields}, Bull. Lond. Math. Soc. \textbf{41} (2009), 1112--1124.

\bibitem{GG}  I. A. Garc\'{\i}a, M. Grau;
\emph{A survey on the inverse integrating factor}, 
Qual. Theory Dyn. Syst. \textbf{9} (2010), no. 1-2, 115--166.

\bibitem{GLV}  H. Giacomini, J. Llibre, M. Viano;
\emph{On the nonexistence, existence, and uniquennes of limit cycles},  
Nonlinearity \textbf{9} (1996), 501--516.

\bibitem{G}  J. Gin\'e;
 \emph{The nondegenerate center problem and the inverse integrating factor},
 Bull. Sci. Math. \textbf{130} (2006), no. 2, 152--161.

\bibitem{G2}  J. Gin\'e;
 \emph{On some open problems in planar differential systems and Hilbert's 
16th problem}, Chaos Solitons Fractals \textbf{31} (2007), no. 5, 1118--1134.

\bibitem{GP}  J. M. Ginoux, L. Petitgirard;
\emph{Poincar\'e's forgotten conferences on wireless telegraphy}, 
Internat. J. Bifur. Chaos Appl. Sci. Engrg. \textbf{20} (2010), no. 11, 3617--3626.

\bibitem{LOV} L. Laura-Guarachi, O. Osuna, G. Villase\~{n}or-aguilar;
\emph{Non-existence of limit cycles via inverse integrating factor}, 
Electron. J. Differential Equations 2011, No. 124, 6 pp.

\bibitem{L} A. Li\'enard;
 \emph{\'Etude des oscillations entretenues}, Rev. G\'en\'erale de 
l'Electricit\'e  \textbf{23} (1928), 901--912.

\bibitem{H}  D. Hilbert;
\emph{Mathematical problems}, Bull. Am. Math. Soc. \textbf{8} (1902) , 437-479.

\bibitem{Po} H. Poincar\'e;
 \emph{M\'emoire sur les courbes d\'efinies par une \'equation diff\'erentielle}, 
I, II, J. Math. Pures Appl. \textbf{7} (1886), 375--422; (1882) \textbf{8}, 
251--296; \emph{Sur les courbes d\'efinies par les \'equations diff\'erentielles} 
III, IV, \textbf{1} (1885), 167--244; \textbf{2} (1886), 155--217.

\bibitem{P}  L. M. Perko;
 \emph{Differential Equations and Dynamical Systems}, Springer-Verlag, 1991.

\bibitem{V} B. van der Pol;
 \emph{On relaxation-oscillations}, Philosophical Magazine 
Ser. 7 \textbf{2} (1926), 978--992.

\bibitem{Y}  Ye Yanqian et al.;
\emph{Theory of Limit Cycles}, Translations of Math. Monographs, 
Vol. 66, Amer. Math. Soc., Providence, 1986.

\end{thebibliography}

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