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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 78, pp. 1--6.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2012/78\hfil Explicit non-algebraic limit cycles]
{Polynomial differential systems with explicit non-algebraic limit cycles}
\author[R. Benterki, J. Llibre \hfil EJDE-2012/78\hfilneg]
{Rebiha Benterki, Jaume Llibre} % in alphabetical order
\address{Rebiha Benterki \newline
D\'{e}partement de Math\'{e}matiques,
Centre Universitaire de Bordj Bou Arr\'{e}ridj,
Bordj Bou Arr\'{e}ridj 34265, El anasser, Algeria}
\email{r\_benterki@yahoo.fr}
\address{Jaume Llibre \newline
Departament de Matematiques,
Universitat Aut\`{o}noma de Barcelona, 08193 Bellaterra, Barcelona,
Catalonia, Spain}
\email{jllibre@mat.uab.cat}
\thanks{Submitted February 13, 2012. Published May 15, 2012.}
\subjclass[2000]{34C29, 34C25}
\keywords{Non-algebraic limit cycle; polynomial vector field}
\begin{abstract}
Up to now all the examples of polynomial differential systems for
which non-algebraic limit cycles are known explicitly have degree
at most 5. Here we show that already there are polynomial
differential systems of degree at least exhibiting explicit
non-algebraic limit cycles. It is well known that polynomial
differential systems of degree 1 (i.e. linear differential
systems) has no limit cycles. It remains the open question to
determine if the polynomial differential systems of degree 2 can
exhibit explicit non-algebraic limit cycles.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\allowdisplaybreaks
\section{Introduction and statement of the main results}\label{s1}
Probably the existence of limit cycles is one of the more difficult
objects to study in the qualitative theory of differential equations
in the plane. There is a huge literature dedicated to this topic,
see for instance the book of Ye et al \cite{Ye}, or the
famous Hilbert 16th problem \cite{Hi} and \cite{Il}.
Publications more closely related to the problem in this article
are \cite{GGT, GG, Al, DL, LL, LM}.
A \emph{polynomial differential system} is a system of the form
\begin{equation}\label{e1}
\begin{gathered}
\dot{x}=P(x,y), \\
\dot{y}=Q(x,y),
\end{gathered}
\end{equation}
where $P(x, y)$ and $Q(x, y)$ are real polynomials in the variables
$x$ and $y$. The \emph{degree} of the system is the maximum of the
degrees of the polynomials $P$ and $Q$. As usual the dot denotes
derivative with respect to the independent variable $t$.
A \emph{limit cycle} of system \eqref{e1} is an isolated periodic
solution in the set of all periodic solutions of system \eqref{e1}.
If a limit cycle is contained in an algebraic curve of the plane,
then we say that it is \emph{algebraic}, otherwise it is called
\emph{non-algebraic}. In other words a limit cycle is algebraic if there
exists a real polynomial $f(x,y)$ such that the algebraic curve
$f(x,y)=0$ contains the limit cycle. In general, the orbits of a
polynomial differential system \eqref{e1} are contained in analytic
curves which are not algebraic.
To distinguish when a limit cycle is algebraic or not, usually, it
is not easy. Thus, the well-known limit cycle of the van der Pol
differential system exhibited in 1926 (see \cite{Po}) was not proved
until 1995 by Odani \cite{Od} that it was non-algebraic. The van
der Pol system can be written as a polynomial differential system
\eqref{e1} of degree $3$, but its limit cycle is not known
explicitly.
These previous years (from 2006 up to now) several papers have been
published exhibiting polynomial differential systems for which
non-algebraic limit cycles are known explicitly. This means that in
some coordinates we have an explicit analytic expression of the
curve containing the non-algebraic limit cycle. The first explicit
non-algebraic limit cycle, due to Gasull, Giacomini and Torregrosa
\cite{GGT}, was for a polynomial differential system of degree $5$.
Of course, multiplying the right hand part of this polynomial
differential system of degree $5$ by $(a x+b y+c)^n$ with $n$ an
arbitrary positive integer, where the straight line $a x+b y+c=0$
must be chosen in such a way that it does not intersect the explicit
limit cycle of the system, we get a polynomial differential system
of degree $5+n$ exhibiting an explicit non-algebraic limit cycle.
Immediately after this first paper appeared the paper of Al-Dosary
\cite{Al} inspired by \cite{GGT} (note that this reference is quoted
in \cite{Al}), providing a similar polynomial differential system of
degree $5$ exhibiting an explicit non-algebraic limit cycle.
Gin\'{e} and Grau \cite{GG} provide a polynomial differential system of
degree $9$ exhibiting simultaneously two explicit limit cycles one
algebraic and another non-algebraic. Note that the paper \cite{GGT}
is also quoted in \cite{GG}.
The aim of this paper is to show that there exist polynomial
differential systems of degree $3$ exhibiting explicit
non-algebraic limit cycles. Thus, our main result is the following
one.
\begin{theorem}\label{thm1}
The differential polynomial system of degree $3$,
\begin{equation}\label{e2}
\begin{gathered}
\dot{x}= x+(y-x)(x^2-x y+y^2), \\
\dot{y}= y-(y+x)(x^2-x y+y^2),
\end{gathered}
\end{equation}
has a unique non-algebraic limit cycle whose expression in polar
coordinates $(r,\theta)$, defined by $x= r \cos \theta$ and $y= r \sin \theta$,
is
\begin{equation}\label{e3}
r(\theta)= e^{\theta } \sqrt{r_*^2-f(\theta)},
\end{equation}
where
\begin{gather*}
r_*= e^{2 \pi} \sqrt{ \frac{f(2\pi)}{e^{4 \pi}-1}}\approx
1.1911644871948721\dots, \\
f(\theta)= 4 \int_0^{\theta} \frac{e^{-2 s}}{2-\sin (2s)}\, ds.
\end{gather*}
Moreover, this limit cycle is a stable hyperbolic limit cycle.
\end{theorem}
The above theorem is proved in section \ref{s2}.
In short, since it is well known that the linear differential
systems (or polynomial differential systems of degree $1$) have no
limit cycles, it remains the following open question:
\noindent\textbf{Open question}. \emph{Are there or not polynomial
differential systems of degree $2$ exhibiting explicit
non-algebraic limit cycles.}
\section{Proof of Theorem \ref{thm1}}\label{s2}
The polynomial differential system \eqref{e2} in polar coordinates
becomes
\begin{equation}\label{e4}
\begin{gathered}
\dot r= r+ \frac{1}{2} (\sin (2 \theta )-2) r^3, \\
\dot \theta= \frac{1}{2} r^2 (\sin (2 \theta )-2).
\end{gathered}
\end{equation}
Taking as independent variable the coordinate $\theta$, this
differential system writes
\begin{equation}\label{e5}
\frac{d r}{d\theta}= r+\frac{2}{r(\sin (2 \theta )-2)}.
\end{equation}
Note that since $\dot \theta<0$, the orbits $r(\theta)$ of the differential
equation \eqref{e5} has reversed their orientation with respect to
the orbits $(r(t),\theta(t))$ or $(x(t),y(t))$ of the differential
systems \eqref{e4} and \eqref{e2}, respectively.
It is easy to check that the solution $r(\theta;r_0)$ of the
differential equation \eqref{e5} such that $r(0;r_0)= r_0$ is
\begin{equation}\label{e6}
r(\theta;r_0)= e^{\theta } \sqrt{r_0^2-f(\theta)},
\end{equation}
where $f(\theta)$ is the function defined in the statement of Theorem
\ref{thm1}.
Clearly the unique equilibrium point of the differential system
\eqref{e2} is the origin of coordinates, which is an unstable node
because its eigenvalues are $1$ with multiplicity two, for more
details see for instance \cite[Theorem 2.15]{DLA}. This
equilibrium point in polar coordinates become $r=0$. This is the
unique point of the plane where the differential equation \eqref{e5}
is not defined. But we can extend the flow of this differential
equation to $r=0$, assuming that at the origin of the plane in polar
coordinates we have an unstable node.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig1}
\end{center}
\caption{The phase portrait in the
Poincar\'{e} disc of the polynomial differential system \eqref{e2}}
\label{aa}
\end{figure}
The periodic orbits $r(\theta;r_0)$ of
\eqref{e5} must satisfy $r(2\pi;r_0)= r_0$. A solution of this
equation is $r_0= r_*$, where $r_*$ is defined in the statement of
Theorem \ref{thm1}. So, if $r(\theta;r_*)>0$ for all
$\theta\in \mathbb{R}$, we shall
have $r(\theta;r_*)>0$ would be a periodic orbit, and consequently a
limit cycle. In what follows it is proved that $r(\theta;r_*)>0$ for all
$\theta\in \mathbb{R}$. Indeed
\begin{align*}
r(\theta;r_*)&= e^{\theta} \sqrt{\frac{e^{4\pi}}{e^{4\pi}-1}f(2\pi)-
f(\theta)} \\
&\geq e^{\theta} \sqrt{f(2\pi)-f(\theta)} \\
&= 2e^{\theta} \sqrt{ \int_\theta^{2\pi} \frac{e^{-2
s}}{2-\sin (2s)}\, ds }>0,
\end{align*}
because $e^{-2 s}/(2-\sin (2s))>0$ for all $s\in \mathbb{R}$.
An easy computation shows that
\[
\frac{d\, r(2\pi;r_0)}{d\, r_0}\big|_{r_0= r_*} =e^{4\pi}>1.
\]
Therefore the limit cycle of the differential equation \eqref{e5} is
unstable and hyperbolic, for more details see
\cite[section 1.6]{DLA}.
Consequently, this is a stable and hyperbolic limit
cycle for the differential system \eqref{e2}.
Clearly the curve $(r(\theta)\cos \theta, r(\theta)\sin \theta)$ in the $(x,y)$
plane with
\[
r(\theta)^2= e^{2\theta } (r_*^2-f(\theta)),
\]
is not algebraic, due to the expression $e^{2\theta } r_*^2$. More
precisely, in cartesian coordinates the curve defined by this limit
cycle is
\[
f(x,y)= x^2+y^2-e^{2 \arctan(y/x)} \Big(r_*^2-4
\int_0^{\arctan(y/x)} \frac{e^{-2 s}}{2-\sin (2s)} \, ds\Big)=0.
\]
If the limit cycle is algebraic this curve must be given by a
polynomial, but a polynomial $f(x,y)$ in the variables $x$ and $y$
satisfies that there is a positive integer $n$ such that
$\partial^n f/(\partial x)^n =0$, and this is not the case because in the derivative
\begin{align*}
\frac{\partial f}{\partial x}&= 2 x+\frac{2y e^{2
\arctan(y/x)} }{x^2+y^2} \Big(r_*^2-4
\int_0^{\arctan (y/x)} \frac{e^{-2 s}}{2-\sin(2 s)} \, ds\Big) \\
&\quad -\frac{4 y}{(x^2+y^2) \big(2-\sin \big(2\arctan(y/x)\big)\big)}
\end{align*}
it appears again the expression
\[
e^{2 \arctan(y/x)} \Big(r_*^2-4
\int_0^{\arctan(y/x)} \frac{e^{-2 s}}{2-\sin (2 s)} \, ds\Big),
\]
which already appears in $f(x,y)$, and this expression will appear
in the partial derivative at any order.
Now we shall prove that the limit cycle given by $r(\theta;r_*)$ is the
unique periodic orbit of the differential system, and consequently
the unique limit cycle. We recall the so called Generalized Dulac's
Theorem, for a proof of it see \cite[Theorem 7.12]{DLA}.
\begin{theorem}\label{thm2}
Let $R$ be an $n$-multiply connected region of $\mathbb{R}^2$ (i.e. $R$ has
one outer boundary curve, and $n-1$ inner boundary curves). Assume
that the divergence function $\partial P/\partial x+\partial
Q/\partial y$ of the $C^1$ differential system $\dot x= P(x,y)$,
$\dot y= Q(x,y)$ has constant sign in the region $R$, and is not
identically zero on any subregion of $R$. Then this differential
system has at most $n-1$ periodic orbits which lie entirely in $R$.
\end{theorem}
We take as new independent variable the variable $\tau$ defined by
$d\tau= (x^2 + y^2) (x^2 - x y + y^2) dt$. Since $(x^2 + y^2) (x^2 -
x y + y^2)$ only vanishes at the origin of coordinates the
differential system \eqref{e2} and the differential system
\begin{equation}\label{e222}
\begin{gathered}
x'= \frac{x+(y-x)(x^2-x y+y^2)}{(x^2 + y^2) (x^2 - x y + y^2)},
\\
y'= \frac{y-(y+x)(x^2-x y+y^2)}{(x^2 + y^2) (x^2 - x y + y^2)},
\end{gathered}
\end{equation}
where the prime denotes derivative with respect to the variable
$\tau$, have the same phase portrait in $R= \mathbb{R}^2\setminus
\{(0,0)\}$. An easy computation shows that the divergence of the
differential system \eqref{e222} is
\[
-\frac{2}{(x^2 + y^2) (x^2 - x y + y^2)}<0 \quad \text{in } R.
\]
So, by Theorem \ref{thm2}, and since $R$ is $2$-multiply connected
region of $\mathbb{R}^2$ it follows that the differential system
\eqref{e222} and consequently the differential system \eqref{e2} has
at most one periodic solution. In short, the unique periodic
solution of system \eqref{e2} is $r(\theta;r_*)$.
This completes the proof of Theorem \ref{thm1}.
Now we shall present the phase portrait of the differential system
\eqref{e2} in the Poincar\'{e} disc, see the Poincar\'{e} compactification
in \cite[Chapter 5]{DLA}.
Since the polynomial $\dot x y-\dot y x= (x^2 + y^2) (x^2 - x y +
y^2)$ has no real linear factors, the compactification of Poincar\'{e}
of the differential system \eqref{e2} has no equilibrium points at
infinity, i.e. the infinity is a periodic orbit. Doing the change of
variables $r=1/\rho$, the infinity of the differential equation
\eqref{e5} passes at the origin, and equation \eqref{e5} becomes
\[
\frac{d \rho}{d\theta}= -\rho -\frac{2 \rho^3}{\sin(2\theta)-2}.
\]
Hence, clearly $\rho=0$ is an stable equilibrium point of this
differential equation, consequently the periodic orbit at infinity
of the differential equation \eqref{e3} is an unstable limit cycle.
Then the phase portrait in the Poincar\'{e} disc of the polynomial
differential system \eqref{e2} is given in Figure \ref{aa}.
\subsection*{Acknowledgments}
The first author is partially supported by
grants MTM2008--03437 from MCYT/FEDER, and 2009SGR--410 from CIRIT
and ICREA Academia.
The second author is partially
supported by grant B03320090002 from the Algerian Ministry of Higher Education
and Scientific Research under project.
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\end{document}