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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 217, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/217\hfil Explicit limit cycles]
{Explicit limit cycles of a family of polynomial differential systems}

\author[R. Boukoucha \hfil EJDE-2017/217\hfilneg]
{Rachid Boukoucha}

\address{Rachid Boukoucha \newline
Department of Technology,
Faculty of Technology,
University of Bejaia,
 06000 Bejaia, Algeria}
\email{rachid\_boukecha@yahoo.fr}

\thanks{Submitted October 4, 2016. Published September 13, 2017.}
\subjclass[2010]{34A05, 34C05, 34CO7, 34C25}
\keywords{Planar polynomial differential system; first integral;
\hfill\break\indent Algebraic limit cycle; non-algebraic limit cycle}

\begin{abstract}
 We consider the family of polynomial differential systems
 \begin{gather*}
 x' = x+( \alpha y-\beta x) (ax^2-bxy+ay^2) ^{n}, \\
 y' = y-( \beta y+\alpha x) (ax^2-bxy+ay^2) ^{n},
 \end{gather*}
 where $a$, $b$, $\alpha $, $\beta $ are real constants and $n$ is positive
 integer. We prove that these systems are Liouville integrable. Moreover, we
 determine sufficient conditions for the existence of an explicit algebraic or
 non-algebraic limit cycle. Examples exhibiting the applicability of our
 result are introduced.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

An important problem in the qualitative theory of differential equations
\cite{9,13,20} is to determine the limit cycles of  systems of the form
\begin{equation}
x'=\frac{dx}{dt}=P(x,y)\quad\text{and}\quad
y'=\frac{dy}{dt}=Q(x,y),
\label{s1}
\end{equation}
where $P(x,y)$ and $Q(x,y)$ are coprime polynomials. Here, the degree of
system \eqref{s1} is denoted by $n=\max \{ \deg P,\deg Q\} $. 
A limit cycle of system \eqref{s1}  is an
isolated periodic solution in the set of all periodic solutions of system 
\eqref{s1}, and it is said to be algebraic if it is contained
in the zero level set of a polynomial function \cite{18}. In 1900 Hilbert
\cite{17} in the second part of his 16th problem proposed to find an
estimation of the uniform upper bound for the number of limit cycles of all
polynomial vector fields of a given degree, and also to study their
distribution or configuration in the plane $\mathbb{R}^2$. 
An  even more difficult problem is to give an explicit expression of them
\cite{1,15}. This has been one of the main problems in the qualitative
theory of planar differential equations in the 20th century. We
solve this last problem for a  system of type  \eqref{s1}.
 Until recently, the only limit cycles known in an explicit way
were algebraic. In \cite{3,12,15} examples of explicit limit cycles which
are not algebraic are given. For instance, the limit cycle appearing in van
der Pol's system is not algebraic as it is proved in \cite{19}.

Let $\Omega $ be a non-empty open and dense subset of $\mathbb{R}^2$. 
We say that a non-locally constant $C^{1}$ function
 $H:\Omega \to \mathbb{R}$ is a first integral of the differential system 
\eqref{s1} in $\Omega $ if $H$  is constant on the
trajectories of the system \eqref{s1}  contained in $\Omega $,
i.e. if
\begin{equation*}
\frac{dH( x,y) }{dt}=\frac{\partial H( x,y) }{
\partial x}P( x,y) +\frac{\partial H( x,y) }{\partial y
}Q( x,y) \equiv 0\quad \text{in }\Omega .
\end{equation*}

Moreover, $H=h$ is the general solution of this equation, where $h$ is an
arbitrary constant. It is well known that for differential systems defined
on the plane $\mathbb{R}^2$ the existence of a first integral determines 
their phase portrait.
There is a lot of literature on the existence of a first integral \cite{2,4}.

A real or complex polynomial $U( x,y) $ is called algebraic
solution of the polynomial differential system \eqref{s1}  if
\begin{equation*}
\frac{\partial U( x,y) }{\partial x}P( x,y) +\frac{
\partial U( x,y) }{\partial y}Q( x,y) =K(
x,y) U( x,y) ,
\end{equation*}
for some polynomial $K( x,y) $, called the cofactor of 
$U(x,y) $. If $U( x,y) $ is non-algebraic the cofactor may not
be algebraic \cite{10,11,16}. If $U$ is real, the curve $U( x,y)=0$ 
is an invariant under the flow of differential system \eqref{s1}
 and the set $\{ ( x,y) \in\mathbb{R}^2,\text{ }U( x,y) =0\} $ 
is formed by orbits of system \eqref{s1}. There are strong relationships between the
integrability of system \eqref{s1} and its number of
invariant algebraic solutions. It is shown \cite{8} that the existence of a
certain number of algebraic solutions for a system implies the Darboux
integrability of the system, that is the first integral is the product of
the algebraic solutions with complex exponents \cite{5,6,7,14}.
 In \cite{21}, it is proved that, if a polynomial system \eqref{s1}
has a Liouvillian first integral, then it can be computed by using the
invariant algebraic solutions and the exponential factors of the system 
\eqref{s1}.

In this paper, we consider the family of the polynomial differential system
of the form
\begin{equation}
\begin{gathered}
x'=\frac{dx}{dt}=x+( \alpha y-\beta x) (ax^2-bxy+ay^2) ^{n}, \\
y'=\frac{dy}{dt}=y-( \beta y+\alpha x) (ax^2-bxy+ay^2) ^{n},
\end{gathered} \label{s2}
\end{equation}
where $a$, $b$, $\alpha $, $\beta $ are real constants and $n$ is positive
integer. We prove that these systems are Liouville integrable. Moreover, we
determine sufficient conditions for a polynomial differential system to
possess an explicit algebraic or non-algebraic limit cycles. Concrete
examples exhibiting the applicability of our result are introduced.

\section{Main result}

Our main result is contained in the following theorem.

\begin{theorem} \label{thm1}
Consider a multi-parameter polynomial differential system \eqref{s2}.
Then the following statements hold.

(1) The curve $U( x,y) =-\alpha ( x^2+y^2)
( ax^2-bxy+ay^2) ^{n}=0$ is an invariant algebraic of system 
\eqref{s2}.

(2) If $\alpha >0$ and $a>\frac{1}{2}| b| $, then
system \eqref{s2}  has the first integral
\begin{equation*}
H( x,y) =( x^2+y^2) ^{n}\exp \Big( \frac{-2n\beta
}{\alpha }\arctan \frac{y}{x}\Big) 
+2n\int_{0}^{\arctan \frac{y}{x}}\Big(
\frac{\exp ( -\frac{2n\beta w}{\alpha }) }
{\alpha ( a-\frac{1}{2}b\sin 2w) ^{n}}\Big) dw.
\end{equation*}

(3) If $\ \alpha >0$, $\beta >0$ and $2a>| b| $
then system \eqref{s2}  has an explicit limit cycle, given in
polar coordinates $( r,\theta ) $ by
\begin{equation*}
r( \theta ,r_{\ast }) 
=\exp \big( \frac{\beta }{\alpha }\theta
\big) \Big( r_{\ast }^{2n}-2n\int_{0}^{\theta }\Big( \frac{\exp ( -
\frac{2n\beta w}{\alpha }) }{\alpha ( a-\frac{1}{2}b\sin
2w) ^{n}}\Big) dw\Big) ^{1/(2n)},
\end{equation*}
where
\begin{equation*}
r_{\ast }=\exp \big( \frac{2\beta \pi }{\alpha }\big)
\Big(\frac{
2n\int_{0}^{2\pi }\big( \frac{\exp ( -\frac{2n\beta w}{\alpha }
) }{\alpha ( a-\frac{1}{2}b\sin 2w) ^{n}}\big) dw}{\exp
( \frac{4n\beta \pi }{\alpha }) -1}\Big)^{1/(2n)}.
\end{equation*}
\end{theorem}

\begin{proof}
(1) We prove that $U( x,y) =-\alpha ( x^2+y^2) (ax^2-bxy+ay^2) ^{n}=0$ 
is an invariant algebraic curve of the
differential system \eqref{s2}.

Indeed, we have
\begin{equation*}
\frac{\partial U( x,y) }{\partial x}P( x,y) +\frac{
\partial U( x,y) }{\partial y}Q( x,y) =U(
x,y) K( x,y) ,
\end{equation*}
where
\begin{align*}
K( x,y)& =2+2n+( 2\beta -2\beta n) (
ax^2-bxy+ay^2) ^{n}\\
&\quad -n\alpha b( ( y^2-x^2)) ( ax^2-bxy+ay^2) ^{n-1}.
\end{align*}

Therefore, $U( x,y) =-\alpha ( x^2+y^2) (ax^2-bxy+ay^2) ^{n}=0$ 
is an invariant algebraic curve of the
polynomial differential systems \eqref{s2}. Hence, statement (1)
is proved.

(2)  To prove our results (2) and (3) we write the
polynomial differential system \eqref{s2}  in polar
coordinates $( r,\theta ) $, defined by $x=r\cos \theta $ and 
$y=r\sin \theta $. Then the system becomes
\begin{equation}
\begin{gathered}
r'=r-\beta ( a-\frac{1}{2}b\sin 2\theta ) ^{n}r^{2n+1},\\
\theta '=-\alpha ( a-\frac{1}{2}b\sin 2\theta )^{n}r^{2n},
\end{gathered} \label{s3}
\end{equation}
where $\theta '=\frac{d\theta }{dt}$, $r'=\frac{dr}{dt}$. 

Taking as new independent variable the coordinate $\theta $, this
differential system reads
\begin{equation}
\frac{dr}{d\theta }=\frac{\beta }{\alpha }r+\frac{-1}{\alpha ( a-\frac{1
}{2}b\sin 2\theta ) ^{n}}r^{1-2n},  \label{s4}
\end{equation}
which is a Bernoulli equation.

By introducing the standard change of variables $\rho =r^{2n}$ we obtain the
linear equation
\begin{equation}
\frac{d\rho }{d\theta }=\frac{2n\beta }{\alpha }\rho +\frac{-2n}{\alpha
( a-\frac{1}{2}b\sin 2\theta ) ^{n}}.  \label{s5}
\end{equation}

The general solution of linear equation \eqref{s5} is
\begin{equation*}
r( \theta ) =\exp \big( \frac{\beta }{\alpha }\theta \big)
\Big( c-2n\int_{0}^{\theta }\Big( \frac{\exp ( -\frac{2n\beta w}{
\alpha }) }{\alpha ( a-\frac{1}{2}b\sin 2w) ^{n}}\Big)
dw\Big) ^{1/(2n)},
\end{equation*}
where $c\in\mathbb{R}$. From these solution we can obtain a first integral 
in the variables $( x,y) $ of the form
\begin{equation*}
H( x,y) =( x^2+y^2) ^{n}\exp \Big(\frac{-2n\beta
}{\alpha }\arctan \frac{y}{x}\Big) 
+2n\int_{0}^{\arctan \frac{y}{x}}\Big(
\frac{\exp ( -\frac{2n\beta w}{\alpha }) }{\alpha ( a-\frac{1
}{2}b\sin 2w) ^{n}}\Big) dw.
\end{equation*}

Since this first integral is a function that can be expressed by quadratures
of elementary functions, it is a Liouvillian function, and consequently
system \eqref{s2}  is Liouville integrable.
The curves $H=h$ with $h\in\mathbb{R}$, which are formed by trajectories
 of the differential system \eqref{s2}, in Cartesian coordinates are written as
\begin{align*}
x^2+y^2&=\Big(h\exp \Big( \frac{2n\beta }{\alpha }\arctan \frac{y}{x}\Big)\\
&\quad - 2n\exp\Big( \frac{2n\beta }{\alpha }\arctan \frac{y}{x}\Big)
\int_{0}^{\arctan \frac{y}{x}}\Big( \frac{\exp ( -\frac{2n\beta w}{
\alpha }) }{\alpha ( a-\frac{1}{2}b\sin 2w) ^{n}}\Big) dw
\Big) ^{1/n},
\end{align*}
where $h\in \mathbb{R}$. Hence, statement (2) is proved.

(3)  Since $\alpha >0$ and $a>\frac{1}{2}| b|$, it follows that
 $-\alpha ( a-\frac{1}{2}b\sin 2\theta )^{n}<0$ for all 
$\theta \in\mathbb{R}$, then $\theta '$ is negative for all $t$, 
which means that each orbit of system \eqref{s2} encircle the singularity at the
origin.

Notice that system \eqref{s2} has a periodic orbit if and
only if equation \eqref{s4}  has a strictly positive $2\pi $
periodic solution. This, moreover, is equivalent to the existence of a
solution of \eqref{s4}  that satisfies 
$r( 0,r_{\ast}) =r( 2\pi ,r_{\ast }) $ and 
$r( \theta ,r_{\ast}) >0$ for any $\theta $ in $[ 0,2\pi ] $.

The solution $r( \theta ,r_{0}) $ of the differential equation 
\eqref{s4}  such that $r( 0,r_{0}) =r_{0}$ is
\begin{equation*}
r( \theta ,r_{0}) =\exp \Big( \frac{\beta }{\alpha }\theta
\Big) \Big( r_{0}^{2n}-2n\int_{0}^{\theta }\Big( \frac{\exp ( -
\frac{2n\beta w}{\alpha }) }{\alpha ( a-\frac{1}{2}b\sin
2w) ^{n}}\Big) dw\Big) ^{1/(2n)},
\end{equation*}
where $r_{0}=r( 0) $.

A periodic solution of system \eqref{s2}  must satisfy the
condition $r( 2\pi ,r_{0}) =r( 0,r_{0}) $, which leads
to a unique value $r_{0}=r_{\ast }$, given by
\begin{equation*}
r_{\ast }=\exp \big( \frac{2\beta \pi }{\alpha }\big) 
\Big(\frac{2n\int_{0}^{2\pi }( \frac{\exp ( -\frac{2n\beta w}{\alpha }
) }{\alpha ( a-\frac{1}{2}b\sin 2w) ^{n}}) dw}{\exp
( \frac{4n\beta \pi }{\alpha }) -1}\Big)^{1/(2n)}.
\end{equation*}

After the substitution of these value $r_{\ast }$ into 
$r( \theta,r_{0}) $ we obtain
\begin{align*}
r( \theta ,r_{\ast }) 
&=\exp \big( \frac{\beta }{\alpha }\theta\big) 
\Big(2n\frac{\exp ( \frac{4n\beta \pi }{\alpha }) }{\exp ( \frac{
4n\beta \pi }{\alpha }) -1}\int_{0}^{2\pi }\Big( \frac{\exp ( -
\frac{2n\beta w}{\alpha }) }{\alpha ( a-\frac{1}{2}b\sin
2w) ^{n}}\Big) dw \\
&\quad - 2n\int_{0}^{\theta }\Big( \frac{\exp ( -\frac{2n\beta w}{\alpha }
) }{\alpha ( a-\frac{1}{2}b\sin 2w) ^{n}}\Big) dw
\Big) ^{1/(2n)}.
\end{align*}

Next we prove that $r( \theta ,r_{\ast }) >0$.
Indeed
\begin{align*}
r( \theta ,r_{\ast }) 
&=\sqrt[2n]{2n}\exp \big( \frac{\beta }{\alpha }\theta \big)
\Big(\frac{\exp ( \frac{4n\beta \pi }{\alpha }) }
{\exp ( \frac{4n\beta \pi }{\alpha }) -1}\int_{0}^{2\pi }( \frac{\exp ( -
\frac{2n\beta w}{\alpha }) }{\alpha ( a-\frac{1}{2}b\sin
2w) ^{n}}) dw \\
&\quad - \int_{0}^{\theta }\Big( \frac{\exp ( -\frac{2n\beta w}{\alpha }
) }{\alpha ( a-\frac{1}{2}b\sin 2w) ^{n}}\Big) dw
\Big) ^{1/(2n)} \\
&\geq \sqrt[2n]{2n}\exp \big( \frac{\beta }{\alpha }\theta \big) 
\Big(\int_{0}^{2\pi }\Big( \frac{\exp ( -\frac{2n\beta w}{\alpha })
}{\alpha ( a-\frac{1}{2}b\sin 2w) ^{n}}\Big) dw\\
&\quad - \int_{0}^{\theta }\Big( \frac{\exp ( -\frac{2n\beta w}{\alpha }
) }{\alpha ( a-\frac{1}{2}b\sin 2w) ^{n}}\Big) dw
\Big) ^{1/(2n)} \\
&=\sqrt[2n]{2n}\exp \big( \frac{\beta }{\alpha }\theta \big) 
\Big(\int_{\theta }^{2\pi }\Big( \frac{\exp ( -\frac{2n\beta w}{\alpha }
) }{\alpha ( a-\frac{1}{2}b\sin 2w) ^{n}}\Big) dw\Big)^{1/(2n)}>0,
\end{align*}
because 
\[
\frac{\exp ( -\frac{2n\beta w}{\alpha }) }{\alpha
( a-\frac{1}{2}b\sin 2w) ^{n}}>0
\]
 for all $s\in\mathbb{R}$.
Moreover, we compute
\begin{equation*}
\frac{dr( 2\pi ,r_{0}) }{dr_{0}}\big| _{r_{0}
=r_{\ast }} =\exp \big( \frac{4\beta n\pi }{\alpha }\big) >1.
\end{equation*}
This is a stable and hyperbolic limit cycle for the differential systems 
\eqref{s2}.
This completes the proof of statement (3) of Theorem  \ref{thm1}.
\end{proof}

\section{Examples}

The following examples illustrate our result.

\begin{example} \label{examp1}\rm
When $a=b=\alpha =\beta =n=1$,  system \eqref{s2} reads
\begin{equation}
\begin{gathered}
x'=x+( y-x) ( x^2-xy+y^2) , \\
y'=y-( y+x) ( x^2-xy+y^2) .
\end{gathered} \label{s6}
\end{equation}
This system  is a cubic system that has a non-algebraic limit cycle whose 
expression in polar coordinates $(r,\theta ) $ is
\begin{equation*}
r( \theta ,r_{\ast }) =e^{\theta }
\Big(r_{\ast }^2-4\int_{0}^{\theta }\Big( \frac{e^{-2\omega }}{2-\sin 2\omega }
\Big) d\omega \Big)^{1/2},
\end{equation*}
where $\theta \in \mathbb{R}$, and the intersection of the limit cycle with 
the $OX_{+}$ axis is the point 
\begin{equation*}
r_{\ast }=\Big(\frac{2e^{4\pi }}{e^{4\pi }-1}\int_{0}^{2\pi }\Big( \frac{2
}{2-\sin 2\omega }e^{-2\omega }\Big) d\omega \Big)^{1/2}
\simeq 1.1912.
\end{equation*}
Moreover,
\begin{equation*}
\frac{dr( 2\pi ,r_{0}) }{dr_{0}}\big|_{r_{0}=r_{\ast }}=\exp ( 4\pi ) >1.
\end{equation*}
This limit cycle is a stable hyperbolic limit cycle. This  results
presented was by  Llibre and Rebiha [3].
\end{example}

\begin{example} \label{examp2} \rm 
When  $a=\alpha =\beta =n=1$ and $b=0$, system \eqref{s2}  reads
\begin{equation}
\begin{gathered}
x'=x-x^{3}+x^2y-xy^2+y^{3}, \\
y'=y-x^{3}-x^2y-xy^2-y^{3}.
\end{gathered} \label{s7}
\end{equation}
This system is a cubic system that has a algebraic
limit cycle whose expression in polar coordinates $( r,\theta ) $
is 
\begin{equation*}
r( \theta ,r_{\ast }) =1,
\end{equation*}
where $\theta \in\mathbb{R}$. In Cartesian coordinates it is written
\begin{equation*}
x^2+y^2=1.
\end{equation*}
Moreover,
\begin{equation*}
 \frac{dr( 2\pi ,r_{0}) }{dr_{0}}\big|_{r_{0}=r_{\ast }}=\exp ( 4\pi ) >1.
\end{equation*}
This limit cycle is a stable hyperbolic limit cycle.
\end{example}

\begin{example} \label{examp3}\rm
When $\alpha =\beta =1$ and $a=b=n=2$, system \eqref{s2}  reads
\begin{equation}
\begin{gathered}
x'=x-4x^{5}+12x^{4}y-20x^{3} 
y^2+20x^2y^{3}-12xy^{4}+4y^{5}, \\
y'=y-4x^{5}+4x^{4}y-4x^{3}y^2-4x^2y^{3}+4xy^{4}-4y^{5},
\end{gathered} \label{s8}
\end{equation}
This system is a quintic system that has a
non-algebraic limit cycle whose expression in polar coordinates 
$(r,\theta ) $ is
\begin{equation*}
r( \theta ,r_{\ast }) 
=e^{\theta }\Big( r_{\ast
}^{4}-4\int_{0}^{\theta }\Big(\frac{\exp ( -4w) }{( 2-\sin
2w) ^2}\Big) dw \Big)^{1/4},
\end{equation*}
where $\theta \in \mathbb{R}$, and the intersection of the limit cycle 
with the $OX_{+}$ axis is the point
\begin{equation*}
r_{\ast }=\exp ( 2\pi ) 
\Big(\frac{4\int_{0}^{2\pi }\big(
\frac{\exp ( -4w) }{( 2-\sin 2w) ^2}\big) dw}{
\exp ( 8\pi ) -1}\Big)^{1/4}\simeq   0.816\,28.
\end{equation*}
Moreover,
\begin{equation*}
 \frac{dr( 2\pi ,r_{0}) }{dr_{0}}|
_{r_{0}=r_{\ast }}=\exp ( 8\pi ) >1.
\end{equation*}
This limit cycle is a stable hyperbolic limit cycle.
\end{example}

\begin{example} \label{examp4} \rm
When $a=b=\alpha =\beta =1$ and $n=3$, 
system \eqref{s2}  reads
\begin{equation}
\begin{gathered}
x'=x-x^{7}+4x^{6}y-9x^{5}y^2+13
x^{4}y^{3}-13x^{3}y^{4}+9x^2y^{5}-4xy^{6}+y^{7}, \\
y'=y-x^{7}+2x^{6}y-
3x^{5}y^2+x^{4}y^{3}+x^{3}y^{4}-3x^2y^{5}+  2xy^{6}-y^{7}.
\end{gathered} \label{s9}
\end{equation}
This system  has a non-algebraic limit cycle whose
expression in polar coordinates $( r,\theta ) $ is
\begin{equation*}
r( \theta ,r_{\ast }) =e^{\theta }
\Big( r_{\ast
}^{6}-6\int_{0}^{\theta }\Big( \frac{8\exp ( -6w) }{( 1-
\frac{1}{2}\sin 2w) ^{3}}\Big) dw\Big)^{1/6},
\end{equation*}
where $\theta \in\mathbb{R}$, and the intersection of the limit cycle with 
the $OX_{+}$ axis is
\begin{equation*}
r_{\ast }=\Big(\frac{\exp ( 12\pi ) }{-1+\exp ( 12\pi
) }\int_{0}^{2\pi }\Big( \frac{48\exp ( -6w) }{(
2-\sin 2w) ^{3}}\Big) dw\Big)^{1/6} \simeq   1.1189.
\end{equation*}
Moreover,
\begin{equation*}
 \frac{dr( 2\pi ,r_{0}) }{dr_{0}}\big|_{r_{0}=r_{\ast }}=\exp ( 12\pi ) >1.
\end{equation*}
This limit cycle is a stable hyperbolic limit cycle.
\end{example}

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