\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small {\em Electronic Journal of
Differential Equations}, Vol. 2003(2003), No. 105, pp. 1--7.\newline 
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu
or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2003 Texas State University-San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE--2003/105\hfil Limit cycles for polynomial systems]
{Limit cycles for a class of polynomial systems and applications}

\author[H. Feng, R. Xu, Q. Liu, \& P. Yang\hfil EJDE--2003/105\hfilneg]
{Hanying Feng, Rui Xu, Qiming Liu, \& Pinghua Yang} 

\address{Shijiazhuang Mechanical Engineering College,
Shijiazhuang 050003, China}
\email[Hanying Feng]{fhanying@yahoo.com.cn}
\email[Rui Xu]{rxu88@yahoo.com.cn}
\email[Qiming Liu]{Llqm2002@yahoo.com.cn}
\email[Pinghua Yang]{huangsz@sjzri.edu.cn}

\date{}
\thanks{Submitted July 2, 2003. Published October 20, 2003.}
\subjclass[2000]{34C07, 37G10}
\keywords{Polynomial system, limit cycle, existence, uniqueness}

\begin{abstract}
 In this paper, we study the existence and uniqueness of
 limit cycles for a particular polynomial system.
 Some known results are extended and improved.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}
As well known, the second part of Hilbert's 16th problem deals
with polynomial differential equations in the plane. Recall that
the second part of Hilbert's 16th problem asks about the following
three questions:
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item Is it true that a planar polynomial vector field has but a
finite number of limit cycles? 
\item Is it true that the number of
limit cycles of a planar polynomial vector field is bounded by a
constant depending on the degree of the polynomials only? 
\item
Given an upper bound on the number of limit cycles in (ii).
\end{enumerate}

There has been a large body of excellent work on the study of the
problems above since 1900 (see, for example, [1-9] and the
references cited therein). However, it remains unsolved even for
quadratic polynomials. There may be a long distance to solve this
problem. So far, most of the previous important works tackle the
quadratic polynomial systems, but there aren't many results
dealing with cubic and higher degree polynomial system. In this
paper, we are concerned with a class of higher degree polynomial
differential system which contains as particular cases certain
more simple systems considered earlier by other authors. The main
purpose of this paper is to study the number and the distribution
of limit cycles of higher degree polynomial differential
equations.  To do so, we consider the following differential
system
\begin{equation}\label{e2.1}
\begin{gathered}
dx/dt=-y^{\alpha}(1+ny)^{\alpha}+dx^{\alpha}(1+ny)^{\alpha}+
lx^{\alpha+1}(1+ny)^{\alpha-1}-my^{\alpha+1}(1+ny)^{\alpha},\\[0.2cm]
dy/dt=x^{\alpha}[(1+ny)^{\alpha}+(ax)^{\alpha}],
\end{gathered}
\end{equation}
where $n\geq m\geq 0$, $\alpha$ is a positive odd integer. Without
loss of generality, we assume $a \leq 0$ (Otherwise, one may make
the transformation $x_1= -x, t_1= -t$).

The paper is organized as follows. In the next section, by
analyzing an equivalent system of (1.1), we discuss the existence
and uniqueness of limit cycles of system (1.1). In section 3, by
applying the main results in section 2, we discuss some more
simple polynomial differential systems considered earlier by some
other authors. Some known results in \cite{gc,ld,sj,zy} are
extended and improved.


\section{The main results and proofs}

In this section, we discuss the existence and uniqueness of limit
cycles of system (1.1) by using the qualitative theory of ordinary
differential equations.

Noting that $y=-1/n$ is a straight line isocline, we set
$x=\overline x (1+n\overline y)$, $y=\overline y$,
$dt=(1+ny)^{1-\alpha}d\overline t,$ then system \eqref{e2.1} can
be transformed into the following equivalent system (denoting
$\overline x$, $\overline y$, $\overline t$ again by $x$, $y$,
$t$, respectively):
\begin{equation}\label{e2.2}
\begin{gathered}
dx/dt=-y^{\alpha}-my^{\alpha+1}+(1+ny)^{\alpha}
[dx^{\alpha}+(l-n)x^{\alpha+1}-na^{\alpha}x^{2\alpha+1}]\equiv
P(x,y),\\[0.2cm]
dy/dt=(1+ny)^{\alpha+1}x^{\alpha}(1+a^{\alpha}x^{\alpha})\equiv
Q(x,y).
\end{gathered}
\end{equation}
We now formulate our main results for systems \eqref{e2.1} and
\eqref{e2.2} as follows.

\begin{theorem} \label{thm2.1}
If $ad [l(\alpha +1)+n\alpha]\leq 0$, then system \eqref{e2.1}
 has
no limit cycle in the whole plane.
\end{theorem}

\begin{proof}
 Take a Dulac's function
$$B(y)=(1+ny)^\frac{-l(\alpha+1)}{n}.$$
Along solutions of system \eqref{e2.2}, we derive that
$$
 \mathop{\rm div}(BP, BQ)=(1+ny)^\frac{n\alpha -l(\alpha +1)}{n}x^{\alpha -1}[ad-(l(\alpha
 +1)+n\alpha)a^{\alpha}x^{\alpha+1}].
$$
Thus, if $ad[l(\alpha +1)+n\alpha]\leq 0$, then $\mathop{\rm
div}(BP, BQ)$ is of fixed sign in the region $y+\frac{1}{n}>0$ or
$y+\frac{1}{n}<0$ and is not identically equal to zero in any
subregion since $\alpha$ is a positive odd integer. By Dulac's
criterion, we know that system \eqref{e2.2} has no limit cycle in
the whole plane. Since system \eqref{e2.1} is equivalent to system
\eqref{e2.2}, we assert that system \eqref{e2.1} has no limit
cycle in the whole plane. This completes the proof.
\end{proof}

By Theorem \ref{thm2.1}, we can assume $ad[l(\alpha
+1)+n\alpha]>0$. By the assumption of $a \leq 0$, we have
$d[l(\alpha +1)+n\alpha]<0$. Hence, in the following, we always
assume $l>-n\alpha/(\alpha +1), a<0,$ and $d<0$.

\begin{lemma} \label{lm2.2}
If system \eqref{e2.2} has limit cycles, then they must intersect
the line $x=-\alpha d /[l(\alpha +1)+n\alpha]$.
 \end{lemma}

\begin{proof} Take a Dulac's function $B(y)=(1+ny)^{\alpha}$.
Then it follows that
$$
 \mathop{\rm div}(BP, BQ)=x^{\alpha -1}(1+ny)^{2\alpha}[\alpha d+(l(\alpha +1)+n\alpha)x].
$$
Hence, if system \eqref{e2.2} has limit cycles, then they must
intersect the line $x=-\alpha d/[l(\alpha +1)+n\alpha]$. The proof
is complete.
\end{proof}

\begin{lemma} \label{lm2.3} If system \eqref{e2.2} has limit cycles around the
origin, then they must not intersect the line $x=-1/a$.
\end{lemma}

\begin{center}
\setlength{\unitlength}{1cm}
\begin{picture}(7,5.5)
\thicklines \put(1,0){\vector(0,1){5}}
\put(0,3.5){\vector(1,0){7}}
\put(2.83,2){\vector(2,3){0.15}}
\put(4,0){\line(0,1){5}} \qbezier(1,1)(2,1)(2.8,2)
\qbezier(2.8,2)(3.6,3)(4,3) \qbezier(4,3)(4.5,3)(5.3,2)
\qbezier(5.3,2)(6.1,1)(6.1,0.9)
\put(0.5,1){\makebox(0.5,0.5)[b]{$R$}}
\put(3.5,2){\makebox(0.5,0.5){$S$}}
\put(0.5,3){\makebox(0.5,0.5){$O$}}
\put(0.5,4.5){\makebox(0.5,0.5)[t]{$y$}}
\put(6.5,3){\makebox(0.5,0.5)[r]{$x$}}
\put(4,4.5){\makebox(2,0.5){$x=-1/a$}}
\end{picture}
\end{center}
\begin{proof} Let $\Gamma$ be a limit cycle around the origin which
intersects the $y$-axis at $R$ and the line $x=-1/a$ on $S$. Since
$Q(x,y)=(1+ny)^{\alpha+1}x^{\alpha}(1+a^{\alpha}x^{\alpha})<0$ on
the right hand of the line $x=-1/a$, $\Gamma$ will go in a right
downward direction after it passes through the point $S$.
Moreover, it is impossible for it to go around the origin. This
contradicts the assumption that $\Gamma$ is a limit cycle around
the origin. The proof is complete.
\end{proof}

\begin{theorem} \label{thm2.4} If $ad\geq [l(\alpha
+1)+n\alpha]/\alpha$, then system \eqref{e2.2} has no limit cycle
around the origin.
\end{theorem}

\begin{proof} By Lemma \ref{lm2.2}, if system
\eqref{e2.2} has a limit cycle around the origin, it must
intersect the line $x=-\alpha d/[l(\alpha +1)+n\alpha]$. By Lemma
\ref{lm2.3}, it must not intersect the line $x=-1/a$. Hence the
line $x=-\alpha d/[l(\alpha +1)+n\alpha]$ must lie in the left
hand of the line $x=-1/a$, i.e. $-\alpha d/[l(\alpha
+1)+n\alpha]<-1/a$. Noticing the assumptions of
$l>-n\alpha/(\alpha+1), a<0, d<0$, the inequality above can be
written in the form $ad<[l(\alpha +1)+n\alpha]/\alpha$. Therefore,
if $ad\geq [l(\alpha +1)+n\alpha]/\alpha$, then system
\eqref{e2.2} has no limit cycle around the origin.The proof is
complete.
 \end{proof}

 From what has been discussed above, we can assume $0<ad<[l(\alpha
+1)+n\alpha]/\alpha$, and we know that if system \eqref{e2.2} has
a limit cycle around the origin, it must lie in the region
$D=\{(x,y) : x<-1/a,\;  y> -1/n\}$.

Let
\begin{equation}\label{e2.3}
\hat x=x,\quad e^{-\hat y}=1+ny,\quad d\hat t=-(1+ny)^\alpha dt.
\end{equation}
On dropping the hats for ease of notation, system \eqref{e2.2}
becomes
\begin{equation}\label{e2.4}
\begin{gathered}
dx/dt=-\varphi(y)-F(x),\\[0.2cm]
dy/dt=g(x),
\end{gathered}
\end{equation}
where
$$
\begin{gathered}
\varphi(y)=(e^{y}-1)^\alpha[(n-m)+me^{-y}]/n^{\alpha+1},\\[0.3cm]
F(x)=dx^\alpha+(l-n)x^{\alpha+1}-na^\alpha x^{2\alpha+1},\\[0.3cm]
g(x)=nx^\alpha(1+a^\alpha x^\alpha).
\end{gathered}
$$
Denote
$$
\begin{gathered}
f(x)=F'(x)=\alpha dx^{\alpha-1}+(\alpha+1)(l-n)x^
\alpha-n(2\alpha+1)a^\alpha x^{2\alpha},\\[0.4cm]
G(x)=\displaystyle\int_0^x g(x)dx=\dfrac{n}{\alpha+1}x^{\alpha+1}+
\dfrac{na^\alpha}{2\alpha+1}x^{2\alpha+1}.
\end{gathered}
$$

\begin{lemma} \label{lm2.5} If $d<0, |d|\ll 1$, then the singular point $O$ of
system \eqref{e2.1} is a stable focus; if $d=0$, then the singular
point $O$ is an unstable focus.
 \end{lemma}

\begin{proof} Let $z=G(x)$. We denote by $x_1(z)>0$ the inverse
function of $G(x)$ in the interval $(0,-1/a)$, and by $x_2(z)$ the
inverse function of $G(x)$ in the half plane $x<0$. Since
$G(x)\sim \frac{n}{\alpha+1}x^{\alpha+1}$ as $x\to 0$, we have
$x_1(z)\sim (\frac{\alpha+1}{n})^{1/(\alpha+1)}z^{1/(\alpha+1)}$
and $x_2(z)\sim -x_1(z)$ as $z\to 0$. Denote $F_i(z)=F(x_i(z))$,
$i=1,2$. It is easy to verify that $\varphi(y)$ and $F_i(z)$
satisfy all the conditions of Corollary 4 in \cite{hm} for $d\leq
0$ and $a<0$.

Since
$$
F_1(z)-F_2(z)\sim
2d\left(\frac{\alpha+1}{n}\right)^{\alpha/(\alpha+1)}z^\frac{\alpha}{\alpha+1}
-2na^\alpha\left(\frac{\alpha+1}{n}
\right)^{(2\alpha+1)/(\alpha+1)}z^\frac{2\alpha+1}{\alpha+1}$$ as
$z\to 0$, it is easy to see that for $0<z\ll 1$, if $d<0$, then
$F_1(z)-F_2(z)<0$ and if $d=0$, then $F_1(z)-F_2(z)>0$. Hence, if
$d<0$, then the origin $O$ is an unstable focus of system
\eqref{e2.4}; if $d=0$, then the origin $O$ is a stable focus.
Noticing the transformation \eqref{e2.3}, it follows that if
$d<0$, then the origin $O$ is a stable focus of system
\eqref{e2.1}; if $d=0$, then the origin $O$ is an unstable focus.
The proof is complete.
\end{proof}

\begin{theorem} \label{thm2.6}
 If $0<ad<[l(\alpha+1)+n\alpha]/\alpha$
and $|d|\ll 1$, then system \eqref{e2.1} has limit cycles around
the origin.
\end{theorem}

\begin{proof}
 By Lemma \ref{lm2.5} and the Hopf's
bifurcation Theorem in \cite{zj}, the conclusion of Theorem
\ref{thm2.6} can be obtained directly. This completes the proof.
\end{proof}

\begin{theorem} \label{thm2.7}
 If $0<ad<[l(\alpha+1)+n\alpha]/\alpha$,
then system \eqref{e2.1} has at most one limit cycle around the
origin, and if it exists, it must be unstable.
\end{theorem}

\begin{proof}
Consider system \eqref{e2.4}. It is easy to determine
$\varphi'(y)>0$, $\varphi(0)=0$ and $xg(x)>0$ for $x<-1/a$ and
$x\neq 0$, and
$$
\Big(\frac{f(x)}{g(x)}\Big)'=-\frac{\alpha}{nx^2(1+a^\alpha
x^\alpha)^2}{\{[n\alpha+l(\alpha +1)]a^\alpha
x^{\alpha+1}+(\alpha+1)da^\alpha x^\alpha +d\} }.
$$
Let
$$
h(x)=[n\alpha+l(\alpha +1)]a^\alpha
x^{\alpha+1}+(\alpha+1)da^\alpha x^\alpha +d.
$$
Then it follows
\begin{align*}
h'(x)& = (\alpha+1)[n\alpha+l(\alpha +1)]a^\alpha
x^\alpha+\alpha(\alpha+1)da^\alpha x^{\alpha-1} \\
&=(1+\alpha)\{[l(\alpha+1)+n\alpha]x+d\alpha\}a^{\alpha}x^{\alpha-1}.
\end{align*}

It is easy to verify that $h(x)$ has a unique point, namely
$x_0=-\alpha d/[n\alpha+l(\alpha+1)]$ at which $h(x)$ attains its
maximum, and $h(x_0)=d\{1-[\frac{\alpha
ad}{l(\alpha+1)+n\alpha}]^\alpha\}$. Noticing that
$0<ad<[l(\alpha+1)+n\alpha]/\alpha$, we know that $h(x_0)<0$,
hence $h(x)<0$. Therefore, it follows that
$(\frac{f(x)}{g(x)})'>0$.

By Lemma \ref{lm2.5}, if $d<0$, then the origin $O$ is a stable
focus of system \eqref{e2.1}. Applying the Zhang Zhifen's Theorem
in \cite{zz}, we have that system \eqref{e2.1} has at most one
limit cycle around the origin, and if it exists, it must be
unstable. This completes the proof.
\end{proof}

\section{The applications of the main results}
In \cite{ld}, the author considered the  system
\begin{equation}\label{e3.1}
\begin{gathered}
dx/dt=-y+dx+x^2+dxy-(m+1)y^2-my^3,\\[0.2cm]
dy/dt=x(1+ax+y),\quad (0\leq m\leq1),
\end{gathered}
\end{equation}
which is a special case of \eqref{e2.1} when  $n=1, \alpha=1$,
$l=1$ in system \eqref{e2.1}. It is easy to derive the following
corollaries by Theorems  \ref{thm2.1}, \ref{thm2.4}, \ref{thm2.6},
\ref{thm2.7}.

\begin{corollary} \label{co3.1}  For system \eqref{e3.1}, if $ad\leq 0$, then
there is no limit cycle in the whole plane; if $ad\geq3$, then
there is no limit cycle around the origin.
 \end{corollary}

\begin{corollary} \label{co3.2} For system \eqref{e3.1}, we have the following
conclusions:
\begin{itemize}
\item[(i)] If $0<ad<3$, then there is at most one limit cycle around
the origin, and if it exists, it must be unstable;

\item[(ii)] If $0<ad<3$ and $|d|\ll 1$, then there is a unique
unstable limit cycle around the origin.
\end{itemize}
\end{corollary}

In \cite{zy}, the author discussed the  quadratic system
\begin{equation}\label{e3.2}
\begin{gathered}
dx/dt=-y+dx+x^2+dxy-y^2\\[0.2cm]
dy/dt=x(1+ax+y),
\end{gathered}
\end{equation}
which is a special case of \eqref{e3.1} when $m=0$ in system
\eqref{e3.1} or a special case of \eqref{e2.1} when $n=1,
\alpha=1$, $l=1$ and $m=0$ in system \eqref{e2.1}. From what has
been discussed above, it is easy to see that above Corollaries
\ref{co3.1}, \ref{co3.2} still hold for system \eqref{e3.2}.

\begin{remark} \label{re10} \rm
Corollaries \ref{co3.1}, \ref{co3.2} not only contain all results
in \cite{ld} and \cite{zy}, but also show sufficient conditions
for the existence of limit cycles which was not involved in
\cite{zy}.
\end{remark}

In \cite{gc}, the authors studied the system
\begin{equation}\label{e3.3}
\begin{gathered}
dx/dt=dx^\alpha(1-ny)^\alpha+lx^{\alpha+1}(1-ny)^{\alpha-1},\\[0.2cm]
dy/dt=x^\alpha[(1-ny)^\alpha+(ax)^\alpha].
\end{gathered}
\end{equation}
On substituting
\begin{equation}\label{e3.4}
\hat x=x,\quad \hat y=-y,\quad \hat t=-t
\end{equation}
into \eqref{e3.3} and dropping the hats for ease of notation,
system \eqref{e3.3} reduces to the equivalent system
\begin{equation}\label{e3.5}
\begin{gathered}
dx/dt=-y^\alpha(1+ny)^\alpha+dx^\alpha(1+ny)^\alpha-lx^{\alpha+1}(1+ny)^{\alpha-1},\\[0.2cm]
dy/dt=x^\alpha[(1+ny)^\alpha+(ax)^\alpha],
\end{gathered}
\end{equation}
which is a special case of \eqref{e2.1} when $m=0$ in system
\eqref{e2.1}. Noticing the negative sign before $l$ in system
\eqref{e3.5} and the transformation \eqref{e3.4}, by Theorems
\ref{thm2.1}, \ref{thm2.4}, \ref{thm2.6}, \ref{thm2.7}, we have
the following results.

\begin{corollary} \label{co3.3} If $ad[l(\alpha+1)-n\alpha]\geq 0$,
then system \eqref{e3.5} has no limit cycle in the whole plane; If
$ad\geq [n\alpha -l(\alpha+1)]/\alpha$, then system \eqref{e3.5}
has no limit cycle around the origin.
\end{corollary}

\begin{corollary} \label{co3.4}
For system \eqref{e3.5}, we have the following conclusions:
\begin{itemize}
\item[(i)] If $0<ad<[n\alpha -l(\alpha+1)]/\alpha$, then system
\eqref{e3.5} has at most one limit cycle around the origin, and if
it exists, it must be stable;

\item[(ii)]  If $0<ad<[n\alpha -l(\alpha+1)]/\alpha$ and $|d|\ll 1$,
then system \eqref{e3.5} has a unique stable limit cycle around
the origin.
\end{itemize}
\end{corollary}

In \cite{sj}, the author discussed the following system:
\begin{equation}\label{e3.6}
\begin{gathered}
dx/dt=-y^\alpha(1-y)^\alpha-dx^\alpha(1-y)^\alpha+lx^{\alpha+1}(1-y)^{\alpha-1},\\[0.2cm]
dy/dt=x^\alpha[(1-y)^\alpha+(ax)^\alpha],
\end{gathered}
\end{equation}
which is a special case of \eqref{e3.5} when $n=1$ in system
\eqref{e3.5}. by Corollary \ref{co3.3} and Corollary \ref{co3.4},
we have the following results directly.

\begin{corollary} \label{co3.5}
If $ad[l(\alpha+1)-\alpha]\geq 0$, then system \eqref{e3.6} has no
limit cycle in the whole plane; If $ad\geq [\alpha
-l(\alpha+1)]/\alpha$, then system \eqref{e3.6} has no limit cycle
around the origin.
\end{corollary}

\begin{corollary} \label{co3.6}
 For system \eqref{e3.6}, we have the following conclusions:
\begin{itemize}
\item[(i)] If $0<ad<[\alpha -l(\alpha+1)]/\alpha$, then system
\eqref{e3.6} has at most one limit cycle around the origin, and if
it exists, it must be stable;

\item[(ii)]  If $0<ad<[\alpha -l(\alpha+1)]/\alpha$ and $|d|\ll 1$,
then system \eqref{e3.6} has a unique stable limit cycle around
the origin.
\end{itemize}
\end{corollary}

In \cite{co}, the author considered the following quadratic system
with a degenerate critical point:
\begin{equation} \label{e3.7}
\begin{gathered}
dx/dt=-y-dx+lx^2+dxy+y^2,\\[0.2cm]
dy/dt=x(1+ax-y),
\end{gathered}
\end{equation}
which is a special case of \eqref{e3.6} when $\alpha=1$ in system
\eqref{e3.6} or a special case of \eqref{e2.1} when $n=1,
\alpha=1$, and $m=0$ in system \eqref{e2.1}. From what has been
discussed above, it is easy to see that above Corollaries
\ref{co3.5}, \ref{co3.6} in which when $\alpha=1$ still hold for
system \eqref{e3.7}.

\begin{remark} \label{re3.1} \rm
Corollaries \ref{co3.3}, \ref{co3.4} and corollaries \ref{co3.5},
\ref{co3.6} contain the main results in \cite{gc} and \cite{sj}
respectively.
\end{remark}

\subsection*{Acknowledgement}
The authors would like to thank the anonymous referee for his/her
helpful comments and suggestions which improved the presentation
of this work.

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\end{document}