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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 84, 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  (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2005/84\hfil Periodic solutions]
{Periodic solutions of polynomial non-autonomous differential equations}
\author[M. A. M. Alwash\hfil EJDE-2005/84\hfilneg]
{Mohamad A. M. Alwash}

\address{Mohamad A. M. Alwash \hfill\break
Department of Mathematics,
West Los Angeles College,
9000 Overland Avenue, Los Angeles, CA 90230-3519, USA}
\email{alwashm@wlac.edu}

\date{}
\thanks{Submitted July 7, 2005. Published July 25, 2005.}
\subjclass[2000]{34C25, 34C07, 34C05}
\keywords{Periodic solutions; polynomial non-autonomous equations;
\hfill\break\indent
 Abel differential equations; limit cycles; uniformly isochronous centers;
\hfill\break\indent
 Hilbert's sixteenth problem}

\begin{abstract}
 We present some results on the number of periodic solutions for
 scalar non-autonomous polynomial equations of degree five.
 We also consider a class of polynomial equations of any degree.
 Our results give upper bounds for the number of limit cycles of
 two-dimensional systems.
\end{abstract}

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

\section{Introduction}

We consider differential equation \begin{equation} \label{e1.1}
\dot{z}:=\frac{dz}{dt}=P_{0}(t)z^{n}+P_{1}(t)z^{n-1}+\dots+
P_{n-1}(t)z+P_{n}(t) \end{equation} where $z$ is a complex-valued function
and $P_{i}$ are real-valued continuous functions. This class of equations has
received some attention in the literature. The number of periodic
solutions of such equations has been studied in
\cite{a3,a5,a6,l1,l2,l3,p1,p2,s1}.

We denote by $z(t,c)$ the solution of \eqref{e1.1} satisfying $z(0,c)=c$.
Take a fixed real number $\omega$, we define the set $Q$ to be the set
of all complex numbers $c$ such that $z(t,c)$ is defined for all $t$ in the
interval $[0,\omega]$; the set $Q$ is an open set. On $Q$ we define the
displacement function $q$ by
$$
q(c)=z(\omega,c)-c.
$$
Zeros of $q$ identify initial points of solutions of \eqref{e1.1} which satisfy
the boundary conditions $z(0)=z(\omega)$. We describe such solutions as
{\em periodic} even when the functions $P_{i}$ are not themselves periodic.
However, if $P_{i}$ are $\omega$-periodic then these solutions are also
$\omega$-periodic.

Note that $q$ is holomorphic on $Q$. The multiplicity of a periodic solution
$\varphi$ is that of $\varphi (0)$ as a zero of $q$. It is useful to work
with a complex dependent variable. The reason is that periodic solutions
cannot then be destroyed by small perturbations of the right-hand side of
the equation. Suppose that $\varphi$ is a periodic solution of multiplicity $k$.
 By applying Rouche's theorem to the function $q$, for any sufficiently small
perturbations of the equation, there are precisely $k$ periodic solutions in
a neighborhood of $\varphi$ (counting multiplicity). Upper bounds to the
number of periodic solutions of \eqref{e1.1} can be used as upper bounds to
the number of periodic solutions when $z$ is limited to be real-valued.
This is the reason that $P_{i}$ are not allowed to be complex-valued.

When $n=3$, equation \eqref{e1.1} is known as the Abel
differential equation. This case is of particular interest because
of a connection with Hilbert's sixteenth problem; see \cite{a5}
for details. It was shown in \cite{l2} and \cite{p2} that when
$P_{0}(t)=1$, then Abel differential equation has exactly three
periodic solutions provided account is taken of multiplicity.
However, local questions related to Hilbert's sixteenth problem
(bifurcation of small-amplitude limit cycles and center
conditions) are reduced to polynomial equations in which $P_{0}$
does have zeros. In this case the results of \cite{l2} and
\cite{p2} no longer hold; indeed Lins Neto \cite{l1} has given
examples which demonstrate that there is no upper bound for the
number of periodic solutions. These examples can be used to show
that there is no upper bound to the number of periodic solutions
when $n \geq 4$ and $P_{0}(t)=1$. On the other hand, systems with
constant angular velocities can be reduced to polynomial
equations. Global results about the number of periodic solutions
can be used to obtain information about the number of limit
cycles; see, for example, \cite{a4} and \cite{c1}

The case $n=4$ was considered in \cite{a3} and \cite{a6} with
$P_{0}(t)=1$. The main concern was the multiplicity of $z=0$ when
the coefficients are polynomial functions in $t$ and in $\cos{t}$
and $\sin{t}$. It was shown in \cite{a6} that the multiplicity is
at most 8 when the coefficients are of degree 2; this result
provides a counterexample to Shahshahani conjecture \cite{s1}. In
\cite{a3}, the methods of Groebner bases were used to study
multiplicity and bifurcation of periodic solutions. In particular,
it was shown that the multiplicity is at most 10 when the
coefficients are polynomial functions of degree 3.

In this paper, we consider the case $n=5$. The aim is to gain
information on the total number of periodic solutions; this is a
global question, while looking at multiplicity leads only to local
results. In Section 2, we describe the phase portrait of
\eqref{e1.1} and recall some results from \cite{l2}. In Section 3,
we present some results on the number of periodic solutions. In
Section 4, we consider the real equation and from the derivatives
of the displacement function, we deduce some results on the number
of real periodic solutions. We also consider a class of equations
with $n \geq 5$ and give an upper bound to the number of real
periodic solutions. This result generalizes a recent result of
Panov \cite{p1}. In the final Section, we return to polynomial
two-dimensional systems. We use the results of Sections 3 and 4 to
give upper bounds for the number of limit cycles.

\section{The Phase Portrait}


If $\varphi(t)$ is a periodic solution of
\begin{equation}
\dot{z}=z^{5}+P_{1}(t)z^{4}+P_{2}(t)z^{3}+P_{3}(t)z^{2}+P_{4}(t)z+
P_{5}(t) \label{e2.1}
\end{equation}
we make the transformation $z \mapsto z-\varphi(t)$; \eqref{e2.1} then
 becomes
\begin{equation}
\dot{z}=z^{5}+P_{1}(t)z^{4}+P_{2}(t)z^{3}+P_{3}(t)z^{2}+P_{4}(t)z. \label{e2.2}
\end{equation}
If $\varphi$ is real, the coefficients of \eqref{e2.2} are also real.
No generality is lost by considering equation \eqref{e2.2} because \eqref{e2.1}
has at least one real periodic solution. In fact, if $n$ is odd and
$P_{0} \equiv 1$ then equation \eqref{e1.1} has at least one real periodic
 solution. This result was given in \cite{p2} for equations with periodic
 coefficients. It can be verified quite easily that the method of
proof in \cite{p2} works whether the coefficients are periodic or
not.

We identify equation \eqref{e2.2} with the quadruple $(P_{1},P_{2},P_{3},P_{4})$
and write $\mathcal{L}$ for the set of all equations of this form.
With the usual definitions of additions and scalar multiplications,
$\mathcal{L}$ is a linear space; it is a normed space if for
 $P=(P_{1},P_{2},P_{3},P_{4})$, we define
 \[
 \|P\| = \max\{\max_{0\leq t \leq \omega}|P_{1}(t)|,
\max_{0\leq t \leq \omega}|P_{2}(t)|,
\max_{0\leq t \leq \omega}|P_{3}(t)|,
\max_{0\leq t \leq \omega}|P_{4}(t)|\}
\]
The displacement function $q$ is holomorphic on the open set $Q$.
Since $z=0$ is a solution, $Q$ contains the origin. Moreover,
$q$ depends continuously on $P$ with the above norm on
$\mathcal{L}$ and the topology of uniform convergence on compact sets
on the set of holomorphic functions.

The positive real axis and the negative real axis are invariant.
Moreover, if $\varphi$ is a non-real solution which is periodic,
then so is $\bar{\varphi}$, its complex conjugate.

In \cite{l2}, it was shown that the phase portrait of \eqref{e2.2}
is as shown in Figure 1 below. We refer to \cite{l2} for the
details. There, the coefficients $P_{i}(t)$ were
$\omega-$periodic. It can be verified that the same methods are
applicable to the study of the number solutions that satisfy
$z(0)=z(\omega)$ whether the coefficients are periodic or not.

\begin{figure}[ht]
\begin{center}
\begin{picture}(150,150)
\put(75,75){\circle{40}}
\put(90,75){\vector(1,0){10}}
\put(60,75){\vector(-1,0){10}}
\put(75,90){\vector(0,1){10}}
\put(75,60){\vector(0,-1){10}}
\put(106,72){\vector(0,-1){10}}
\put(106,78){\vector(0,1){10}}
\put(44,72){\vector(0,-1){10}}
\put(44,78){\vector(0,1){10}}
\put(72,106){\vector(-1,0){10}}
\put(78,106){\vector(1,0){10}}
\put(72,44){\vector(-1,0){10}}
\put(78,44){\vector(1,0){10}}
\put(125,100){\vector(-1,1){10}}
\put(100,115){\vector(1,-1){10}}
\put(25,100){\vector(1,1){10}}
\put(50,120){\vector(-1,-1){10}}
\put(125,50){\vector(-1,-1){10}}
\put(100,30){\vector(1,1){10}}
\put(25,50){\vector(1,-1){10}}
\put(50,30){\vector(-1,1){10}}
\put(95,90){\vector(-1,-1){10}}
\put(55,90){\vector(1,-1){10}}
\put(95,58){\vector(-1,1){10}}
\put(55,58){\vector(1,1){10}}
\put(94,70){\line(1,0){35}}
\put(94,80){\line(1,0){35}}
\put(56,70){\line(-1,0){35}}
\put(56,81){\line(-1,0){35}}
\put(70,96){\line(0,1){35}}
\put(80,96){\line(0,1){35}}
\put(70,54){\line(0,-1){35}}
\put(80,54){\line(0,-1){35}}
\put(93,83){\line(1,1){35}}
\put(86,90){\line(1,1){35}}
\put(57,83){\line(-1,1){35}}
\put(65,92){\line(-1,1){35}}
\put(93,67){\line(1,-1){35}}
\put(86,59){\line(1,-1){35}}
\put(58,66){\line(-1,-1){35}}
\put(65,58){\line(-1,-1){35}}
\put(132,70){\makebox(0,0)[bl]{$G_{0}$}}
\put(130,130){\makebox(0,0)[bl]{$G_{1}$}}
\put(130,100){\makebox(0,0)[bl]{$H_{0}$}}
\put(90,125){\makebox(0,0)[bl]{$H_{1}$}}
\put(70,140){\makebox(0,0)[bl]{$G_{2}$}}
\put(2,72){\makebox(0,0)[bl]{$G_{4}$}}
\put(15,125){\makebox(0,0)[bl]{$G_{3}$}}
\put(5,95){\makebox(0,0)[bl]{$H_{3}$}}
\put(50,125){\makebox(0,0)[bl]{$H_{2}$}}
\put(50,15){\makebox(0,0)[bl]{$H_{5}$}}
\put(15,20){\makebox(0,0)[bl]{$G_{5}$}}
\put(5,50){\makebox(0,0)[bl]{$H_{4}$}}
\put(70,5){\makebox(0,0)[bl]{$G_{6}$}}
\put(85,15){\makebox(0,0)[bl]{$H_{6}$}}
\put(130,20){\makebox(0,0)[bl]{$G_{7}$}}
\put(130,50){\makebox(0,0)[bl]{$H_{7}$}}
\put(70,72){\makebox(0,0)[bl]{$D$}}
\thicklines
\end{picture}
\end{center}
\caption{Phase Portrait}
\end{figure}

Note that the radius, $\rho$, of the disc $D$ depends only on
$\|P\|$ and $\omega$.
 If $z=r e^{i\theta}$ then the sets
$G_{k},k=0,1,\dots,7$, which are the arms in the figure, are
defined by
\[
G_{k}= \{z| r>\rho, \frac{k\pi}{4}-\frac{a}{r} < \theta < \frac{k\pi}{4}+\frac{a}{r}\}
\]
where $a=\max\{6,6\|P\|\}$. Between the arms are the sets
$H_{k},k=0,1,\dots,7$, which are defined by
\[
H_{k}= \{z| r>\rho, \frac{k\pi}{4}+\frac{a}{r} \leq \theta \leq \frac{(k+1)\pi}{4}
-\frac{a}{r}\}
\]
For even $k$, trajectories can enter $G_{k}$ only across $r=\rho$,
and for odd $k$, trajectories can leave $G_{k}$ only across $r=\rho$.
No solution can become infinite in $H_{k}$ as time either increases or
decreases. Every solution enters $D$. Solutions become unbounded
if and only if they remain in one of the arms $G_{k}$, tending to infinity
 as $t$ increases if $k$ is even and as $t$ decreases if $k$ is odd.
For each $k$, there is a unique curve $C_{k}$ on the bottom of $G_{k}$ such
that the solution $z(t,c)$ remains in $G_{k}$ for as long as it is defined
if and only if $c \in C_{k}$.

Let $q(P,c)=z_{P}(\omega,c)-c$, where $z_{P}(t,c)$ is the solution of
 $P \in \mathcal{L}$ satisfying $z_{P}(0,c)=c$. Suppose that $(P_{j})$ and
$(c_{j})$ are sequences in $\mathcal{L}$ and $\mathbb{C}$,
respectively, such that $q(P_{j},c_{j})=0$. If $P_{j} \to  P$ and
$c_{j} \to  c$ as $j \to  \infty$, then either $q(P,c)=0$, in this
case $z_{P}(t,c)$ is a periodic solution, or $z_{P}(t,c)$ is not
defined for the whole interval $0 \leq t \leq \omega$. In the
later case, we say that $z_{P}(t,c)$ is a {\em singular periodic
solution}. We also say that $P$ has a singular periodic solution
if $c_{j} \to  \infty$; in this case there are $\tau$ and $c$ such
that the solution $z_{P}$ with $z_{P}(\tau)=c$ becomes unbounded
at finite time as $t$ increases and as $t$ decreases. We summarize
the results of \cite{l2} when applied to \eqref{e2.2}.

\begin{proposition} \label{prop2.1}
\begin{itemize}
\item[(i)] Let $\mathcal{A}$ be the subset of $\mathcal{L}$ consisting of all
equations which have no singular periodic solutions.
The set $\mathcal{A}$ is open in $\mathcal{L}$. All equations in the same
components of $\mathcal{A}$ have the same number of periodic solutions.

\item[(ii)] The equation $\dot{z}=z^{5}$ has exactly five periodic solutions.
\item[(iii)] The number of periodic solutions of equation \eqref{e2.2} is odd.
\item[(iv)] For each $k$, there is just one solution that crosses $r=\rho$
at a given time and becomes infinite without leaving $G_{k}$
(under reversed time if $k$ is odd).
\end{itemize}
\end{proposition}

\section{Number of Periodic Solutions}

We call the solution $z=0$ a {\em center} if $z(t,c)$ is periodic
for all $c$ in a neighborhood of $0$. When $P_{0}$ has zeros then
there are equations with a center. For cubic equations, this is
related to the classical center problem of polynomial
two-dimensional systems; we refer to \cite{a5} for details.
However, when $P_{0}$ has no zeros then $z=0$ is never a center.
This result was proven in \cite{a6} for the case $n=4$.
 We give a brief proof, with $n=5$, for the sake of completeness.

\begin{theorem} \label{thm3.1}
The solution $z=0$ is isolated as a periodic solution of \eqref{e2.2}.
\end{theorem}

\begin{proof}
Suppose, if possible, that there is a open set
$U \subset \mathbb{C}$ containing the origin such that all solutions
starting in $U$ are periodic. Then $q \equiv 0$ in the component of
its domain of definition containing the origin. But the real zeros
of $q$ are contained in the disc $D$. Thus
\[
\inf\{c \in \mathbb{R}: c>0, z(t,c)\mbox{ is not defined for }
0 \leq t \leq \omega\} < \infty
\]
It follows that there is a real singular periodic solution;
but a positive real periodic solution which tends to infinity can do
so only as $t$ increases. This is a contradiction, and the result follows.
\end{proof}

Now, we give the result about the number of periodic solutions.

\begin{theorem} \label{thm3.2}
 Suppose that $r^{3}-r P_{2}(t) - P_{3}(t) \geq 0$ and
$r^{3}-r P_{2}(t) + P_{3}(t) \geq 0$ for positive $r$ and
$0 \leq t \leq \omega$. Then equation \eqref{e2.2} has exactly
 five periodic solutions.
\end{theorem}

\begin{proof} With $z=re^{i\theta}$, we have
\[
\dot{\theta}=r^{4}\sin{4\theta}+r^{3}P_{1}(t) \sin{3\theta}+
r^{2}P_{2}(t) \sin{2\theta}+r P_{3}(t)\sin{\theta}.
\]
If $|c| > \rho$ and is real then the real solution $z(t,c)$ remains outside
the disk $D$ as $t$ increases and will become infinite.
Solutions that enter $G_{0}$ or $G_{4}$ will leave $G_{0}$ or $G_{4}$,
except the solution that enters at the intersection of $C_{0}$ and $C_{4}$
with the real axis; this solution is real because any solution which is
once real is always real. Therefore, the unique solution that becomes
infinite described in part (iv) of Proposition \ref{prop2.1} is a real solution
if $k=0$ or $k=4$. On the other hand, no real solution is unbounded as
$t$ increases and decreases. Hence, no singular periodic solution
enters $G_{0}$ or $G_{4}$ because singular periodic solutions are
unbounded both as $t$ increases and decreases.
Thus, a singular periodic solution enters $D$ from $G_{1}$ or $G_{3}$
and leaves $D$ to $G_{2}$. Hence, for a singular periodic solution
$\dot{\theta} > 0$ at $\theta = \frac{\pi}{3}$ and $\dot{\theta} < 0$
at $\theta = \frac{2\pi}{3}$. On the other hand,
\begin{gather*}
\dot{\theta}(\frac{\pi}{3})=\frac{-\sqrt{3}}{2}r(r^{3}-rP_{2}(t)-P_{3}(t)),\\
\dot{\theta}(\frac{2\pi}{3})=\frac{\sqrt{3}}{2}r(r^{3}-rP_{2}(t)+P_{3}(t))
\end{gather*}
Under the above hypotheses, $\dot{\theta}(\frac{\pi}{3}) < 0$ and
$\dot{\theta}(\frac{2\pi}{3}) > 0$. Therefore, no singular periodic
solution can enter $D$ from $G_{1}$ or $G_{3}$ and leaves $D$ to $G_{2}$.
Since the phase portrait is symmetric about the $x-$axis, no singular
periodic solution can enter $D$ from $G_{5}$ or $G_{7}$ and leaves $D$
to $G_{6}$. It follows that the equation does not have a singular
periodic solution.

Now, consider the class of equations
\[
\dot{z}=z^{5}+s P_{1}(t)z^{4}+s P_{2}(t)z^{3}+s P_{3}(t)z^{2}+s P_{4}(t)z,
\]
with $0 \leq s \leq 1$.
If $r^{3}-r P_{2}(t) - P_{3}(t) \geq 0$ and
$r^{3}-r P_{2}(t) + P_{3}(t) \geq 0$ then
$r^{3}-s r P_{2}(t) - s P_{3}(t) \geq 0$ and
$r^{3}-s r P_{2}(t) + s P_{3}(t) \geq 0$ for $0 \leq s \leq 1$.
Therefore, any equation in this family does not have singular
periodic solutions. The equation $\dot{z}=z^{5}$ belongs to this
family and has five periodic solutions.
By part (i) of Proposition \ref{prop2.1}, each of these equations has
five periodic solutions.
\end{proof}

\begin{corollary} \label{coro3.3}
If $P_{3}(t) \equiv 0$, and $P_{2}(t) \leq 0$ then \eqref{e2.2} has five
periodic solutions.
\end{corollary}

\section{Real Periodic Solutions}

Consider the equation
\[
\dot{x}=f(x,t)
\]
where $x \in \mathbb{R}$ and $f$ is as smooth as is required in the argument.
With $f_{k}=\frac{\partial^{k}f}{\partial x^{k}}$, we define
\begin{gather*}
E(t,c)=\exp\big[\int_{0}^{t}f_{1}(x(t,c),\tau) d\tau\big], \\
D(t,c)=E(t,c)f_{2}(x(t,c),t),\\
G(t,c)=\int_{0}^{t}D(\tau,c) d\tau
\end{gather*}
 From \cite{l1}, we have the following formulae for the first three
derivatives of $q(c)$,
\begin{gather*}
q'(c)=E(\omega,c)-1,\\
q''(c)=E(\omega,c) \int_{0}^{\omega}D(t,c)dt,\\
q'''(c)=E(\omega,c)[\frac{3}{2}(G(\omega,c))^{2}+
\int_{0}^{\omega}(E(\omega,c))^{2}f_{3}(x(t,c),t)dt] \,.
\end{gather*}
Formulae for the fourth and fifth derivatives of $q$ are given in \cite{a6}.
Their use is not as direct as that of the first three derivatives;
simply $f_{4}\geq 0$ does not imply that $q^{(iv)} \geq 0$.

If the $n$-th derivative of a function does not change sign on an interval,
then the function has at most $n$ zeros in that interval. Using this fact
and the formulae for the derivatives of $q$, we prove the following.

\begin{theorem} \label{thm4.1} Consider the equation
\begin{equation}
 \dot{x}=x^{5}+P_{1}(t)x^{4}+P_{2}(t)x^{3}+P_{3}(t)x^{2}+P_{4}(t)x+
P_{5}(t) \label{e4.1}
\end{equation}
with $x \in \mathbb{R}$. \\
(i) If $P_{2}(t) \geq 0.4 (P_{1}(t))^{2}$ then  \eqref{e4.1} has
at most three real periodic solutions.\\
(ii) If $P_{1}(t) \geq 0$, $P_{2}(t) \geq 0$, and $P_{3}(t) \geq 0$,
then  \eqref{e4.1} has at most two positive periodic solutions.
\end{theorem}

\begin{proof}  (i) Since $f_{3}=6(10x^{2}+4P_{1}x+P_{2})$, it follows that
 $f_{3} \geq 0$ if $16 (P_{1})^{2}-40P_{2} \leq 0$. Hence, $q'''(c) > 0$
 if $P_{2} \geq 0.4 P_{1}^{2}$. Therefore, $q$ has at most three zeros.

 (ii) The conditions imply that $f_{2} \geq 0$ for positive $x$.
This implies that $q''(c) > 0$ when $c>0$.
\end{proof}

Now, we consider the equation
\begin{equation}
\dot{x}=x^{n}+P_{1}(t)x^{m}+P_{2}(t)x^{3}+P_{3}(t)x^{2}+P_{4}(t)x +
P_{5}(t), \label{e4.2}
\end{equation}
with $n>m>3$. Using the ideas of cross-ratio, it was shown in \cite{p1}
that equation \eqref{e4.2} has at most three periodic solutions when
$n$ is odd and $P_{1} \equiv P_{2} \equiv 0$. The method used in the
proof of Theorem \ref{thm4.1}. can be used to prove the following
generalization of this result.

\begin{theorem} \label{thm4.2}
\begin{itemize}
\item[(i)] If $n$ and $m$ are odd, $P_{1}(t) \geq 0$, and $P_{2}(t) \geq 0$
then  \eqref{e4.2} has at most three periodic solutions.
\item[(ii)] If $P_{1}(t) \geq 0$, $P_{2}(t) \geq 0$, and $P_{3}(t) \geq 0$
then equation \eqref{e4.2} has at most two positive periodic solutions.
\item[(iii)] If $n$ and $m$ are odd, $P_{1} \geq 0$, $P_{2}(t) \geq 0$
and $P_{3}(t) \leq 0$ then \eqref{e4.2} has at most two negative
periodic solutions.
\item[(iv)] If $P_{1}(t) \geq 0$ and $P_{2}(t) \geq 0$ then \eqref{e4.2} has
at most three positive periodic solutions.
\item[(v)] If $n$ and $m$ are even, $P_{1} \geq 0$, $P_{2}(t) \equiv 0$,
and $P_{3}(t) \geq 0$ then \eqref{e4.2} has at most two real periodic solutions.
\end{itemize}
\end{theorem}

\section{Number of Limit Cycles}

Consider the system
\begin{equation}
\begin{gathered}
\dot{x}=\lambda x - y +x(R_{n-1}(x,y)+R_{n-2}(x,y)+\dots+R_{1}(x,y))\\
\dot{y}=x+ \lambda y +y(R_{n-1}(x,y)+R_{n-2}(x,y)+\dots+R_{1}(x,y)),
\end{gathered}\label{e5.1}
\end{equation}
where $R_{i}$ is a homogeneous polynomial of degree $i$. The system
in polar coordinates becomes
\begin{gather*}
\dot{r}=r^{n}R_{n-1}(\cos{\theta},\sin{\theta})+
r^{n-1}R_{n-2}(\cos{\theta},\sin{\theta})+\dots+
r^{2}R_{1}(\cos{\theta},\sin{\theta})+\lambda r\\
\dot{\theta}=1.
\end{gather*}
Some necessary conditions for a center are given in \cite{a4}. It is clear
that the origin is the only critical point and if it is a center then
it is a uniformly isochronous center. Limit cycles of \eqref{e5.1}
correspond to positive $2\pi-$periodic solutions of
\[
\frac{dr}{d\theta}=R_{n-1} r^{n} + R_{n-2} r^{n-1} + \dots
+ R_{1} r^{2} + \lambda r
\]
Now, we consider the case $n=5$. In the special case $R_{3} \equiv 0$ and
$R_{1} \equiv 0$, the center conditions were given in \cite{a1,a2,v1}.
The following result follows from Theorems \ref{thm3.2} and \ref{thm4.1}.

\begin{theorem} \label{thm5.1}
Consider system \eqref{e5.1} with $n=5$ and $R_{4} \equiv 1$.
\begin{itemize}
\item[(i)] If $c^{3}-cR_{2}(\cos{\theta},\sin{\theta})-
R_{1}(\cos{\theta},\sin{\theta}) \geq 0$, and
$c^{3}-cR_{2}(\cos{\theta},\sin{\theta})+
R_{1}(\cos{\theta},\sin{\theta}) \geq 0$ for positive $c$ and
$0 \leq \theta \leq 2 \pi$, then the system has at most four limit cycles.

\item[(ii)] If $R_{2}(\cos{\theta},\sin{\theta}) \geq 0.4 (R_{3}
(\cos{\theta},\sin{\theta}))^{2}$,
then the system has at most two limit cycles.

\item[(iii)] If $R_{1} \equiv R_{3} \equiv 0$, and
$R_{2}(\cos{\theta},\sin{\theta}) \geq 0$, then the system has at
most two limit cycles.
\end{itemize}
\end{theorem}

Finally, we consider the case
\begin{equation}
\begin{gathered}
\dot{x}=\lambda x - y +x(R_{n-1}(x,y)+R_{m-1}(x,y)+R_{2}(x,y)+R_{1}(x,y))\\
\dot{y}=x+\lambda y +y(R_{n-1}(x,y)+R_{m-1}(x,y)+R_{2}(x,y)+R_{1}(x,y)),
\end{gathered}\label{e5.2}
\end{equation}
with $n>m>3$. In polar coordinates, this system reduces to
\[
\frac{dr}{d\theta}=R_{n-1} r^{n} + R_{m-1} r^{m} + R_{2} r^{3} + R_{1} r^{2}
+ \lambda r .
\]
If a function $R_{i}$ does not change sign, then it is necessary
to assume that $i$ is even. The following result follows directly
from Theorem \ref{thm4.2}.

\begin{theorem} \label{thm5.2}
\begin{itemize}
\item[(i)] Suppose that $n$ and $m$ are odd numbers, and $R_{n-1} \equiv 1$.
If $R_{m-1}(\cos{\theta},\sin{\theta}) \geq 0$ and
$R_{2}(\cos{\theta},\sin{\theta}) \geq 0$ then system \eqref{e5.2} has
at most two limit cycles.
\item[(ii)] If $R_{n-1} \equiv 1$, $R_{m-1}(\cos{\theta}, \sin{\theta}) \geq 0$,
and $R_{2}(\cos{\theta}, \sin{\theta}) \geq 0$, then system \eqref{e5.2}
has at most three limit cycles.
\end{itemize}
\end{theorem}

\begin{remark} \label{rmk5.3} \rm
 If the leading coefficient $R_{n-1}$ does not vanish anywhere then
the transformation of the independent variable
\[
\theta \mapsto \exp(\int_{0}^{\theta} R_{n-1}(\cos{u},\sin{u}) du)
\]
reduces the polar equation into a similar equation but with a leading
coefficient equals one.
\end{remark}

\subsection*{Acknowledgments}
I am very grateful to the Department of Mathematics,
University of California, Los Angeles for the hospitality.
The author is very grateful to the referee for the useful
remarks and suggestions.


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