\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 101, pp. 1--15.\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 2007 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2007/101\hfil Complex centers]
{Complex centers of polynomial differential equations}
\author[ M. A. M. Alwash\hfil EJDE-2007/101\hfilneg]
{Mohamad Ali M. Alwash}
\address{Department of Mathematics,
West Los Angeles College and University of California, Los Angeles\\
9000 0verland Avenue, Los Angeles, CA 90230-3519, USA}
\email{alwash@math.ucla.edu and alwashm@wlac.edu}
\thanks{Submitted March 19, 2007. Published July 25, 2007.}
\subjclass[2000]{34C05, 34C07, 34C25, 37C10, 13P10}
\keywords{Polynomial differential equations; periodic solutions; multiplicity;
\hfill\break\indent centers; Pugh problem; Groebner bases}
\begin{abstract}
We present some results on the existence and nonexistence of centers
for polynomial first order ordinary differential equations with
complex coefficients. In particular, we show that binomial differential
equations without linear terms do not have complex centers.
Classes of polynomial differential equations, with more than two terms,
are presented that do not have complex centers. We also study the
relation between complex centers and the Pugh problem.
An algorithm is described to solve the Pugh problem for equations
without complex centers. The method of proof involves phase plane
analysis of the polar equations and a local study of periodic solutions.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{remark}[theorem]{Remark}
\section{Introduction}
Consider the differential equation
\begin{equation} \label{e1.1}
\dot{z}:=\frac{dz}{dt}=A_{N}(t) z^{N}+A_{N-1}(t)z^{N-1}+\ldots+A_{1}(t)z
\end{equation}
where $z$ is complex and $A_{i}(t)$ are continuous functions in $t$.
Let $z(t,c)$ be the solution of \eqref{e1.1} such that $z(0,c)=c$.
For a fixed real number $\omega$, we say that $z(t,c)$ is \emph{periodic}
when $z(0,c)=z(\omega,c)$. If the functions $A_{i}(t)$ are periodic with
period $\omega$, then a periodic solution of equation \eqref{e1.1} is
a periodic function with period $\omega$. The \emph{multiplicity} of a
periodic solution $\varphi(t)$ of \eqref{e1.1} is the multiplicity of
$\varphi(0)$ as a zero of the displacement function $q\mapsto z(\omega,c)-c$.
Note that $q$ is defined and analytic in an open set containing the origin.
The solution $z=0$ is called a \emph{center} for the differential equation if
all solutions starting in a neighborhood of $0$ are $\omega$-periodic.
A center is called a \emph{complex center} when the coefficients are
complex functions, and it is called a \emph{real center} when the coefficients
are real functions. If the coefficients are real functions and $A_{N}(t)$
does not change sign, then $z=0$ is not a center (see \cite{A1}).
There are real centers when $A_{N}(t)$ changes sign (see \cite{A5}, \cite{A6},
and \cite{A15}); the problem is related to the classical center-focus problem
of polynomial two-dimensional systems. The case $N=3$, with
$A_{3}(t) \equiv 1$, was considered recently in \cite{A7}; where it was shown
that the equation could have a center at the origin if the coefficients
are complex valued. Two unexpected properties of the Abel differential
equation are given in \cite{A7}; there is no upper bound for the number
of periodic solutions, and the center
variety is formed by infinitely many connected components.
The work in \cite{A7} is mainly motivated by a
problem stated in \cite{A9}. It was shown in \cite{A9} that if the coefficients
are real and bounded by a constant $C$, then there are at most
$8\exp\{(3C+2)\exp[\frac{3}{2}(2C+3)^{N}]\}$ periodic solutions. The
problem in \cite{A9} is to prove an analogue of this result for complex
equations, that is, for complex $z$ and complex coefficients $A_{i}$.
This paper is partly motivated by the results of \cite{A7}.
Another motivation for the work in this paper is the local study
of the Pugh problem about equations with real coefficients.
We recall that Pugh problem (see \cite{A14}) is to
find an upper bound for the number of periodic solutions in terms
of $N$. It was shown in \cite{A10} that there are no upper bounds
for the number of periodic solutions for $N \geq 4$. Upper bounds
can be found for some particular classes; (see \cite{A1} and its
references). Hence, one should seek upper bounds for the number of
periodic solutions in terms of $N$ and the degrees of the real
polynomial functions $A_{i}(t)$. A local version of Pugh problem
is to find an upper bound for the multiplicity of a periodic
solution in terms of $N$ and the degrees of the polynomials
$A_{i}(t)$. This problem was considered in \cite{A13}, with $N=4$.
It was conjectured in \cite{A13}, that if $A_{4}\equiv 1$,
$A_{1}\equiv0$, $A_{2}$ and $A_{3}$ are of degree $k$ then the
multiplicity of the origin is at most $k+3$. It was shown in
\cite{A4}, that when $k=2$ then the multiplicity of the origin is
at most $8$ and there is a unique equation with this maximum
multiplicity. It was shown in \cite{A3}, that the multiplicity of
the origin is at most $10$ when the degrees of $A_{2}(t)$ and
$A_{3}(t)$ are $2$ and $3$, respectively. Having determined the
maximum multiplicity, the next step is to construct equations with
this number of periodic solutions. This is done by making a
sequence of perturbations in $A_{2}(t)$ and $A_{3}(t)$, each of
which reduces the multiplicity of the origin by one; a periodic
solution thus bifurcates out of the origin. This bifurcation task
was considered in \cite{A3} and \cite{A4}. The local problem can
be considered with the use of Groebner bases method. This task is
considered in the last section; the case $N=4$ and $A_{1}\equiv 0$
is studied. The problem reduces to study the solvability of system
of polynomial equations in many variables. Since the solvability
in the theory of Groebner bases is over the field of complex
numbers, it becomes necessary to consider equations with complex
coefficients. That is to consider complex centers.
All the known equations with complex centers have linear terms. On
the other hand, computations for equations without linear part
demonstrate that $z=0$ is not a center for polynomial
coefficients. These remarks lead us to conjecture that polynomial
differential equations without linear terms do not have centers at
the origin, at least when the coefficients are polynomial
functions in $t$.
\begin{conjecture} \label{conj1}
Assume that $A_{j}(t)$, for $j=2,3,\dots,N-1$, are polynomial
functions in $t$. The solution $z=0$ is not a center for the
differential equation
\[
\dot{z}=z^{N}+A_{N-1}(t) z^{N-1}+\dots+A_{3}(t) z^{3} + A_{2}(t)
z^{2} .
\]
\end{conjecture}
In the statement of the conjecture, we are not including equations
in which the coefficients are trigonometric polynomials. The
results of Section 5 about such equations demonstrate that the
maximum possible multiplicity of the origin when the coefficients
are complex is higher than the maximum multiplicity when the
coefficients are restricted to be real numbers.
First,we present classes of equations with complex coefficients
that do not have centers at the origin. Our main results in this
direction are the following:
\begin{theorem} \label{thmA}
Assume that the differential equation \eqref{e1.1} has an
invariant line $\theta = \alpha$, and either $A_{N}(t) > \gamma
>0$, or $A_{N}(t) < \gamma < 0$. If $\gamma \cos((N-1) \alpha) >
0$ then the equation can have only a finite number of periodic
solutions on the invariant line. In particular, $z=0$ is not a
center.
\end{theorem}
\begin{proposition} \label{propB}
Consider continuous functions $A(t), B(t)$ and $C(t)$ and let
\begin{gather*}
\lambda_{1}=\int_{0}^{\omega} C(t) dt,\\
\lambda_{2}=\int_{0}^{\omega} B(t) dt, \\
\lambda_{3}=\int_{0}^{\omega}[A(t)+(M-L)B(t) (\int_{0}^{t} C(s) ds) ]dt .
\end{gather*}
The solution $z=0$ of the differential equation
\begin{equation} \label{e1.2}
\dot{z}=A(t) z^{N}+B(t) z^{M} +C(t) z^{L},
\end{equation}
with $1(\frac{2}{(N-1)\gamma \omega \cos((N-1)\alpha)})^{\frac{1}{N-1}}
\]
and such that the inequality
\[
\dot{r}> \frac{\gamma \cos((N-1) \alpha)}{2}r^{N}
\]
is satisfied when $r \geq a$ and $\theta=\alpha$. Consider a solution,
$re^{i\theta}$ with initial condition
$r_{0}e^{i\alpha}$. If $r_{0} \geq a$, then the inequality implies that
\[
r(t) \geq \frac{1}{(r_{0}^{1-N}-0.5(N-1)\gamma
\cos((N-1)\alpha) t)^{\frac{1}{N-1}}}.
\]
Therefore, if $r_{0} \geq a$, then the solution becomes unbounded at a
point $t_{0}$ with
\[
t_{0} \leq \frac{2 r_{0}^{1-N}}{(N-1)\gamma \cos((N-1)\alpha)} \leq \omega .
\]
Hence, the solution is undefined
on the interval $[0,\omega]$. Therefore, solutions start outside the disk
$r \leq a$ are not periodic, and any solution
leaves the disk $r \leq a$ stays outside the disk as time increases.
If there is an infinite sequence of periodic solutions
$z(t,r_{n}e^{i \alpha})$. The numbers $r_{n}$ are inside the
disk $r \leq a$. Hence, let $r_{n}\rightarrow r_{0}$.
If $z(t,r_{0}e^{i\alpha})$ is a periodic solution, then all
solutions starting in a neighborhood of $r_{0}e^{i\alpha}$ are periodic.
But $q$ is an analytic function. It follows
that $q \equiv 0$. Therefore, $z=0$ is a center for the equation \eqref{e1.2}.
We define a real number $R$ by
\[
R=\sup\{r:z(t,re^{i\alpha}) \text{ is } periodic\}.
\]
It is clear that $R<\infty$. If the solution $z(t,Re^{i\alpha})$ is defined,
then it is a periodic solution. Since $q$ is analytic, it follows from the
property of continuous dependence of a solution on its initial value,
that $z(t,re^{i\alpha})$ is periodic for $r$ in a neighborhood of $R$.
This is contrary to the definition of $R$. Therefore, $z(t,Re^{i\alpha})$
is undefined on the interval $[0,\omega]$ and becomes unbounded at
$t_{0}\epsilon [0,\omega]$ and leaves the disk $r \leq a$. But
$z(t,Re^{i\alpha})$ leaves $r \leq a$, whence
$z(t,re^{i\alpha})$ leaves $r \leq a$ for $r < R$ and close to $R$.
These solutions will not return to the disk
$r \leq a$ because $\dot{r}(t)>0$, and hence they are not periodic solutions.
This is a contradiction to the
assumption that $z=0$ is a center.
If $z(t,r_{0}e^{i\alpha})$ is undefined, then it leaves the disk $r \leq a$.
Therefore, $z(t,r_{n}e^{i\alpha})$ leaves $r \leq a$ for large enough $n$.
This is again a contradiction to the periodicity of $z(t,r_{n}e^{i\alpha})$.
\end{proof}
\begin{corollary} \label{coroA}
Assume that $A_{N}(t) > \gamma >0$.
\begin{enumerate}
\item Let $A_{j}(t)=a_{j}(t)+ib_{j}(t)$, where
$a_{j}$ and $b_{j}$ are real continuous functions. If there exists
an integer $k$ such that
\[
a_{j}\sin(j \frac{2k\pi}{N-1})=b_{j} \cos(j \frac{2k\pi}{N-1})
\]
for $j=1,2,\dots,N-1$, then $z=0$ is not a center for \eqref{e1.1}.
\item If the coefficients $A_{j}$, $j=1,2,\dots,N-1$, are real functions
then $z=0$ is not a center for \eqref{e1.1}.
\item If $N-1$ is a multiple of $4$, $A_{j}$ are pure imaginary
functions for all odd $j$, and $A_{j}$ are real for all even $j$, then
$z=0$ is not a center for equation \eqref{e1.1}.
\end{enumerate}
\end{corollary}
\begin{proof}
To prove the first part, let $\alpha =\frac{2k \pi}{N-1}$.
The condition in the statement implies that
\begin{gather*}
a_{j} \sin(j\alpha)=b_{j} \cos(j\alpha),\\
\sin((N-j-1)\alpha)=-\sin(\alpha),\quad
\sin((N-1)\alpha)=0.
\end{gather*}
Hence, $\dot{\theta}(\alpha)=0$.
The line $\theta = \alpha$ is an invariant line. Any solution has a point
on this line stays on the line as long as it
is defined.
On the other hand,
\[
\cos((N-1)\alpha)=1.
\]
Hence, $A_{N}(t) \cos((N-1)\alpha) = A_{N}(t) > \gamma$.
The conditions of Theorem \ref{thmA} are satisfied and the result
follows.
The second part follows directly from the first part with $k=0$.
It is known that $z=0$ is not a center when the
coefficients are real functions (see \cite{A1}). To prove the third part,
we take $k=\frac{N-1}{4}$. This
implies that $\alpha =\frac{2k \pi}{N-1}=\frac{\pi}{2}$.
Thus $\sin(m\alpha)=0$ if $m$ is even, and
$\cos(m\alpha)=0$ if $m$ is odd.
\end{proof}
\begin{remark} \label{rmk2} \rm
Equations \eqref{e1.1}, with real coefficients, have been studied
in \cite{A11} and \cite{A12} using the methods of complex analysis
and topological dynamics. However, some results of \cite{A11} and
\cite{A12} do not hold when the coefficients are allowed to be
complex. For example, Abel differential equation, $N=3$, has $3$
periodic solutions when its coefficients are real. On the other
hand, it may have an infinite number of periodic solutions when
the coefficients are complex (see, \cite{A7}). Equations with real
coefficients satisfy other properties; in particular, the real
axis is an invariant set and the complex conjugate of a periodic
solution is also a periodic solution. The phase portrait presented
in \cite{A11}, with some modifications, could be used for
equations with complex coefficients. The next task is to obtain
some global results on the number of periodic solutions as in
\cite{A1} and \cite{A2}; we defer this exploitation to another
paper.
\end{remark}
\section{Proof of Proposition \ref{propB}}
\begin{proof}
For $0 \leq t \leq \omega$ and $c$ in a neighborhood of $0$, we write
\[
z(t,c)=\sum_{n=1}^{\infty} d_{n}(t)c^n,
\]
where $d_{1}(0)=1$ and $d_{n}(0)=0$ if $n>1$. Thus
\[
q(c)=(d_{1}(\omega)-1)c+\sum_{n=2}^{\infty}d_{n}(\omega)c^{n}.
\]
The multiplicity is $K$ if and only if
\[
d_{1}(\omega)=1,d_{2}(\omega)=d_{3}(\omega)=\dots=d_{K-1}(\omega)
=0,d_{K}(\omega)\neq 0.
\]
The origin is a center when $d_{1}(\omega)=1$ and $d_{n}(\omega)=0$ for
all $n>1$. The functions $d_{n}(t)$ are
determined by substituting the sum into the equation \eqref{e1.2} and
comparing coefficients of powers of $c$. The following
recursive sequence of differential equations is obtained with the initial
equations.
\[
\dot{d}_{n}=A S_{N} +B S_{M} + C S_{L},
\]
where
\[
S_{K}=\sum_{j_{1}+j_{2}+\dots+j_{K}=n}^{} d_{j_{1}}d_{j_{2}}\dots d_{j_{K}},
\]
with $K \epsilon \{N, M, L\}$. To solve these equations, we integrate
repeatedly. It is clear that $\dot{d}_{1}=0$ and
hence $d_{1}\equiv 1$. The next non-zero equation is
\[
\dot{d}_{L}=C (d_{1})^{L}.
\]
It gives
\[
d_{L}(t)=\int_{0}^{t} C(s)ds .
\]
The formula for $d_{M}$ has two possibilities. If $M+1\neq 2L$, then
\[
\dot{d}_{M}=B (d_{1})^{M}.
\]
This implies
\[
d_{M}(t)=\int_{0}^{t} B(s) ds.
\]
In the case $M+1=2L$, the equation becomes
\[
\dot{d}_{M}=B(d_{1})^{M}+L C d_{L} (d_{1})^{L-1}).
\]
We integrate this equation to obtain
\[
d_{M}(t)=\int_{0}^{t} B(s) ds + \frac{L}{2} (\int_{0}^{t} C(s)ds)^2.
\]
Consequently, the formula for $d_{N}$ has two possibilities. The equation is
\[
\dot{d}_{N}=A (d_{1})^{N}+B M d_{L}(d_{1})^{M-1}+C L
d_{M}(d_{1})^{L-1}.
\]
We integrate this equation to obtain $d_{N}(t)$. If $M+1 \neq 2L$, then
\[
d_{N}(t)=\int_{0}^{t}A(s) ds+(M-L) \int_{0}^{t} (B(s) \int_{0}^{s} C(u) du)ds
+ L \int_{0}^{t}B(s) ds \int_{0}^{t} C(s) ds.
\]
In the case $M+1=2L$, the formula becomes
\begin{align*}
d_{N}(t)&=\int_{0}^{t} A(s) ds+(M-L) \int_{0}^{t} (B(s) \int_{0}^{s} C(u) du)ds
\\
&\quad +L \int_{0}^{t} B(s) ds \int_{0}^{t} C(s) ds + \frac{L^{2}}{6}
(\int_{0}^{t} C(s) ds)^{3} .
\end{align*}
The multiplicity is $L$ if $\int_{0}^{\omega} C(t) dt \neq 0$.
The multiplicity is $M$ if $\int_{0}^{\omega} C(t) dt=0$ but
$\int_{0}^{\omega} B(t) dt \neq 0$. Finally, if the multiplicity is
greater than $M$, then
\[
d_{N}(\omega)=\int_{0}^{\omega}A(t) dt + (M-L) \int_{0}^{\omega} (B(t)
\int_{0}^{t} C(s) ds)dt.
\]
The assumption in the statement of Proposition \ref{propB}, implies
that $d_{N}(\omega) \neq 0$. Therefore, the multiplicity is at most $N$.
\end{proof}
\begin{corollary} \label{coroB} \quad
\begin{enumerate}
\item
For any continuous functions $A(t)$ and $B(t)$ with
$\int_{0}^{\omega} A(t) dt \neq 0$, the origin is not a
center for the equation
\[
\dot{z}=A(t) z^{N}+B(t) z^{M}
\]
when $1 \gamma \omega \neq 0$. Hence, the multiplicity is
at most $N$.
Now we prove the second part. The first necessary condition for a center
is $e^{\int_{0}^{\omega} D(t) dt} = 1$.
We make the transformation $w=e^{\int_{0}^{t} D(s) ds} z$, and obtain
\[
\dot{w}= A_{1}(t) w^{N} + B_{1} (t) w^{M} + C_{1} (t) w^{L},
\]
where $A_{1}$, $B_{1}$, and $C_{1}$ are as defined in the statement of
the Corollary. Initial conditions and
multiplicities of periodic solutions are unchanged under this transformation.
Now the result follows from
Proposition \ref{propB}.
\end{proof}
\begin{remark} \label{rmk3} \rm
If the differential equation has a linear term, then the equations for $d_{k}$
are more complicated. Instead, we will have linear differential equations
in which the right-hand side depends also on $d_{k}$. We consider such cases
in the next section.
\end{remark}
\section{Proof of Propositions \ref{propC} and \ref{propD}}
\begin{proof}[Proof of Proposition \ref{propC}]
We follow the procedure of the previous section. In this case, we obtain
a sequence of linear differential equations.
\begin{gather*}
\dot{d}_{1}=A d_{1},\\
\dot{d}_{k}=A d_{k},\,\, 2 \leq k \leq (N-1),\\
\dot{d}_{N}=(d_{1})^{N}+A d_{N}
\end{gather*}
together with the initial conditions
\[
d_{1}(0)=1;\,d_{k}(0)=0,\, 2 \leq k \leq N.
\]
Solving these initial value problems, gives
\begin{gather*}
d_{1}(t)=e^{\int_{0}^{t} A(s) ds},\\
d_{k}(t)\equiv 0,\,\, 2 \leq k \leq (N-1), \\
d_{N}(t) = \int_{0}^{t} A(s)ds \int_{0}^{t} e^{(N-1)\int_{0}^{s} A(u)
du} ds.
\end{gather*}
The origin is of multiplicity $N$ if and only if $d_{1}(\omega)=1$ and
$d_{N}(\omega) \neq 0$. These two
conditions give
\begin{gather*}
e^{\int_{0}^{\omega} A(t) dt }=1, \\
\int_{0}^{\omega} e^{(N-1)\int_{0}^{t} A(s) ds} dt \neq 0.
\end{gather*}
If these two integrals vanish then the solution
$z=0$ is a center. This follows from the general solution of this equation.
The general solution is given by
\[
w(t)=e^{(1-N) \int_{0}^{t} A(s) ds} [(1-N) \int_{0}^{t} e^{(N-1)
\int_{0}^{s} A(u) du} ds + C],
\]
where $w=z^{1-N}$ and $C$ is a constant. Since $w(0)=w(\omega)$, all the
solutions are periodic. Hence, $z=0$ is a center.
\end{proof}
It is clear that the conditions for a center are not satisfied by any real
function $A(t)$. We give two equations that
have centers at the origin. In the first equation, we have
\[
A(t)=C+ \frac{2}{N-1} \tan (t+c_{0}) .
\]
Since the zeros of the complex function $\cos$ are real, the function $A(t)$
is a periodic function of period $2 \pi$,
when $c_{0}$ is a complex non-real number. The origin is of multiplicity
one when
\[
e^{\int_{0}^{2 \pi} [C+ \frac{2}{N-1}\tan(t+c_{0}) ]dt} =1.
\]
This gives
\[
e^{ 2 \pi C} e^{\frac{1}{N-1}
[\ln(\sec^{2}(2\pi+c_{0}))-\ln(\sec^{2}(c_{0}))]} = 1.
\]
Since $\sec(t+c_{0}) $ is of period $2 \pi$, the condition
becomes $e^{2 \pi C} =1$. Hence, $C=pi$ for an integer $p$. The second
condition is
\[
\int_{0}^{2\pi} e^{(N-1)pit} \sec^{2}(t+c_{0}) dt=0.
\]
If $(N-1)p$ is odd, then a change of variables $t \mapsto t-\pi$ gives
\[
\int_{\pi}^{2\pi} e^{(N-1)pit} \sec^{2}(t+c_{0}) dt
=-\int_{0}^{\pi} e^{(N-1)pit} \sec^{2}(t+c_{0}) dt=0.
\]
Therefore,
\[
\int_{0}^{2\pi} e^{(N-1)pit} \sec^{2}(t+c_{0}) dt=0.
\]
The conditions for a center given in Proposition \ref{propC} are
satisfied. Hence, all the solutions $z(t,c)$, with $c$ is in a neighborhood
of $0$, are $2\pi$-periodic. This proves
the first part of Corollary \ref{coroC}.
The second part in Corollary \ref{coroC}, with $N=3$, is similar to the result
of \cite{A7}. In fact, it was shown in
\cite{A7} that $z=0$ is a center for the equation
\[
\dot{z} = z^{3} +(C_{0} + C_{1} e^{-it} + C_{2} e^{it})z
\]
if and only if $C_{0}=ki$ for an integer $k$, $C_{1} C_{2} \neq 0$, and
$4i \sqrt{C_{1}C_{2}}$ is a zero for $J_{2|k|}$.
Now, we prove the second part of Corollary \ref{coroC}. The first condition for a
center in Proposition \ref{propC} is
\[
e^{\int_{0}^{2\pi} (C - xi \sin t)dt}=e^{2C\pi}.
\]
But, $e^{2C \pi}=1$ if and only if $C=pi$, for an integer $p$.
The second condition for a center becomes
\[
\int_{0}^{2\pi}e^{pit-xi \sin t} dt = \int_{0}^{2\pi} \cos(pt - x
\sin t) dt + i \int_{0}^{2\pi} \sin(pt - x \sin t) dt.
\]
The imaginary part of this quantity is zero; it is an
integral of an odd $2\pi-$periodic function over the interval $[0,2\pi]$.
The real part is $2\int_{0}^{\pi}\cos(pt - x \sin t) dt$.
This integral is the integral form of the Bessel's function. Therefore,
the real part is zero if $x$ is a zero of the Bessel's function $J_{p}$.
We recall that the integral form and power series
expansion of $J_{p}$ are defined by
\[
J_{p}(x)= \int_{0}^{\pi} \cos(pt - x \sin t) dt= \sum_{j=0}^{\infty}
(-1)^{j}\frac{(\frac{x}{2})^{p+2j}}{j!(p+j)!}.
\]
The power series representation of Bessel's function is used in \cite{A7}.
Consequently, their proof is much
longer than our proof.
%\end{proof}
\begin{proof}[Proof of Proposition \ref{propD}]
With the change of variables $z \mapsto e^{\int_{0}^{t}q(s) ds} z$,
the equation becomes
\[
\dot{z}=\sum_{k=2}^{N} p'(t)\,f_{k}(p(t))\,z^{k}.
\]
From the expansion of Section 3, we can show inductively that the
coefficients $d_{k}$ are functions of $p(t)$. This
implies that $z(t,c)$ is a function of $p(t)$. Therefore, in a neighborhood
of the origin, all solutions are
$\omega$-periodic and $z=0$ is a center.
\end{proof}
\begin{remark} \label{rmk7} \rm
If the coefficient of $z^{N}$ in the statement of Proposition \ref{propD}
satisfies the condition
\[
p'(t)\,f_{N}(p(t))=e^{(N-1)\,\int_{0}^{t}q(s)ds}
\]
then the coefficient of $z^{N}$ equals $1$. This condition is satisfied
when $p(t)=\frac{e^{2 \pi i t}}{2 \pi i}$,
$f_{N} \equiv 1$, and $q=2 \pi i$. The period in this case is $N-1$.
\end{remark}
\section{The Pugh problem}
The use of computer algebra has led to significant progress in the
investigation of the properties of polynomial differential
systems. In this section, we describe an application of computer
algebra to find the maximum possible multiplicity of periodic
solutions of polynomial differential equations. In particular, we
present solutions to the local Pugh problem \cite{A14} and
Shahshahani conjecture \cite{A13}. We mention that Pugh problem
was listed as a part of problem 13 in Steve Smale list of 18 open
problems for the next century. This problem was considered as a
version of Hilbert sixteenth problem. Hilbert sixteenth problem is
to estimate the number of limit cycles of polynomial
two-dimensional systems. Research related to Hilbert sixteenth
problem has derived enormous benefit from the availability of
computer algebra.
Consider the differential equation
\begin{equation} \label{e5.1}
\dot{z}=z^{4}+A(t)z^{3}+B(t)z^{2}
\end{equation}
We show that this local problem can be solved using the method of
Groebner bases. We give an automatic means of finding the maximum
possible multiplicity of a periodic solution. The algorithm
involves computing Groebner bases. Computer algebra systems, such
as Maple, can be used to implement this algorithm. If the equation
does not have a complex center, then the local Pugh problem is
solvable by our procedure. The algorithm for computing the maximum
possible multiplicity is then described in this section. We apply
the algorithm for equations in which the coefficients $A(t)$ and
$B(t)$ are polynomial functions in $t$, and in $\cos t$ and $\sin
t$. The case $N=4$ is considered. However, the method works for
any $N$. In the case that the coefficients are polynomial
functions in $t$, we show that the maximum possible multiplicity
is the same whether the coefficients are complex or are restricted
to be real. When the coefficients are trigonometric polynomials,
cases are described where the maximums are not equal. Moreover, we
show that the origin is not a complex center in each of the
equations considered; these results provide evidences to support
our conjecture.
We follow the same procedure of Section 3. For equation \eqref{e5.1},
$d_{1}(t)\equiv1$ and the equations satisfied by
the $d_{n}(t)$ (for $n>1$) are
\begin{equation} \label{e5.2}
\dot{d}_{n}=\sum_{i+j+k+l=n}^{}d_{i}d_{j}d_{k}d_{l}+A\,
\sum_{i+j+k=n}^{}d_{i}d_{j}d_{k}+ B\,
\sum_{i+j=n}^{}d_{i}d_{j}.
\end{equation}
These equations were integrated by parts repeatedly and the formulae
for $d_{n}$, with $n\leq 8$, are given in
\cite{A4}. The calculations become extremely complicated as $n$ increases.
It is impossible to accomplish these
computations by hand except in the simplest cases.
The next step, in computing the multiplicity, is to consider the quantities
\begin{equation*}
\eta_{n}=d_{n}(\omega)
\end{equation*}
Note that $\eta_{n}$ is a polynomial function in the coefficients of the
polynomials $A$ and $B$. The multiplicity of the origin is $k$ if
\[
\eta_{2}=\eta_{3}=\dots =\eta_{k-1}=0,\eta_{k} \neq 0.
\]
We write $\mu_{\rm max}(\mathcal{C})$ for the maximum possible
multiplicity of $z=0$ for equations in a class $\mathcal{C}$. For
the class of equations in which the coefficients are polynomial
functions of degree $m$, Pugh problem is to find $\mu_{\rm max}$ in
terms of $m$.
Now, we give the formulae of $\eta_{n}$ with $n\leq 4$, for
equation \eqref{e5.1}. We use a tilde over a function to
denote its indefinite integral:
\[
\widetilde{f}(t)=\int_{0}^{t}f(s)ds
\]
\begin{proposition} \label{prop5}
For equation \eqref{e5.1}, the quantities $\eta_{2},\eta_{3}$, and
$\eta_{4}$ are as follows.
\[ \eta_{2}=\int_{0}^{\omega}B dt, \quad
\eta_{3}=\int_{0}^{\omega}A dt, \quad
\eta_{4}=\omega+\int_{0}^{\omega}(\widetilde{B}A)dt
\]
\end{proposition}
\begin{proof}
The formulae were obtained by solving equations \eqref{e5.2} recursively.
The computations leading to the formulae proceed by sequences of
judiciously chosen integrations by parts; for any functions $f$ and $g$,
we make use of the identity
\[
\widetilde{f\widetilde{g}}=\widetilde{f}\widetilde{g}
-\widetilde{\widetilde{f}g}.
\]
These are elementary though together they form a complicated web.
To obtain $\eta_{n}$, we reduce $d_{n}(\omega)$ modulo the ideal
generated by $d_{2}(\omega),\dots,d_{n-1}(\omega)$.
\end{proof}
\subsection*{The Algorithm}
We shall write $\mu_{\rm max}({\rm real})$ when the coefficients of
the polynomials $A$ and $B$ are real numbers, and $\mu_{\rm
max}(complex)$ when the coefficients are complex numbers. It
follows from the result in \cite{A1}, that $\mu_{\rm max}({\rm
real}) < \infty$. We call the set of equations that have this
maximum multiplicity, the maximum variety, $V_{\rm max}$.
Similarly, we define $V_{\rm max}({\rm real})$ and $V_{\rm
max}(complex)$.
To find $\mu_{\rm max}$, first we integrate recursively to compute the
functions $d_{n}(t)$. Then we consider the expressions
$d_{n}(\omega)$, which are polynomial functions in the
coefficients of $A$ and $B$. To obtain $\eta_{n}$ we reduce
$d_{n}(\omega)$ modulo the ideal generated by
$d_{2}(\omega),\dots,d_{n-1}$. We stop until the system
$\eta_{2}=\eta_{3}=\dots=\eta_{k}=0$ has no real solutions. In
this case $\mu_{\rm max}=k$. From the theory of Groebner bases, the
Groebner basis of the ideal $\langle
\eta_{2},\eta_{3},\dots,\eta_{k} \rangle$ is $\langle 1 \rangle$
if and only if the system $\eta_{2}=\eta_{3}=\dots=\eta_{k}=0$ has
no complex solutions. So, we have to verify that this maximum
multiplicity can be attained by certain real values of
coefficients. The procedure gives an upper bound for $\mu_{\rm max}$.
However, for the equations in which the coefficients are
polynomial functions of $t$ that we will consider,
$\mu_{\rm max}({\rm real}) = \mu_{\rm max}(complex) < \infty$.
The algorithm for computing $\mu_{\rm max}$ can be summarized as follows:
\begin{itemize}
\item Input: functions
$A(t)$ and $B(t)$ which are polynomials in $t$, or in $\cos t$ and $\sin t$.
\item Integrate to compute $d_{n}(t)$.
\item Compute $d_{n}(\omega)$.
\item Find $\eta_{n}$ by reducing $ d_{n}(\omega)$ modulo
$\langle \eta_{2},\eta_{3},\dots,\eta_{n-1} \rangle$.
\item Stop when the Groebner basis
$\langle \eta_{2},\eta_{3},\dots,\eta_{k}\rangle$ is $\langle 1 \rangle$.
\item Output: $\mu_{\rm max}=k$.
\end{itemize}
For the details related to Groebner bases, we refer to \cite{A8}. We use
the software package Maple 8 to compute Groebner bases.
Some of the Groebner bases are not given, but they are available upon request.
Now, we consider the first class of coefficients. Here, we assume that
$\omega=1$. Let
\begin{gather*}
B(t) = C_{1}+at+bt^{2}, \\
A(t)=C_{2}+ct+dt^{2}+et^{3}.
\end{gather*}
If $\eta_{2}=\eta_{3}=0$, then
\[
12C_{1}+6a+4b+3c=0, \quad 12C_{2}+6d+4e+3f=0.
\]
This class of equations was used in \cite{A3} to construct
equations with $10$ real periodic solutions. We substitute the
values of $C_{1}$ and $C_{2}$ from these equations and then
compute the Groebner basis of the ideal $\langle
\eta_{4},\eta_{5},\dots,\eta_{10} \rangle $. This basis is
$\langle 1 \rangle $. Therefore, the maximum possible multiplicity
is $10$. To find the equations that have this maximum
multiplicity, we compute the Groebner basis $\langle
\eta_{4},\eta_{5},\dots,\eta_{9}\rangle$. This set of equations
has $15$ solutions, counting multiplicity, and at least one of the
solutions is real. We summarize the result for this class in the
following lemma.
\begin{lemma} \label{lem1}
If $A(t)$ and $B(t)$ are polynomial functions in $t$ of degrees $3$ and $2$,
respectively, then
$\mu_{\rm max}({\rm real})=\mu_{\rm max}(complex)=10$. Moreover,
$V_{\rm max}$ is a zero-dimensional ideal.
\end{lemma}
The Groebner basis is computed with the graded reverse
lexicographic term order and with respect to the list
$[b,c,d,a,e]$; this term order usually gives more compact Groebner
basis. Then, the basis is changed to the lexicographic term order,
which is the most suitable to eliminate variables from a set of
equations. The last equation in this basis is a polynomial in $e$
of degree $15$; it has at least one real solution. The other
variables are given explicitly as functions of $e$.
The next class has the coefficients
\begin{gather*}
B(t)= C_{1}+at,\\
A(t)=C_{2}+bt+ct^{2}+dt^{3}+et^{4}+ft^{5}.
\end{gather*}
Let $2C_{1}+a=0$ and $60C_{2}+30b+20c+15d+12e+10f=0$ for
$\eta_{2}=\eta_{3}=0$. We compute the Groebner basis
as in the first case. This gives
$\langle \eta_{4},\eta_{5},\dots,\eta_{10}\rangle=\langle 1 \rangle$.
Moreover,
$\langle \eta_{4},\eta_{5},\dots,\eta_{9}\rangle$ has a polynomial equation
in $f$ of degree $6$, which has
two real solutions and four complex non-real solutions.
The other variables are given as functions of $f$.
The result for this class are given in the following lemma.
\begin{lemma} \label{lem2}
If $A(t)$ and $B(t)$ are polynomial functions in $t$ of degrees $5$ and $1$,
respectively, then
$\mu_{\rm max}({\rm real})=\mu_{\rm max}(complex)=10$. Moreover,
$V_{\rm max}$ is a zero-dimensional ideal.
\end{lemma}
In the last class of polynomials in $t$, we let
\begin{gather*}
A(t) = C_{1}+bt+ ct^{2} + d t^{3} , \\
B(t)=C_{2}+at.
\end{gather*}
If $\eta_{2}=\eta_{3}=0$, then
\[
12C_{1}+6b+cb+3d=0, 2C_{2}+a=0.
\]
We substitute the values of $C_{1}$ and $C_{2}$ from these
equations and then compute the Groebner basis of the ideal
$\langle \eta_{4},\eta_{5},\dots,\eta_{8}\rangle$. This basis is
$\langle 1 \rangle$. Therefore, the maximum possible multiplicity
is $8$. To find the equations that have this maximum multiplicity,
we compute the Groebner basis $\langle
\eta_{4},\eta_{5},\dots,\eta_{7}\rangle$. The last equation, which
is a polynomial in $a$, has two real solutions and four complex
non-real solutions. The other variables are given in terms of $a$.
\begin{lemma} \label{lem3}
If $A(t)$ and $B(t)$ are polynomial functions in $t$ of degrees $3$
and $1$, respectively, then
$\mu_{\rm max}({\rm real})=\mu_{\rm max}(complex)=8$.
Moreover, $V_{\rm max}$ is a zero-dimensional ideal.
\begin{gather*}
\langle 8695641600 \,b-6086949120\, d-773773\, a^{5}+3151791360\, a^{2},\\
8695641600 c+13043462400
d+773773\,a^{5}-3151791360\, a^{2},\\
-50295245\, a^{4}+37439568 \,d^{2}+118513886400\, a,\\ -120772800 \,d+70343 \,d
a^{3}, 773773\, a^{6}-3151791360\, a^{3}+3130430976000 \rangle
\end{gather*}
\end{lemma}
Now, we consider classes of coefficients which are polynomial functions
in $\cos t$ and $\sin t$. Here, we take $\omega=2 \pi$.
\begin{lemma} \label{lem4}
If $A(t)$ and $B(t)$ are homogeneous polynomial functions in $\cos t$ and
$\sin t$ of degree $1$ or $2$, then $\mu_{\rm max}({\rm real})=6$ and
$\mu_{\rm max}(complex)=7$. The sets $V_{\rm max}({\rm real})$
and $V_{\rm max}(complex)$ are not
zero-dimensional ideals.
\end{lemma}
In the case $A(t)=c \cos t + d \sin t$, and $B(t)=a \cos t + b \sin t$.
It is clear that $\eta_{2}=\eta_{3}=0$. From
Maple, we have:
\[
\langle \eta_{4},\eta_{5},\eta_{6}\rangle= \langle a^{2}+b^{2},bc-ad-2
\rangle.
\]
This set has complex solutions but does not have a real solution. Moreover
\[
\langle \eta_{4},\eta_{5},\eta_{6},\eta_{7}\rangle = \langle 1 \rangle .
\]
For the other case $A(t)=c \cos^{2} t + d \cos t \sin t + c_{1} \sin^{2} t$
and $B(t)=a \cos^{2} t + b \cos t \sin t + a_{1} \sin^{2} t$.
We have $\eta_{2}=a+a_{1}$ and $\eta_{3}=c+c_{1}$. If $a_{1}=-a$ and
$c_{1}=-c$, then Maple gives
\[
\langle \eta_{4},\eta_{5},\eta_{6}\rangle= \langle 4a^{2}+b^{2},bc-ad-8
\rangle, \]
and
\[
\langle \eta_{4},\eta_{5},\eta_{6},\eta_{7}\rangle = \langle 1 \rangle.
\]
A similar argument proves the Lemma.
If one of the coefficients $A(t)$ and $B(t)$ contains only terms with of
even degrees and the other coefficient
contains only terms of odd degrees, then it follows from \ref{prop5}
Proposition that $\mu_{\rm max}({\rm
real})=\mu_{\rm max}(complex)=4$. When the coefficients are homogeneous
polynomials of degree $3$, the Groebner
basis becomes very large.
\begin{lemma} \label{lem5}
Let $A(t)$ be a homogeneous polynomial of degree $3$, and $B(t)$
is a homogeneous polynomial of degree $1$. The solution $z=0$ is
not a center. Moreover, $\mu_{\rm max}({\rm real})=8$, and
$\mu_{\rm max}(complex)=10$.
\end{lemma}
We take $B(t)= a \cos t + b \sin t$. Using the identities
$\sin^{3} t = \sin t(1- \cos^{2} t)$ and
$\cos^{3}(t)=\cos t(1-\sin^{2} t)$, we can write a homogeneous polynomial
of degree $3$ in the following form:
\[
A(t)=c \cos t + d \sin t + e \cos t \sin^{2} t + f \sin t \cos^{2} t.
\]
The Groebner basis of $\langle \eta_{4},\eta_{5},\eta_{6},\eta_{7} \rangle$
has a real solution; we can take
$a=d=1,c=f=0, e^{2}=432$, and $36\,b=e$. On the other hand,
$\langle \eta_{4},\dots,\eta_{8}\rangle$ does not have a real
solution but $\langle \eta_{4},\dots,\eta_{9} \rangle \neq \langle
1 \rangle$, and $\langle \eta_{4},\dots, \eta_{10} \rangle
=\langle 1 \rangle $. On the other hand, $\eta_{8}$ has the form
\[
\eta_{8}=\pi\,((5/8)\,a^{4}+(5/8)\,b^{4}+(5/4)\,a^{2}\,b^{2} +(1/96)\,e^{2}
+(1/96)\,f^{2}) .
\]
It is clear that $\eta_{8} \neq 0$, when the coefficients are restricted
to be real numbers.
\subsection*{Acknowledgments}
I would like to thank Dr. Colin Christopher for a very fruitful
email discussion.
I am very grateful to the Simmons Hall residential scholar program
at MIT for their hospitality.
\begin{thebibliography}{00}
\bibitem{A1} M. A. M. Alwash;
\emph{Periodic solutions of Abel differential equation},
J. Math. Anal. Appl., 329(2007)1161-1169.
\bibitem{A2} M. A. M. Alwash;
\emph{Periodic solutions of polynomial non-autonomous differential equations},
Electron. J. Differential Equations, 2005(2005), no. 84, 1-8.
\bibitem{A3} M. A. M. Alwash;
\emph{Periodic solutions of a quartic differential equations and Groebner bases},
J. Comp. Appl. Math., 75(1996), 67-76.
\bibitem{A4} M. A. M. Alwash and N.G. Lloyd;
\emph{Periodic solutions of a quartic nonautonomous equation},
Nonlinear Analysis, 11(1987), 809-820.
\bibitem{A5} M. A. M. Alwash and N.G. Lloyd;
\emph{Non-autonomous equations related to polynomial two-dimensional systems},
Proc. Royal Soc. Edinburgh, 105(1987), 129-152.
\bibitem{A6} M. Briskin and Y. Yomdin;
\emph{Tangential version of Hilbert 16th problem for the Abel
equation}, Mosc. Math. J., 5(2005), 23-53.
\bibitem{A7} A. Cima, A. Gasull, and F. Manosas;
\emph{ Periodic orbits in complex Abel equations},
J. Diff. Eqns., 232(2007), 314-328.
\bibitem{A8} D. A. Cox and B. Sturmfels;
\emph{Applications of Computational Algebraic Geometry}, American
Mathematical Society (1998).
\bibitem{A9} Y. Ilyashenko;
\emph{Hilbert-type numbers for genralized Abel
equations, growth and zeros of holomorphic functions},
Nonlinearity, 13(2000), 1337-1342.
\bibitem{A10} A. Lins Neto;
\emph{On the number of solutions of the equation
$\frac{dx}{dt}=\sum_{j=0}^{n}a_{j}(t)x^{j}, 0 \leq t \leq 1$, for
which $x(0)=x(1)$}, Invent. Math., 59(1980), 67-76.
\bibitem{A11} N. G. Lloyd;
\emph{The number of periodic solutions of the equation
$\dot{z}=z^{N}+P_{1}(t)z^{N-1}+\dots+P_{N}(t)$}, Proc. London
Math. Soc., 27(1973), 667-700.
\bibitem{A12} V. A. Pliss;
\emph{Nonlocal Problems in the Theory of Oscillations},
Academic Press, New York, (1966).
\bibitem{A13} S. Shahshahani;
\emph{Periodic solutions of polynomialfirst order differential equations},
Nonlinear Analysis, 5(1981), 157-165.
\bibitem{A14} S. Smale;
\emph{Mathematical problems for the next century},
Mathematics: Frontiers and Perspectives, AMS(2000), 271-294.
\bibitem{A15} Y. Yomdin;
\emph{The center problem for the Abel equations, compositions of functions,
and moment conditions}, Mosc. Math. J., 3(2003), 1167-1195.
\end{thebibliography}
\end{document}