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\markboth{\hfil Transformation to Li\'{e}nard form \hfil EJDE--2000/76}
{EJDE--2000/76\hfil W. A. Albarakati,  N. G. Lloyd, \& J. M. Pearson \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol.~{\bf 2000}(2000), No.~76, pp.~1--11. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)}
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  Transformation to Li\'{e}nard form  
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\thanks{ {\em Mathematics Subject Classifications:} 34CO7.
\hfil\break\indent
{\em Key words:} Ordinary differential equations, polynomial systems, 
Li\'{e}nard systems.
\hfil\break\indent
\copyright 2000 Southwest Texas State University. \hfil\break\indent
Submitted October 31, 2000. Published December 22, 2000.} }
\date{}
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\author{ W. A. Albarakati,  N. G. Lloyd, \& J. M. Pearson }
\maketitle

\begin{abstract} 
 We show that certain two-dimensional differential systems can be
 transformed to a system of Li\'{e}nard type. This enables known criteria for
 the existence of a centre for Li\'{e}nard systems to be exploited, so
 extending the range of techniques which are available for proving that
 conditions which are known to be necessary for a centre are also
 sufficient.
\end{abstract}

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}


\section{Introduction}
In the investigation of two-dimensional dynamical systems
\begin{equation}
\dot{x}=P(x,y),\quad \dot{y}=Q(x,y), \tag{1}
\end{equation}
the twin questions of the conditions under which a critical point is a centre
and the number of limit cycles are often encountered. Recall that a
\emph{limit cycle} is an isolated closed orbit, while a critical point is a
\emph{centre} if all orbits in its neighbourhood are closed.

Much of the published work refers to specific classes of systems. These may be
polynomial systems of a fixed degree (cubic systems, for example - that is,
systems in which $P$ and $Q$ are cubic polynomials) or they may be of a
specific form, for example
\begin{equation}
\dot{x}=y-F(x),\quad \dot{y}=-g(x). \tag{2}
\end{equation}
Systems of this form are said to be of Li\'{e}nard type. They are well
understood, and there is a very extensive literature on them, not least
because they often arise in applications.

Systems of the form (1) in which $P$ and $Q$ are polynomials are of particular
interest, and several authors have transformed such systems to Li\'{e}nard
form in order to exploit the many known results on the existence of limit
cycles and their number. For example, Coppel [5,6] and Cherkas and Zhilevich
[2] transformed various quadratic systems in this way to prove that there is
at most one periodic orbit encircling the origin. Kooij [9] obtained the same
conclusion for some cubic systems in a similar fashion, though in this case a
sequence of transformations was required. Other instances are found in [7] and
[18], where the results are relevant to models of ecological competition.
These are just a few of the examples in the literature and are indicative of
the variety of instances where the technique has proved useful. Our interest
here is not so much in the number of limit cycles, but rather in the use of
transformation to a Li\'{e}nard form to prove that a critical point is a centre.

Experience has shown that in the search for centre conditions, necessity and
sufficiency should be treated separately; the literature is littered with
incomplete sets of conditions which are claimed to be both necessary and
sufficient. Proof of necessity often involves extensive use of computer
algebra, and it is the availability of such systems which has led to many of
the recent advances. The eventual requirement is to eliminate variables from
the focal values, which are polynomials in the coefficients arising in $P$ and
$Q$. In many cases these are polynomials of very high degree with coefficients
which are very large integers. As pointed out in [20] the computations are on
occasion beyond the scope of the available elimination algorithms. A
particular instance is the system
\begin{equation}
\begin{aligned}
\dot{x}&=y,\\
\dot{y}&=-x+a_1x^2+a_2xy+a_3y^2+a_4x^3+a_5x^2y+a_6xy^2+a_7y^3.
\end{aligned}\tag{3}
\end{equation}
This system was first investigated by Kukles and has been the subject of a
number of papers, including [3,8,10,11 and 21].

A variety of methods have been developed for proving the sufficiency of centre
conditions, and in practice several are used in combination. The simplest
criterion for a centre is when the system is Hamiltonian:
\begin{equation}
P_x+Q_y=0 \tag{4}
\end{equation}
in a neighbourhood of the origin. The other classical condition is that of
symmetry. The origin is a centre if the system is invariant under the
transformation $(x,y,t)\longmapsto(-x,y,-t)$ or $(x,y,t)\longmapsto(x,-y,-t);$
the system is symmetric in the $y$-axis in the first case and in the $x$-axis
in the second. Clearly the same conclusion holds if the system is symmetric in
any line through the origin. This is most easily tested when the system is
written in complex form:
\[
i\dot{z}=-z+\sum A_{k,j}z_{k}\bar{z}^{j}.
\]
The origin is a centre if there exists $\theta$ such that $A_{kj}
e^{i(k-j-1)\theta}$ is real for all $k,j$ (see [13]).

Condition (4) can be generalised. The origin is a centre if there is an
integrating factor (sometimes called a Dulac function) $B$ such that
$(BP)_x+(BQ)_y=0$ in a neighbourhood of the origin. Geometrically this
happens when there is a transformation of time, depending on the space
variables $x$ and $y,$ to a Hamiltonian system. The task is to find an
integrating factor and a systematic way of doing so is described in [14]. A
function $C$ is said to be \emph{invariant} if there exists a polynomial $L$
such that $\dot{C}=CL$. The idea is to find integrating factors which are
products of the form $C_1^{k_1}C_2^{k_2}...,$ where each $C_{i}$ is an
invariant polynomial or exponential of a polynomial. The original idea goes
back to Darboux in the late nineteenth century and systems with such an
integrating factor are said to be \emph{Darboux integrable}. Though it would
be attractive to develop a fully automatic method to find such integrating
factors it was found that a modicum of user intervention is desirable and so
the process described in [15] is semi-automatic in the sense that a number of
procedures are developed which the user calls as required.

However, not all centre conditions can be proved to be sufficient by means of
the above approaches. A method was developed by Cherkas, and described in [4],
in which the polynomial system is transformed to Li\'{e}nard form. As noted
above, because of the independent interest in Li\'{e}nard systems, there is a
very large literature, and known results can then be used. In this paper we
extend the scope of the transformation described in [4] to cover a wider class
of polynomial systems, including the Kukles system (3). The transformed system
need not be, and is not usually, polynomial. This approach extends the range
of techniques which are available for proving that centre conditions which are
known to be necessary are also in fact sufficient.

The centre conditions for a Li\'{e}nard system on which this method depends
are in fact derived, albeit at one remove, from the symmetry of a related
system. Systems which can be so transformed are sometimes said to display
\emph{generalised symmetry. }Centres can be classified depending on whether
they arise from symmetry in a line (time-reversible systems) or are Darboux
integrable or have generalised symmetry.

System (2) is derived from the second order equation
\begin{equation}
\ddot{x}+f(x)\dot{x}+g(x)=0 \tag{5}
\end{equation}
with $f(x)=F'(x)$. It is usual to suppose that $g(x)$sgn $x>0$ for $x$
small and $x\neq0$. Sometimes the corresponding system in the phase plane is
used:
\begin{equation}
\dot{x}=y,\quad\dot{y}=-f(x)y-g(x). \tag{6}
\end{equation}
The relevant results on Li\'{e}nard systems are summarised in the following,
where $F(x)=\int_0^xf(\xi)d\xi$ and $G(x)=\int_0^xg(\xi)d\xi$.

\begin{lemma}
\begin{enumerate}
\item [(i)]The origin is a centre for (6) if and only if there is an analytic
function $\Phi$ with $\Phi(0)=0$ such that $G(x)=\Phi(F(x))$.

\item[(ii)] The origin is a centre for (6) if and only if there is a function
$z(x)$ with $z'(0)<0$ such that
\[
F(z)=F(x)\text{ and }G(z)=G(x).
\]
\end{enumerate}
\end{lemma}

\noindent Proofs of these results are given in [4], for instance.\smallskip

In some cases, it is possible to transform a polynomial system to a
generalised Li\'{e}nard system:
\[
\dot{x}=h(y)-F(x),\quad\dot{y}=-g(x)
\]
where $h(0)=0$ and $h'(0)\neq0$. Lemma 1 holds for this system.\smallskip
\section{The Transformation}
In this paper we consider polynomial systems of the general form
\begin{equation}
\dot{x}=\Sigma p_{k}(x)y^{k},\quad\dot{y}=\Sigma q_{k}(x)y^{k} \tag{7}
\end{equation}
where $p_{k}(x)$ and $q_{k}(x),$ $(k=0,...)$ are polynomials. We could in
exactly the same way consider systems which are polynomial in $x$ with
coefficients which are polynomials in $y$. The Kukles system (3) is a very
specific instance of this form.\smallskip

Cherkas considered systems of this form with $p_{k}=0$ for $k\geq2$ and
$q_{k}=0$ for $k\geq3;$ his approach was discussed in [4], and is summarized
as follows.

\begin{lemma}
Suppose that $p_1(x)\neq0$ for $-\alpha<x<\beta$, where $\alpha,\beta>0$.
For $-\alpha<x<\beta$ the system
\[
\dot{x}=p_0(x)+p_1(x)y,\quad \dot{y}=q_0(x)+q_1(x)y+q_2(x)y^2
\]
can be transformed to a system of the form (6) with
\[\begin{gathered}
f=-(p_0'-p_0p_1'+q_1-2p_0q_2p_1^{-1})\phi\,,\\
g=(p_0q_1-q_0p_1-p_0^2q_2p_1^{-1})\phi^2,
\end{gathered}
\]
where $\phi(x)=(p_1(x))^{-1}\exp\left(  -\int_0^xq_2(t)/p_1
(t)dt\right)  $.
\end{lemma}

The transformation can be achieved by a change of independent variable. Let
$dt/d\tau=\phi(x);$ routine calculation then leads to the desired result. This
transformation was used in both [4] and [12] to confirm the sufficiency of the
centre conditions previously shown to be necessary. In [4] we used part (ii)
of Lemma 1, while in [12] we used part (i).\smallskip

However, Lemma 2 does not cover systems such as
\begin{equation}
\dot{x}=p_0+p_1y,\quad\dot{y}=q_0+q_1y+q_2y^2+q_3y^3 \tag{8}
\end{equation}
(where, of course, the $p_{k}$ and $q_{k}$ are functions of $x$ alone).
Cherkas [1] was able to transform such systems to Li\'{e}nard form when
$p_0=0$ but required a particular solution in order to be able to do so; the
transformation he used was
\[
y=Z(x)Y(Y+1)^{-1}
\]
where $Z$ is a particular solution of
\[
\frac{dy}{dx}=\frac{q_0+q_1y+q_2y^2+q_3y^3}{p_1y}.
\]
The transformation of such systems to Li\'{e}nard form is also discussed by
Sadovskii [17].

The purpose here is to present a different approach to systems of the form
(8). We use a transformation of the form
\begin{equation}
y=\frac{a(x)Y}{1+b(x)Y} \tag{9}
\end{equation}
where $a$ and $b$ are differentiable functions to be chosen, with $a(0)\neq0$.
The transformation is invertible in a neighbourhood of the origin.
We consider systems
\begin{equation}
\begin{aligned}
\dot{x}&=p_1(x)y,\\
\dot{y}&=q_0(x)+q_1(x)y+q_2(x)y^2+q_3(x)y^3,
\end{aligned} \tag{10}
\end{equation}
where $q_0(0)=0$ (so the origin is a critical point). Since such systems
generalise the form (3), they are sometimes said to be of \emph{Kukles type.
}System (8) can easily be transformed to (10), so our consideration of the
latter is without loss of generality.

A routine calculation leads to
\begin{equation}
\begin{aligned}
\dot{x}&=P_1(x)Y,\\
\dot{Y}&=Q_0(x)+Q_1(x)Y+Q_2(x)Y^2+Q_3(x)Y^3,
\end{aligned}\tag{11}
\end{equation}
where
\begin{gather*}
P_1    =ap_1,\quad Q_0=a^{-1}q_0,\;Q_1=q_1+3a^{-1}bq_0,\\
Q_2    =aq_2+2bq_1+3b^2a^{-1}q_0-a'p_1
\end{gather*}
and
\[
Q_3=a^2q_3+abq_2+b^2q_1+b^3a^{-1}q_0+(ab'-a'b)p_1.
\]

If $a,b$ are chosen so that $Q_3=0,$ then (11) can be further transformed to
a Li\'{e}nard system in accordance with Lemma 2. This requires $u=b/a$ to
satisfy the differential equation
\begin{equation}
u'p_1=-(q_0u^3+q_1u^2+q_2u+q_3). \tag{12}
\end{equation}
We therefore lose no generality by taking $a(x)\equiv1$. The difficulty is
that it may not be possible to solve equation (12) explicitly - and an
explicit solution would be required to be able to use results such as those
given in Lemma 1. In some situations, of course, an explicit solution can be
found (a particular instance is when $q_1=q_3=0)$.

\begin{lemma}
Suppose that $b/a$ satisfies equation (12). Then the system (10) can be
transformed to Li\'{e}nard form by means of (9).
\end{lemma}

System (11) is itself of Li\'{e}nard form if both $Q_2$ and $Q_3$ are
zero. The functions $a$ and $b$ are then determined by a pair of differential
equations. Again, an explicit solution may not be possible.

The remaining possibility is to choose $a$ and $b$ so that $Q_1=Q_2=0$.
The transformed system is then of the form
\begin{equation}
\begin{gathered}
\dot{x}=u,\\
\dot{u}=-g(x)-f(x)u^3,
\end{gathered}
\tag{13}
\end{equation}
and the question arises whether results analogous to those given in Lemma 1
apply. The functions $a$ and $b$ are now given by the relations
\begin{equation}
\begin{gathered}
[c]{l}
b=-\tfrac{1}{3}q_0^{-1}q_1a,\\
a'=ap_1^{-1}\left(  q_2-\tfrac{1}{3}q_1^2q_0^{-1}\right)  .
\end{gathered}
\tag{14}
\end{equation}
Thus
\begin{equation}
a(x)=\exp\left[  \int_0^x(3q_0q_2-q_1^2)q_0^{-1}p_1
^{-1}\right]  . \tag{15}
\end{equation}
We suppose that
\begin{enumerate}
\item [(i)]$p_1(0)\neq0$,

\item[(ii)] $(q_1(x))^2(q_0(x))^{-1}$ tends to a finite limit as
$x\to 0$.
\end{enumerate}

Both these are very natural conditions: (i) states that the $y$-axis is not
invariant, which must be the case if there are closed orbits surrounding the
origin, and (ii) is certainly satisfied by (3), for example.

\begin{theorem}
System (10) can be transformed to (13), with $a,b$ given by (15) and (14)
respectively, where
\[
g=-q_0p_1^{-1}a^{-2}
\]
and
\[
f=\tfrac{1}{3}a\left(  \frac{q_1}{q_0}\right)  '-p_1
^{-1}a\left(  q_3-\tfrac{1}{3}q_1q_2q_0^{-1}+\tfrac{2}{27}q_1
^3q_0^{-2}\right)  .
\]
\end{theorem}

The origin is certainly a centre for (11) when the system is symmetric in the
$x$-axis. This requires $Q_1=Q_3=0,$ which exactly corresponds to
$f(x)=0$. The following result is a straightforward observation.

\begin{theorem}
Suppose that the origin is a critical point of (10) of focus type. Given
$p_1,q_0,q_1,q_2,$ the origin is a centre for (10) if
\[
q_3=\tfrac{1}{3}p_1\left(  \frac{q_1}{q_0}\right)  '-\tfrac
{1}{3}q_1q_2q_0^{-1}+\tfrac{2}{27}q_1^3q_0^{-2}.
\]
\end{theorem}

It might be expected that a transformation
\[
y=\frac{a(x)Y+c(x)}{1+b(x)Y}
\]
would lead to a more general result. This is not the case. The function $c(x)$
does not appear in $Q_3,$ and an additional term arises in the equation for
$\dot{x}$:
\[
\dot{x}=P_0(x)+P_1(x)Y.
\]
Then $c$ is determined by the need for $P_0(x)=0,$ and no generalisation is achieved.

We now turn to systems of the form (13); we consider
\begin{equation}
\begin{array}
[c]{l}
\dot{x}=y,\\
\dot{y}=-g(x)-f(x)\phi(y),
\end{array}
\tag{16}
\end{equation}
where $y\phi(y)>0$. There is no analogue in this case of the form (2).

We suppose that $f$ and $g$ are $C^{1},$ $g(0)=0$ and $g(x)$sgn $x>0$ for
$x\neq0$. Let $G(x)=\int_0^xg(\xi)d\xi,$ and define
\[
u=\sqrt{2G(x)}\text{ sgn }g(x).
\]
The transformation $x\longmapsto u$ has an inverse; let this be $x=\xi(u)$.
Let $k(x)=f(x)/g(x),$ and define $k^{\ast}(u)=uk(\xi(u))$. In terms of $y$ and
$u,$ system (16) is
\begin{equation}
\begin{array}
[c]{l}
\dot{u}=u^{-1}g(\xi(u))y,\\
\dot{y}=-g(\xi(u))-f(\xi(u))\phi(y).
\end{array}
\tag{17}
\end{equation}
Now $u^{-1}g(\xi(u))\to 1$ as $u\to 0$. The orbits of (17) are
the same as those of
\begin{equation}
\begin{array}
[c]{l}
\dot{u}=y,\\
\dot{y}=-u-k^{\ast}(u)\phi(y).
\end{array}
\tag{18}
\end{equation}
If $k^{\ast}$ is an odd function then (18) is unchanged under the
transformation $(u,y,t)\longmapsto(-u,y,-t);$ it follows that the origin is a centre.

Conversely, compare (18) with the system
\begin{equation}
\dot{u}=y,\quad\dot{y}=-u-K(u)\phi(y), \tag{19}
\end{equation}
where $K(u)=\frac{1}{2}(k^{\ast}(u)-k^{\ast}(-u))$. The origin is a centre for
(19), again by symmetry in the $y$-axis. But system (18) is rotated with
respect to (19) in a neighbourhood of the origin. This follows from the fact
that the vector product of the two vector fields is $\frac{1}{2}
y\phi(y)(k^{\ast}(u)+k^{\ast}(-u)),$ which is of one sign in a neighbourhood
of the origin. Hence, if the origin is a centre, $k^{\ast}$ is odd. It
follows, as in [4], that there is a unique function $z(x)$ satisfying
$z'(0)<0$ and $z(0)=0,$ such that
\[
G(z)=G(x),\quad k(z)=k(x).
\]

\begin{theorem}
Suppose that $f,g,\phi$ are $C^{1}$ functions, with $xg(x)>0$ $(x$ small,
$x\neq0)$ and $y\phi(y)>0$ ($y$ small, $y\neq0)$. Then the origin is a centre
for system (16) if and only if there is a unique function $z(x)$ with
$z(0)=0,$ $z'(0)<0,$ such that
\[
G(z)=G(x)\text{ and }k(z)=k(x),
\]
where $G$ and $k$ are as defined above.
\end{theorem}

It might be thought that the ideas described above could be used in relation
to systems more general than (10). The simplest example would be
\[
\dot{x}=p_1y+p_2y^2,\quad\dot{y}=q_0+q_1y+q_2y^2+q_3
y^3+q_4y^{4},
\]
where the $p_{k}$ and $q_{k}$ are again functions of $x$. The transformation
(9) gives
\begin{align*}
\dot{x}  &  =P_1(x)Y+P_2(x)Y^2,\\
\dot{Y}  &  =Q_0(x)+Q_1(x)Y+Q_2(x)Y^2+Q_3(x)Y^3+Q_4(x)Y^{4},
\end{align*}
where
\begin{align*}
P_1  &  =a^2p_1,\quad P_2=a^2(ap_2+bp_1),\\
Q_0  &  =q_0,\quad Q_1=aq_1+4bq_0,\\
Q_2  &  =a^2q_2+3abq_1+6b^2q_0-aa'p_1,\\
Q_3  &  =a^3q_3+2a^2bq_2+3ab^2q_1+4b^3q_0-a(ap_2
+bp_1)a'+a(ab'-a'b)p_1,\\
Q_4  &  =a^{4}q_4+a^3bq_3+a^2b^2q_2+ab^3q_1+b^{4}
q_0+a(ab'-a'b)(ap_2+bp_1).
\end{align*}
If we can choose $a,b$ so that $P_2=Q_4=0,$ then we can use Theorem 4 to
obtain the desired form. We can ensure that $P_2=0$ by choosing
$b=-ap_2p_1^{-1}$. However, then
\[
Q_4=a^{4}(q_4-q_3p_2p_1^{-1}+q_2p_2^2p_1^{-2}-q_1p_2
^3p_1^{-3}+q_0p_4^{4}p_1^{-4}).
\]
We have $Q_4=0$ only if
\[
p_1^{4}q_4-p_1^3p_2q_3+p_1^2p_2^2q_2-p_1p_2^3
q_1-p_4^{4}q_0=0,
\]
a relation which is not usually satisfied. As noted previously no further
benefit is derived from using (16) instead of (9).

The approach which we have described is designed to extend the range of the
techniques available for proving that a critical point is a centre. We
conclude by giving a simple illustration of the use of the ideas which we have
described. Ordinary differential equations are used extensively to model
biological population dynamics (see [14] for example); they are appropriate
when spatial detail is less significant than changes in populations with time.
The systems used often consist of polynomial equations, and their analysis is
helped by considering just two taxonomic categories (see [19]). In [16] a
model in which intraprophic predation is taken into account is described and
analysed. In nondimensional form the differential system that arises is
\[
\acute{x}=xR(x,y),\quad\dot{y}=yS(x,y),
\]
where
\begin{align*}
R(x,y) &  =\xi(-\kappa\eta x+y)(1+\eta x+y)^{-1}-\delta,\\
S(x,y) &  =1-\varepsilon y-\xi x(1+\eta x+y)^{-1},
\end{align*}
and $\xi,\kappa,\eta,\delta$ and $\varepsilon$ are parameters.

After transforming the origin to a critical point in the first quadrant and
rescaling time the system is of the form
\begin{align*}
\dot{x} &  =p_0(x)+yp_1(x),\\
\dot{y} &  =q_0(x)+yq_1(x)+y^2q_2(x)+y^3q_3(x).
\end{align*}
For simplicity, we choose $\kappa$ so that $p_0=0$. The functions $p_1$
and $q_{i}(i=0,...,3)$ are all linear:
\begin{align*}
p_1(x)  & =Cx+k_5,\quad q_0(x)=k_4x,\quad q_1(x)=k_2+k_3x,\\
q_2(x)  & =k_0-k_1x,\quad q_3(x)=-\varepsilon,\quad
\end{align*}
where $C$ and the $k_{i}$ are functions of the parameters in $R$ and $S$. The
origin is a critical point of focus type if $k_2^2+4k_4k_5<0$. We can
then use Theorem 5 to deduce that it is a centre when
\begin{align*}
k_2(2k_2^2-9k_4k_5) &  =0\\
k_2(2k_2k_3-3k_0k_4-3Ck_4) &  =0\\
2k_2k_3^2-3k_4(k_0k_3-k_1k_2)+9\varepsilon k_4^2 &  =0\\
k_3(2k_3^2+9k_1k_4) &  =0.
\end{align*}
 From the first of these equations, $k_2=0$ or $2k_2^2-9k_4k_5=0$.
The latter is inconsistent with the requirement that $k_2^2+4k_4
k_5<0$. Here we must have $k_2=0$. We conclude that the system has a
centre if
\[
k_2=0,\;2k_3^2+9k_1k_4=0,\;3\varepsilon k_4=k_0k_3.
\]

\begin{thebibliography}{99} {\frenchspacing

\bibitem{c1} L. A. Cherkas, Conditions for the equation $yy'=\sum
_{i=0}^3p_{i}(x)y$ to have a center, \emph{Differ. Uravn., }\textbf{14}
(1978), 1133-1138.

\bibitem{c2} L. A. Cherkas \& L. I. Zhilevich, Some criteria for the absence of
limit cycles and for the existence of a single limit cycle, \emph{Differ.
Uravn., }\textbf{6 }(1970), 891-897.

\bibitem{c3} C. J. Christopher \& N. G. Lloyd, On the paper of Jin and Wang
concerning the conditions for a centre in certain cubic systems, \emph{Bull.
London Math. Soc., }\textbf{22} (1990), 5-12.

\bibitem{c4} C. J. Christopher, N. G. Lloyd \& J\ M\ Pearson, On Cherkas's method
for centre conditions, \emph{Nonlinear World, }\textbf{2 }(1995), 459-469.

\bibitem{c5} W. A. Coppel, Some quadratic systems with at most one limit cycle,
\emph{Dynam. Report. Expositions Dynam. Systems (N.S.), }\textbf{2} (1989), 61-88.

\bibitem{c6} W. A. Coppel, A new class of quadratic systems, \emph{J.
Differential Equations, }\textbf{92 }(1991), 360-372.

\bibitem{g1} A Gasull \& A Guillamon, Non-existence, uniqueness of limit cycles
and center problems in a system that includes predator-prey systems and
generalised Li\'{e}nard equations, \emph{Differential Equations Dynam.
Systems,}\textbf{ 3 }(1995), 345-366.

\bibitem{j1} X Jin \& D Wang, On the conditions of Kukles for the existence of a
center, \emph{Bull. London Math. Soc., }\textbf{22} (1990), 1-4.

\bibitem{k1} R. E. Kooij, \emph{Limit cycles in polynomial systems, }PhD Thesis,
Technical University of Delft (1993).

\bibitem{l1} N. G. Lloyd \& J. M. Pearson, Conditions for a centre and the
bifurcation of limit cycles in a class of cubic systems,
in: \emph{Bifurcations of Planar Vector Fields, }Eds. J. P. Fran\c{c}oise and
R Roussarie, Springer, Berlin (1990), 230-242.

\bibitem{l2} N. G. Lloyd \& J. M. Pearson, Computing centre conditions for
certain cubic systems, \emph{J. Comput. Appl. Math., }\textbf{40} (1992), 323-336.

\bibitem{l3}N. G. Lloyd \& J. M. Pearson, Five limit cycles for a simple cubic
system, \emph{Publ. Mat., }Vol. \textbf{41} (1997), 199-208.

\bibitem{l4} N. G. Lloyd \& J. M. Pearson, Bifurcation of limit cycles and
integrability of planar dynamical systems in complex form, \emph{J. Phys. A,
}\textbf{32} (1999), 1973-1984.

\bibitem{m1} J. D. Murray, \emph{Mathematical Biology}, Springer-Verlag, Berlin,
Heidelberg (1989).

\bibitem{m2} J. M. Pearson, N. G. Lloyd \& C J Christopher, Algorithmic
derivation of centre conditions, \emph{SIAM Rev., }(1996), 619-636.

\bibitem{p1} Jonathan Pitchford \& John Brindley, Intratrophic predation in
simple predator-prey models, \emph{Bull. Math. Biol., }\textbf{60 }(1998), 937-953.

\bibitem{s1} A. P. Sadovskii, Solution of the center-focus problem for a cubic
system of nonlinear oscillations, \emph{Differ. Uravn., }\textbf{33 }(1997), 236-244.

\bibitem{s2} Jitsuro Sugie, Kie Kohono \& Rinko Miyazaki, On a predator-prey
system of Holling type, \emph{Proc. Amer. Math. Soc., }\textbf{125 }(1997), 2041-2050.

\bibitem{t1} J. E. Truscott \& J Brindley, Equilibria, stability and excitability
in a general class of plankton population models, \emph{Phil. Trans. R. Soc.
Lond. A,}\textbf{ 347} (1994), 703-718.

\bibitem{w1} Dongming Wang, Polynomial systems for certain differential
equations, \emph{J. Symbolic Comput., }\textbf{28 }(1999), 303-315.

\bibitem{w2} Yongbin Wu, Guowi Chen \& Xinan Yang, Kukles systems with two fine
foci, \emph{Ann. of Diff. Eqs., }\textbf{15} (1999), 422-437.

}\end{thebibliography}

\noindent{\sc W. A. Albarakati }\\
Department of Mathematics, King Abdulaziz University, \\
Jeddah,  Saudi Arabia \smallskip

\noindent{\sc Noel G. Lloyd }\\
Department of Mathematics, University of Wales, \\
Aberystwyth, Ceredigion SY23 3BZ,  UK\\
e-mail: ngl@aber.ac.uk \smallskip

\noindent{\sc Jane M. Pearson }\\ 
Department of Mathematics, University of Wales, \\
Aberystwyth, Ceredigion SY23 3BZ, UK \\
e-mail: jmp@aber.ac.uk


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