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\headline={\ifnum\pageno=1 \hfill\else%
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\def\rightheadline{EJDE--1998/13\hfil Limit cycles from Kukles isochrones
\hfil\folio}
\def\leftheadline{\folio\hfil B. Toni
\hfil EJDE--1998/13}
\def\pretitle{\vbox{\eightrm\noindent\baselineskip 9pt %
Electronic Journal of Differential Equations,
Vol.\ {\eightbf 1998}(1998), No.~13, pp.~1--10.\hfil\break
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\hfill\break
ftp (login: ftp) 147.26.103.110 or 129.120.3.113\bigskip} }
\topmatter
\title
Branching of periodic orbits from Kukles isochrones
\endtitle
\thanks
{\it 1991 Mathematics Subject Classifications:} 34C15, 34C25, 58F14, 58F21,
58F30.
\hfil\break\indent
{\it Key words and phrases:} Limit cycles, Isochronous system, Linearization,
Perturbations.
\hfil\break\indent
\copyright 1998 Southwest Texas State University and
University of North Texas.\hfil\break\indent
Submitted April 22, 1998. Published May 13, 1998.
\endthanks
\author B. Toni \endauthor
\address Bourama Toni \newline
Facultad de Ciencias,
Universidad Autonoma del Edo de Morelos\newline
Cuernavaca 62210, Morelos, Mexico.
\endaddress
\email toni\@servm.fc.uaem.mx
\endemail
\abstract
We study local bifurcations of limit cycles from isochronous (or linearizable)
centers. The isochronicity has been determined using the method of Darboux
linearization, which provides a birational linearization for the
examples that we analyze.
This transformation simplifies the analysis by avoiding the complexity of
the Abelian integrals appearing in other approaches. As an application of
this approach, we show that the Kukles isochrone (linear and nonlinear)
has at most one branch point of limit cycles. Moreover, for each isochrone,
there are small perturbations with exactly one continuous family of limit
cycles.
\endabstract
\endtopmatter
\document
\heading 1. Introduction \endheading
In this paper we address the bifurcations of limit cycles (isolated periodic
orbits) for polynomial perturbations of polynomial integrable vector fields.
When the unperturbed system is isochrone (linearizable), the linearization is
known, and is a birational transformation in the phase plane, which is, in
general, a Darboux linearization, i.e., a linearizing transformation
involving polynomial maps and their complex powers, \cite{5}.
Specifically, we consider an autonomous polynomial perturbation $(p,q)$ of a
plane vector field in the form
$$
\Cal X_\epsilon :=(P(x,y)+\epsilon p(x,y))\frac{\partial}{\partial x}
+(Q(x,y)+\epsilon q(x,y))\frac{\partial}{\partial y},
\tag{$\Cal P_\epsilon$}$$
where
$$\gather
P(x,y)=-y+\sum_{2\leq i+j \leq n}{P_{ij}x^{i}y^{j}},\quad
Q(x,y)=x+\sum_{2\leq i+j \leq n}Q_{ij}x^{i}y^{j}\\
p(x,y)=\sum_{i=1}^{n}{\sum_{k=0}^{i}{a_{i-k,k}x^{i-k}y^{k}}},\quad
q(x,y)=\sum_{i=1}^{n}{\sum_{k=0}^{i}{b_{i-k,k}x^{i-k}y^{k}}},
\endgather
$$
and $\lambda_{ij}=(a_{ij},b_{ij},p_{ij},q_{ij})\in {\Bbb R}^4$, with
$\epsilon$ a small parameter. When
$\epsilon=0$, we assume further that the unperturbed vector field $(\Cal X_0)$
has an isochronous
center at the origin $O\in {\Bbb R}^2$, i.e., all orbits in a sufficiently
small neighborhood $\Cal A$ of the origin are closed and have the same
period. The largest such neighborhood is called an
{\it isochronous period annulus}. For a closed orbit $\gamma_0 \in \Cal A$,
it is interesting to study the creation of limit cycles from $\gamma_0$
on passing from $\epsilon=0$ to small nonzero values of
$\epsilon$. Recall that a limit cycle is a periodic orbit isolated in the set
of
closed orbits of the vector field.
For fixed $\lambda_{ij}$, there is a neighborhood $U$ of the origin in
${\Bbb R}^2$ on which the flow associated with \thetag{$\Cal P_{\epsilon}$}
exists for all initial values in $U$. Assume, furthermore, that $U$ is small
enough so that a Poincar\'e
return mapping $\delta(r,\epsilon)$ is defined on $U$, with the distance
coordinate $r$. The solution $\gamma_{\epsilon}(t)$ starting at $(r,0)$,
$r>0$, intersects the positive $x-$axis for the first time at some point
$(\delta(r,\epsilon),0)$. Let $\Sigma =\{(x,0)\in U, x>0 \}$ denote the
transversal section or Poincar\'e section of $U$. By transversality the
mapping $\delta$ is analytic, and can be expanded as the
convergent Taylor series
$$\delta(r,\epsilon)=r+\sum_{k\ge 1}{\delta_k(\epsilon) r^k}.
\tag{1-1}$$
On $\Sigma$ we define the displacement function
$d(r,\epsilon):=\delta(r,\epsilon)-r$. Of course, the zeros of $d(r,\epsilon)$
correspond to periodic orbits of $(\Cal P_\epsilon)$ intersecting $\Sigma$.
Assuming that the period annulus $\Cal A$ is parametrized by $r$, then
$d(r,0)\equiv 0$. We reduce the analysis to that of finding the roots of a
suitable bifurcation function derived from the displacement function. This is
achieved by investigating the number and position of the periodic orbits in the
isochronous period annulus $\Cal A$ that survive after perturbation by giving
birth to a continuous family $\gamma{_\epsilon}$ of limit cycles of the
perturbed system.
In section two below, we describe our {\it isochrone reduction} method for
studying both the first order bifurcations of limit cycles in autonomous
perturbations of a polynomial isochronous system, and the branching of periodic
orbits from isochrones. Section three is entirely devoted to Kukles isochrones.
We show that at most one local family of limit cycles bifurcates at first order
from these isochrones. Moreover, there exist perturbations which exhibit
exactly one such perturbation. Each limit cycle is asymptotic to a circle whose
radius is a simple positive zero of the bifurcation function.
\heading 2. Isochrone reduction \endheading
Under the previous assumptions, consider an element $r_*\in \Sigma$ such that
$$d_{\epsilon}(r_*,0)=0,\quad \text{and } d_{r \epsilon}(r_*,0)\neq 0\,,
\tag2-1$$
i.e., $r_*$ is a simple zero of $d_{\epsilon}$; the subscripted $\epsilon$ and
$r$ denote partial derivatives. Thus, by the Implicit Function Theorem, there
exits a smooth function $r=\omega(\epsilon)$ defined in some neighborhood of
$\epsilon=0$, such that $\omega(0)=r_*$ and $d(\omega(\epsilon),\epsilon) \equiv
0$. The curve $r=\omega(\epsilon)$ corresponds to a local family of limit
cycles emerging from the periodic trajectory $\gamma_{r_*}$ of the unperturbed
system which meets $\Sigma$ at $r_*$. A difficulty arises in the calculations
and analysis of the partial derivatives of $d(r,\epsilon)$. Of course, for
$d_{\epsilon} (r,0)\equiv 0$, or if one of the zeros is not simple, then
higher-order derivatives must be computed. Actually, in $\Cal A$,
$d_{\epsilon}(r,0)=0$ for all values of $r$, and so we cannot apply the Implicit
Function Theorem. However, from the perturbation of the Taylor series
$$
d(r,\epsilon)=\epsilon d_{\epsilon}(r,0)+O(\epsilon^2)=
\epsilon(d_{\epsilon}(r,0)+O(\epsilon))=\epsilon
B(r,\epsilon),
\tag2-2$$
with $B(r,\epsilon):= d_{\epsilon}(r,0)+O(\epsilon)$, we define a reduced
displacement function by
$$B(r):=d_{\epsilon}(r,0),
\tag2-3$$
for small real values of $\epsilon$. Clearly, if
$B(\omega(\epsilon),\epsilon)\equiv 0$ then $d(\omega(\epsilon),\epsilon) \equiv
0$ and the Implicit Function Theorem does apply to $B$. In other words, a
simple zero of $B$ corresponds to the appearance of a local family $r=\omega
(\epsilon)$ of periodic orbits. Such a zero, $r_*$, of $B$ is called {\it a
branch point of periodic orbits} for the system \thetag{$\Cal P_\epsilon$}. The
corresponding periodic orbit $\gamma_{r_*}$ is said to {\it survive} or to {\it
persist} after perturbation.
If $r_*$ is a simple root of $B(r)$ of order $k$, i.e.,
$\partial_\epsilon^k d(r_*,0)=0,\quad \partial_r\partial_\epsilon^k (r_*,0)\neq 0,$
with $\partial_\epsilon^i d(r_*,0)\equiv 0$, for $i=0,\dots,(k-1)$,
then writing the perturbation Taylor series in the form
$$d(r,\epsilon)=\epsilon^k(\partial_\epsilon^k d(r_*,0)/k!+O(\epsilon)):=
\epsilon^k B^k(r,\epsilon)
\tag2-4$$
yields $B^k(r_*,0)=0$ and $B^k_r (r_*,0)\neq 0$. Applying the Implicit
Function Theorem to $B^k$, we see that by continuity, there is a number
$\epsilon_1>0$ and a unique smooth function $r=\omega (\epsilon)$ with
$|\epsilon|<\epsilon_1$ such that $\omega(0)=r_*$ and
$d(\omega(\epsilon),\epsilon)\equiv 0$. If $r_*$ is a root of multiplicity
$m$, it follows from the Weierstrass Preparation theorem \cite{6} that
there at most $m$ distinct smooth functions $r=\omega_i(\epsilon).$
In the case of an isochronous period annulus the isochronal assumption
is essential to our approach. It is well known (see, e.g., \cite{5}) that
the origin of \thetag{$\Cal P_{\epsilon}$} is isochronous if and
only if there exists an analytic change of coordinates
$$(\Cal T_l):\ (u(x,y),v(x,y))=(x + o(|(x,y)|),y + o(|(x,y)|))
$$
in its neighborhood, reducing the system to a linear isochrone. Once we know
explicitly \thetag{$\Cal T_l$}, we reduce the autonomous perturbation of the
nonlinear isochrone to that of a linear one; we then derive a simple expression
of the bifurcation function $B$. Practically speaking, consider the perturbed
system \thetag{$\Cal P_\epsilon$}. Through \thetag{$\Cal T_l$}, \thetag{$\Cal
P_\epsilon$} is simplified to the weakly linear system
$$\dot v= Av +\epsilon h(v),
\tag{$\bar \Cal P_\epsilon$}$$
with $v:=(X,Y) \in \Bbb R^2$, $A=\pmatrix 0&-1\\1&0 \endpmatrix$, and
$h(v):=(h_1(v),h_2(v))$.
The determination of the branch points of the periodic orbits of \thetag{$\Cal
P_\epsilon$} proceeds as follows. We first reduce the appropriate displacement
function to a bifurcation function and apply the Implicit Function Theorem and
its related corollaries. We next identify the bifurcation function in terms of
the reduced perturbed system; the branch points are its simple zeros.
\proclaim{Theorem 2.1}
Consider a weakly linear system in the form \thetag{$\bar \Cal P_\epsilon$}.
Assume the unperturbed system has a period annulus parametrized by $r$.
A branch point of periodic orbits of \thetag{$\bar \Cal P_\epsilon$} is a
simple zero of the function
$$B(r):=\int_0^{2\pi}{\left (h_1(r\cos t,r\sin t)\cos t+h_2(r\cos t,r\sin t)
\sin t \right)dt},
$$
where $r$ is taken in an interval of $(0,\infty)$.
\endproclaim
\demo{Proof} Given $d(r,\epsilon)$, the associated displacement function,
defined globally on the Poincar\'e section $\Sigma$, the bifurcation function is
defined as $B(r):=d_{\epsilon}(r,0)$ for small values of $\epsilon$. Using a
periodic orbit $\gamma$ with integral curve
$\gamma_{\epsilon}(r,t):=(X(t,r,\epsilon),Y(t,r,\epsilon))$ starting at $(r,0)$
we obtain
$$d_{\epsilon}(r,0)=\dot X (T(r,0),r,0)T_{\epsilon}(r,0)+X_{\epsilon}(T(r,0),r,0).
\tag2-5$$
At $r=0$ we have $\dot X (T(r,0),r,0)=-Y(0,r,0)=0$. Thus, we obtain
$d_{\epsilon}(r,0)=X_{\epsilon}(T(r,0),r,0)$. Looking for
$X_{\epsilon}(T(r,0),r,0)$ amounts to integrating the variational equation
$$\gathered
\dot X_{\epsilon}=-Y_{\epsilon}+h_1(X,Y),\\
\dot Y_{\epsilon}=X_{\epsilon}+h_2(X,Y),\\
X_{\epsilon}(0,r,0)=Y_{\epsilon}(0,r,0)=0\,.
\endgathered\tag2-6
$$
In matrix form it is expressed as
$$\gathered
\dot W=A W+H(t),\\
W(0)=0\,,
\endgathered
\tag2-7$$
where $A$ is as given above, and
$$H(t)=\pmatrix
h_1(X(t,r,\epsilon),Y(t,r,\epsilon))\\
h_2(X(t,r,\epsilon),Y(t,r,\epsilon))
\endpmatrix.
\tag2-8$$
By the method of variation of constants, we get
$$\aligned
W(T(r,0))=&\left (X_{\epsilon}(T(r,0),r,0),Y_{\epsilon}(T(r,0),r,0)\right)\\
=&\Phi(T(r,0))\int_0^{T(r,0)}{\Phi^{-1}(s)H(\gamma(r,s))}\,ds,
\endaligned
\tag2-9$$
where $\Phi(t)$ denotes the principal fundamental matrix solution of
$\dot W=AW$ at $t=0$. We have
$$
\Phi(t)=e^{tA}=\pmatrix
\cos t & - \sin t\\
\sin t & \cos t
\endpmatrix
\tag2-10$$
and $H(\gamma(r,t))=\pmatrix
h_1(r \cos t,r \sin t)\\
h_2(r \cos t,r \sin t)
\endpmatrix.$
Hence, for $T(r,0)=2\pi$, it follows
$$
\pmatrix
X_{\epsilon}(2\pi,r,0)\\
Y_{\epsilon}(2\pi,r,0)
\endpmatrix
=\pmatrix \int_0^{2\pi}{\left (h_1(r \cos s,r \sin s)\cos s+h_2(r \cos s,r \sin s)\sin s \right)ds}\\
\int_0^{2\pi}{\left (-h_1(r \cos s,r \sin s)\sin s+h_2(r \cos s,r \sin s)\cos s
\right)ds} \endpmatrix \tag2-11
$$
Thus we obtain
$$\aligned
B(r)=d_{\epsilon}(r,0)=& X_{\epsilon}(2\pi,r,0)\\
=&\int_0^{2\pi}{\left (h_1(r \cos s,r \sin s)\cos s+h_2(r \cos s,r \sin s)
\sin s\right)ds}.
\endaligned
\tag2-12$$
\qed \enddemo
\subheading{Perturbations of the linear isochrone}
We consider a perturbation of degree $n$ of the linear isochrone and prove
the following theorem.
\proclaim{Theorem 2.2}
From the linear isochrone, to first order, no more than
$(n-1)/2$, (resp. $(n-2)/2$) continuous families of limit cycles can bifurcate
in the direction of any autonomous polynomial perturbation of degree $n$, where
$n$ is odd (resp. even). And we can construct small perturbations with the
maximum number of limit cycles. Moreover the limit cycles are asymptotic to the
circles whose radii are simple positive roots of the bifurcation function.
\endproclaim
\demo{Proof}
Using the expressions of $p$ and $q$ as polynomials of degree $n$ in
\thetag{$\Cal P_{\epsilon}$}, we compute
the bifurcation function $B$ and obtain
$$B(r)=\sum_{i=1}^n{r^i \sum_{k=0}^i{\left(\int_0^{2\pi}{(a_{i-k,k}\cos t+
b_{i-k,k}\sin t)\cos^{i-k}t \sin^k t}\,dt\right)}}.\tag2-13$$
This can be simplified using the well known rules
$\int_0^{2\pi}{\cos^m t \sin^n t dt}=0$, for $m$ or $n$ odd (including $0$).
As a result
$$B(r)=r\sum_{s=1,s\text{ odd}}^N{r^{s-1}c_s},
\tag2-14$$
where
$$N=\cases n,&\text{for $n$ odd}\\
n-1,&\text{for $n$ even}
\endcases \tag2-15
$$
and $c_s$ is the nonzero constant
$$c_s=(a_{s0}+b_{0s})+\sum_{k=1,\text{ k odd}}^{s-2}
{\left (b_{s-k,k}+a_{s-k-1,k+1}\right)
\int_0^{2\pi}{\cos^{s-k} t \sin^{k+1} t\, dt}}.
\tag2-16$$
Therefore the upper bound of the number of simple zeros of $B(r)$
is $M(n)=(n-1)/2$ for $n$ odd and $(n-2)/2$ for $n$ even. Perturbations with
the maximum
number are constructed as in the cubic case below.
As $\epsilon \longrightarrow 0$, the weakly linear system
$(\bar \Cal P_{\epsilon})$ tends to the linear
isochrone whose solution curves are circles $x^2+y^2=r^2$. Therefore the
periodic orbits (limit cycles)
are asymptotic to these circles as $\epsilon \longrightarrow 0$. \qed
\enddemo
\remark{Remark}
Note that as an application to first order, no limit cycles can emerge from
periodic trajectories of
the linear isochrone after a quadratic autonomous perturbation, in agreement
with a result in \cite{1} (Section 3.1: The linear isochrone). For a
cubic autonomous perturbation, we obtain the following
\endremark
\proclaim{Corollary 2.3}
From a periodic trajectory $\gamma_0$ in the period
annulus $\Cal A$ of the linear isochrone, at most one continuous family of limit
cycles bifurcate from $\gamma_0$ in the direction of the cubic autonomous
perturbation $(p,q)$. The maximum number one is attained if and only if the
coefficients satisfy the condition $c_0 c_2<0$, where $c_0$ and $c_2$ are given
below. In this instance, this family emerges from the real simple roots of the
quadratic function
$$\Delta (r):=c_0+c_2r^2.$$
\endproclaim
\demo{Proof}
The corresponding bifurcation function is given by
$$
B(r)=(r\pi)\left ((a_{10}+b_{01})+\frac{r^2}{4}(3a_{30}+a_{12}+b_{21}+
3b_{03})\right).
\tag2-17$$
Thus, the roots of the quadratic $\Delta (r):=a_{10}+b_{01}+
\frac{r^2}{4}(3a_{30}+a_{12}+b_{21}+3b_{03})$
yield the continuous families of limit cycles that bifurcate from the period
annulus at the origin of the
linear isochrone. Define
$$
c_2:=\frac{1}{4}(3a_{30}+a_{12}+b_{21}+3b_{03}),\quad c_0:=(a_{10}+b_{01})\,.
\tag2-18
$$
If $c_0=0$ and $c_2\neq 0$, then the origin is the only root of the
polynomial $\Delta (r)$; however,
But $c_0 \neq 0$ implies $r^2=-c_2/c_0$. Therefore, the condition
$c_0 c_2<0$ gives exactly two
real roots of opposite signs that must be simple. Only the positive root is
accounted for. Moreover one
may construct perturbations with condition $c_0 c_2<0$. Hence the corollary
is proven \qed \enddemo
In \cite{5}, there are several examples of systems for which the isochronous
strata are known and an algebraic linearizing transformation is given
explicitly. Thus, there are many problems that can be solved using our
approach. For the sake of illustration, we choose to address the
bifurcations of limit cycles from Kukles isochrones.
\head 3. First order bifurcations from Kukles isochrones \endhead
We consider the reduced Kukles system in the form
$$
\aligned
\dot x=& -y\,,\\
\dot y=&x+a_1x^2+a_2xy+a_3y^2+a_4x^3+a_5x^2y+a_6xy^2,
\endaligned \tag{$\Cal K$}
$$
parametrized by $\lambda =(a_1,a_2,a_3,a_4,a_5,a_6)\in {\Bbb R}^6$, see
\cite{2,4}. From \cite{7}, the
following theorem gives the Kukles isochrone, and actually shows that it does
possess a birational
linearizing transformation as required.
\proclaim{Kukles isochrone} The origin is an isochronous center of
\thetag{$\Cal K$} if and only if the
system is linear or can be brought, through rescaling of $(x,y)$ and $t$ to the form
$$
\aligned
\dot x= & -y\,,\\
\dot y= & x+3 xy +x^3.
\endaligned \tag{$\Cal K_0$}
$$
Moreover, a rational linearizing change of coordinates of the system
\thetag{$\Cal K_0$} is given by
$$(u(x,y),v(x,y))=\left(\frac{x}{x^2+y+1},\frac{x^2+y}{x^2+y+1}\right).
\tag{$\Cal T_l$}$$
\endproclaim
\head First Order Perturbations\endhead
Consider a one-parameter cubic autonomous perturbation $(\Cal K_{\epsilon})$
of the Kukles nonlinear isochrone \thetag{$\Cal K_0$} in the form
$$
\aligned
\dot x =& -y+\epsilon p(x,y)\\
\dot y =& x+3xy+x^3+\epsilon q(x,y),
\endaligned \tag{$\Cal K_{\epsilon}$}
$$
where the parameter $\epsilon \in {\Bbb R}$, and $p$ and $q$ are polynomials
of degree 3. From the linearizing change of coordinates \thetag{$\Cal T_l$},
and by resetting
$(u(x,y),v(x,y))=(f^*(x,y),g^*(x,y)$, we derive the inverse transformation
$(\Cal T_l^{-1})$
$$
x(u,v)=f(u,v)=\frac{u}{1-v},\quad y(u,v)=g(u,v)=\frac{v-(u^2+v^2)}{(1-v)^2}\,.
\tag3-1
$$
The system \thetag{$\Cal K_\epsilon$} is transformed via \thetag{$\Cal T_l$}
into the weakly
linear system
$$
\aligned
\dot u=& -v+\epsilon \bar p(u,v)\,,\\
\dot v=& u+\epsilon \bar q(u,v)\,,
\endaligned \tag{$\bar \Cal K_\epsilon$}
$$
where we have
$$
\bar p(u,v)=f^*_x(u,v)p+f^*_y(u,v)q\,,\quad
\bar q(u,v)=g^*_x(u,v)p+g^*_y(u,v)q\,,
\tag3-2
$$
with
$$
\aligned
f^*_x(u,v)=& \frac{\partial f^*}{\partial x}(u,v)=1-2u^2-v
\quad \text{and } f^*_y(u,v)=\frac{\partial f^*}{\partial y}(u,v)=-u(1-v)\,,\\
g^*_x(u,v)=& \frac{\partial g^*}{\partial x}(u,v)=2u(1-v), \quad
\text{and } g^*_y(u,v)=\frac{\partial g^*}{\partial y}(u,v)=(1-v)^2\,.
\endaligned \tag3-3
$$
Calculation of the bifurcation function yields
$$\aligned
B(r)=&\int_0^{2\pi}{(\bar p(r \cos t,r \sin t)\cos t+\bar q(r \cos t,r \sin t)
\sin t)}dt\\
=&\sum_{i=1}^3{r^i \sum_{k=0}^i{(R_1^{ik} a_{i-k,k}+R_2^{ik} b_{i-k,k})}},
\endaligned
$$
with
$$\aligned
R_1^{ik}=& \int_0^{2\pi}{\frac{\cos^{i-k+1} t(\sin t-r)^k (1-2r^2+r \sin t)}
{(1-r \sin t)^{i+k}}}dt\\
R_2^{ik}=& \int_0^{2\pi}{\frac{\cos^{i-k} t(\sin t-r)^{k+1}}{(1-r \sin t)^{i+k-1}}}dt.
\endaligned \tag3-4
$$
For computational reasons, the expression of $B(r)$ is better expressed as
$$B(r)=(\bar R_1+\bar R_2+\bar R_3),
\tag3-5$$
with
$$\aligned
\bar R_1=& r (a_{10}R_1^{10}+a_{01}R_1^{11}+b_{10}R_2^{10}+b_{01}R_2^{11})\,,\\
\bar R_2=& r^2 (a_{20}R_1^{20}+a_{11}R_1^{21}+a_{02}R_1^{22}+b_{20}R_2^{20}+b_{11}R_2^{21}+b_{02}R_2^{22})\\
\bar R_3=& r^3 (a_{30}R_1^{30}+a_{21}R_1^{31}+a_{12}R_1^{32}+a_{03}R_1^{33}+b_{30}R_2^{30}+b_{21}R_2^{31}\\
& +b_{12}R_2^{32}+b_{03}R_2^{33}).
\endaligned \tag3-6
$$
Results from the theory of Residues were used to derive Equation (3-6).
The powers of $\cos t$ are odd in the cases
$$
(i,k)\in \cases \left\{(2,0);(3,1);(1,1);(2,2);(3,3)\right\}, &
\text{for }R_{1}^{ik}\\
\left\{(1,0);(2,1);(3,0);(3,2)\right\},& \text{for }R_{2}^{ik},
\endcases \tag3-7
$$
yielding zero integrals. Thus, we concentrate on the computation of
$R_{l}^{ik}$ with even powers of
cosine, that is,
$$
(i,k) \in \cases \left\{(1,0);(2,1);(3,0);(3,2)\right\} &\text{for }
R_{1}^{ij}\\
\left\{(2,0),(3,1),(1,1),(2,2),(3,3)\right\}& \text{for }R_{2}^{ij}\,.
\endcases \tag 3-8
$$
The integrals $R_1^{ik}$ and $R_2^{ik}$ are rational functions of $\sin t$.
Through the change of variable $\sin t=\frac{1}{2i}\left(z-\frac{1}{z}\right)$,
these integrals consist of terms of the form
$$
T_n^h=\frac{2^{n-h}}{r^n i^{1+h+n}}\int_{C}{\frac{a(z)}{b(z)}}dz,
\tag3-9$$
with
$$\aligned
a(z)=& z^{n-h-1}(z^2-1)^h \quad (\text{a polynomial in $z$ of degree $N=n+h-1$}),\\
b(z)=& (z-z_1)^n(z-z_2)^n\quad (\text{a polynomial in $z$ of degree $M=2n$})\\
=&(z^2-2\rho i z-1)^n,
\endaligned \tag3-10
$$
and $z_{1,2}=\mp (\sqrt{1-\rho^2}+\rho i)$ with $\rho=1/r$. We may
therefore apply the following well-known lemma, \cite{3}.
\proclaim{Lemma 3.1}
Let $a(z)=\sum_{k=0}^{N}{a_k z^{N-k}}$ and $b(z)=\sum_{k=0}^{N}{b_k z^{N-k}}$
be polynomials in $z$ of respective degrees $N$ and $M$, with $a_0\neq 0$,
$b_0 \neq 0$. Let $C$ be a simple closed contour enclosing all zeros of
$b(z)$. Then
$$
\int_{C}{\frac{a(z)}{b(z)}}\,dz=\cases
2i\pi\frac{a_0}{b_0}, & \text{if }M-N=1\,,\\
0, & \text{if }M-N \geq 2\,.\endcases
\tag3-11$$
\endproclaim
This lemma implies that
$$\aligned
T_n^h=& \cases 2\pi(-r)^{-n},& \text{if $n=h,$}\\
0, & \text{if $n\geq h+1$}. \endcases
\endaligned \tag3-12
$$
For the remaining cases, corresponding to $n<(h+1)$, we use the fact that
$$
\int_{C}{F_1(z)}dz=2i\pi {\underset {z=0} \to {\text{Res}}}(\frac{1}{z^2}F_1
\left(\frac{1}{z}\right))=\frac{(1-z^2)^h}{z^{h+1-n}(z^2-2i\rho z-1)^n}\,,
\tag3-13$$
with $F_1(z)=\frac{a(z)}{b(z)}$. Thus $z=0$ is a pole of order $m=h+1-n$.
The residue at this pole is computed by means of classical residue techniques.
We then prove the following theorem.
\proclaim{Theorem 3.2} Define $A_0$ and $A_2$ by
$$\gather
A_2= 4a_{10}+4b_{01}+2b_{20}-15a_{30}+4b_{21}+a_{11} \text{ and }\\
A_0= 2a_{10}+2b_{01}+4b_{20}+2b_{02}-8a_{11}-18a_{30}-6b_{21}-2a_{12}\,.
\endgather
$$
From $\gamma_0$, a periodic trajectory in the period annulus of the non-linear
isochrone $(\Cal K_0)$, one continuous family of limit cycles bifurcates in the
direction of the cubic perturbation $(p,q)$ if and only if the coefficients
satisfy the condition $A_0 A_2<0$. When this condition is met, this family
emerges exactly from the real simple positive root of the quadratic function
$$\Lambda(r):= A_0+A_2r^2.$$
Moreover, at most one such family of limit cycles emerge for fixed $(p,q)$.
\endproclaim
\demo{Proof}
Computation of the previous integrals in \thetag{3-9} yields
$$
T_n^h=\cases 2\pi(-r)^{-n}, \quad\text{if }
(n,h) \in \left\{(1,1);(3,3);(5,5)\right\}\,,&\\
0, \quad\text{if }(n,h) \in \left\{(1,0);(3,0);(3,1);(3,2);(5,0);(5,1);
(5,2);(5,3);(5,4)\right\}\,. &\endcases
\tag3-14$$
and, respectively, for $(n,h) \in \left\{(1,2);(1,3);(3,4);(3,5)\right\}$,
we have
$$
T_1^2=\frac{-2\pi}{r^2};\quad
T_1^3=(-2\pi)\frac{2-r^2}{r^3};\quad
T_3^4=\frac{-6\pi}{r^4};\quad
T_3^5=\pi\frac{(24-r^2)}{r^5}\,.
\tag3-15$$
Thus,
$$
B(r)=\left( \bar R_1+\bar R_2+\bar R_3 \right)=(\pi/\xi)\left(A_2r^2+A_0\right),
\tag3-16
$$
with
$$\aligned
A_2=& 4a_{10}+4b_{01}+2b_{20}-15a_{30}+4b_{21}+a_{11}\text{ and }\\
A_0=& 2a_{10}+2b_{01}+4b_{20}+2b_{02}-8a_{11}-18a_{30}-6b_{21}-2a_{12}\,.
\endaligned \tag3-17
$$
The branch points of periodic orbits are the positive roots of the quadratic
function $\Lambda (r):=A_2r^2+A_0=0$. Hence the result.
\qed\enddemo
\subhead Concluding remarks\endsubhead
The integral $B(r)$ is the first variation of the displacement function with
respect to the bifurcation parameter, and its simple zeros are the branch points
of periodic orbits. If it vanishes identically the higher variations have to be
computed and analyzed. How many variations are sufficient to make the final
conclusions about the limit cycles is highly nontrivial. That is the core
content of Bautin's result for quadratic systems that inspired \cite{1}.
Moreover, we wish to emphasize here that the method of {\it Isochrone
Reduction}, described above, might be applied with much success to various
isochronous strata (e.g. quadratic, symmetric cubic, Hamiltonian isochrones)
and in the more general case of Darboux linearizable systems. To our knowledge
it is the first application of Darboux linearization of isochronous centers to
the study of bifurcations of limit cycles.
\subhead Acknowledgments \endsubhead
We are very grateful to Professor C. Rousseau for our fruitful discussions.
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\enddocument