\input amstex
\documentstyle{amsppt}
\loadmsbm
\magnification=\magstephalf \hcorrection{1cm} \vcorrection{-6mm}
\nologo \TagsOnRight \NoBlackBoxes
\headline={\ifnum\pageno=1 \hfill\else%
{\tenrm\ifodd\pageno\rightheadline \else
\leftheadline\fi}\fi}
\def\rightheadline{EJDE--1999/35\hfil Limit cycles from polynomial isochrones
\hfil\folio}
\def\leftheadline{\folio\hfil B. Toni 
 \hfil EJDE--1999/35}

\def\pretitle{\vbox{\eightrm\noindent\baselineskip 9pt %
 Electronic Journal of Differential Equations,
Vol. {\eightbf 1999}(1999), No.~35, pp.~1--15.\hfil\break
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\hfill\break 
ftp ejde.math.swt.edu (login: ftp)\bigskip} }

\topmatter
\title
Higher order branching of periodic orbits from polynomial isochrones
\endtitle
\author B. Toni\endauthor
\thanks 
{\it 1991 Mathematics Subject Classifications:} 34C15, 34C25, 58F14, 58F21, 
58F30.\hfil\break\indent
{\it Key words and phrases:} Limit cycles, Isochrones, Perturbations, 
Cohomology Decomposition.
\hfil\break\indent
\copyright 1999 Southwest Texas State University  and
University of North Texas.\hfil\break\indent
Submitted August 29, 1999. Published September 20, 1999.
\endthanks

\address 
Dr. B. Toni  \newline 
Facultad de Ciencias \newline
Universidad Aut\'onoma Del Estado de Morelos \newline
Av. Universidad 1001, Col. Chamilpa \newline
Cuernavaca 62210, Morelos, Mexico. \newline
Tel: (52)(73) 29 70 20. Fax: (52) (73) 29 70 40. \newline
\endaddress
\email toni\@servm.fc.uaem.mx
\endemail

\abstract
We discuss the higher order local bifurcations of limit cycles from polynomial
isochrones (linearizable centers) when the linearizing transformation is
explicitly known and yields a polynomial perturbation one-form.  Using a method
based on the relative cohomology decomposition of polynomial one-forms
complemented with a step reduction process, we give an explicit formula for the
overall upper bound of branch points of limit cycles in an arbitrary $n$ degree
polynomial perturbation of the linear isochrone, and provide an algorithmic
procedure to compute the upper bound at successive orders.

We derive a complete analysis of the nonlinear cubic Hamiltonian isochrone and
show that at most nine branch points of limit cycles can bifurcate in a cubic
polynomial perturbation.  Moreover, perturbations with exactly two, three, four,
six, and nine local families of limit cycles may be constructed.
\endabstract
\endtopmatter

\document 
\heading 1. Introduction \endheading 
If a planar system with an annulus of periodic orbits is subjected to an 
autonomous polynomial perturbation, an interesting question is do any of the 
periodic orbits survive giving birth to limit cycles (isolated periodic orbits). 

In this paper we address this problem in the case of an isochronous 
annulus of periodic orbits (all orbits have the same constant period), and 
the unperturbed system is explicitly linearizable by a birational transformation of Darboux 
form, i.e. involving polynomial maps and their complex powers \cite{6}. The 
usual method for the perturbation is to use the Poincar\'e-Andronov-Melnikov 
integral of the perturbation one-form (divided if necessary by the integrating factor) 
along the closed orbits of the unperturbed system. In general such an integral 
is a transcendental function, and any question about its zeros is highly nontrivial. 

The approach in this paper as in \cite{10} is to apply an explicit linearizing transformation, and 
solve the perturbation problem in the new coordinates by reducing it to computing 
the integral of a rational one-form $R_1(u,v)du+R_2(u,v)dv$ over the family of 
concentric circles $u^2+v^2=r^2$. Using this idea a complete analysis at first order
has been given in \cite{10} for the linear isochrone under an arbitrary degree polynomial perturbation, 
and for the reduced Kukles system subjected to one-parameter arbitrary cubic polynomial perturbation. 
Here we discuss higher order perturbations, first for the linear isochrone at any order and then the more 
general case when the polynomial perturbation remains polynomial under the linearizing 
transformation. Our approach is based on the relative cohomology decomposition of polynomial one-forms 
\cite{9}. As an application we give a complete analysis for cubic planar Hamiltonian systems 
with an isochronous center subjected to one-parameter arbitrary cubic polynomial perturbation.

More precisely, consider an autonomous polynomial perturbation $(p,q)$ of a 
plane vector field in the form
$$
\Cal P_\epsilon :=(P(x,y)+\epsilon p(x,y))\frac{\partial}{\partial x}
+(Q(x,y)+\epsilon q(x,y))\frac{\partial}{\partial y},
\tag1-1$$
where 
$$\gather
P(x,y)=-y+\sum_{2\leq i+j \leq n}{P_{ij}x^{i}y^{j}},\quad 
Q(x,y)=x+\sum_{2\leq i+j \leq n}Q_{ij}x^{i}y^{j}\\
p(x,y)=\sum_{i=1}^{n}{\sum_{k=0}^{i}{p_{i-k,k}x^{i-k}y^{k}}},\quad 
q(x,y)=\sum_{i=1}^{n}{\sum_{k=0}^{i}{q_{i-k,k}x^{i-k}y^{k}}},
\endgather
$$
with $\lambda^n=(P_{ij},Q_{ij},p_{ij},q_{ij}, 1\leq i+j \leq n)$  the set of system
coefficients, and 
$\epsilon$ a small parameter. When
$\epsilon=0$, we assume further that the unperturbed vector field $(\Cal P_0)$ 
has an {\it isochronous period annulus} ${\Bbb A}$.

For fixed $\lambda^n$, 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 $\delta(r,\epsilon,\lambda^n)$ 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
$(\delta(r,\epsilon,\lambda^n),0)$ after time $T(r,\epsilon)$. Let $\Sigma =\{(x,0)\in U, x>0 \}$ 
denote the transversal or Poincar\'e section of $U$. By transversality and blowing 
up arguments the mapping $\delta$ is analytic. On $\Sigma$ we define the displacement function
$$
d(r,\epsilon,\lambda^n):=\delta(r,\epsilon,\lambda^n)-r=\sum_{i=1}^k
{d_i(r,\lambda^n)\epsilon^i}+O(\epsilon^{k+1}),
\tag1-2$$
where $d_i(r,\lambda^n)=\frac{1}{i!}\frac{\partial^i d(r,\epsilon,\lambda^n)}{\partial
\epsilon^i}|_{\epsilon=0}$.
The isolated zeros of $d(r,\epsilon,\lambda^n)$ correspond to limit cycles (isolated periodic orbits) 
of $(\Cal P_\epsilon)$ intersecting $\Sigma$.
In the period annulus ${\Bbb A}$, $d(r,0,\lambda^n)\equiv 0$.  We reduce the analysis to that of 
finding the roots of a suitable bifurcation function derived from the displacement function. For the higher 
order bifurcation analysis we need to determine $d_k(r,\lambda^n)$ under the assumptions that 
$d_j(r,\lambda^n)\equiv 0$ for $j<k$. 

Below in section two we describe our improved {\it isochrone reduction} method introduced in \cite{10} 
where we proved that to first order at most $\frac{n-1}{2}$ 
(resp. $\frac{n-2}{2}$) local families of limit cycles bifurcate from 
a polynomial perturbation of odd (resp. even) degree $n$ of the linear isochrone. In section three, using the 
relative cohomology decomposition of polynomial one-forms along with a so-called step-reduction process, we 
effectively compute the explicit formula for the maximum number of branch points of limit cycles in an 
arbitrary $n$ degree polynomial perturbation of the linear 
isochrone. This upper bound is three (resp. five) in a quadratic (resp. cubic) perturbation. An algorithmic  
construction for the upper bounds at successive orders is presented.
Section four addresses the cubic Hamiltonian isochrones. We show that, from these isochrones, at most nine 
local families of limit cycles bifurcate in a cubic polynomial perturbation.  Moreover, in all cases, one may 
construct in the usual way perturbations with the maximum number. As shown in \cite{10} each
limit cycle is asymptotic to a circle whose
radius is a simple positive zero of the bifurcation function.

\heading 2. Isochrone reduction \endheading
\smallskip
The isochrone reduction technique has been introduced in \cite{10}. We recall it for the sake
of 
completeness, and present here a partially generalized version. Consider $r_*\in \Sigma$ 
a simple zero of $d_1(r,\lambda^n)$. Thus, by the Implicit Function Theorem, there
exits a smooth function $r=r(\epsilon)$ defined in some neighborhood of
$\epsilon=0$ such that $r(0)=r_*$ and $d(r(\epsilon),\epsilon,\lambda^n) \equiv
0$.  The curve $r=r(\epsilon)$ corresponds to a local family of limit
cycles emerging from the periodic trajectory $\gamma(r_*)$ of the unperturbed
system which meets $\Sigma$ at $r_*$. For
$d_1(r,0,\lambda^n)\equiv 0$, or if one of the zeros is not simple, then
higher order derivatives must be computed.  Actually, in $\Bbb A$,
$\partial_{r}d(r,0,\lambda^n)=0$ for all values of $r$, and so we cannot apply the Implicit
Function Theorem.  However, from the perturbation of the Taylor series
$$
d(r,\epsilon,\lambda^n)=\epsilon d_{1}(r,0,\lambda^n)+O(\epsilon^2)=
\epsilon(d_{1}(r,0,\lambda^n)+O(\epsilon))=\epsilon 
B_{1}^n(r,\lambda^n), 
\tag2-1$$
with $B_1^n(r,\lambda^n):= d_{1}(r,0,\lambda^n)+O(\epsilon)$, we define a reduced 
displacement function by
$$B_{1}^n(r,\lambda^n):=d_1(r,\lambda^n),
\tag2-2$$
for small real values of $\epsilon$.  Clearly, if
$B_1^n(r(\epsilon),\lambda^n)\equiv 0$ then $d(r(\epsilon),\epsilon,\lambda^n) \equiv
0$ and the Implicit Function Theorem does apply to $B_1^n$. A simple zero $r_*$ of 
$B_1^n$ is called {\it a first order 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 $B_1^n(r,\lambda^n)$ of order $k$, 
then the corresponding perturbation Taylor series
$$d(r,\epsilon,\lambda^n)=\epsilon^k(d_k(r,\lambda^n)+O(\epsilon)):=
\epsilon^k B_k^n(r,\lambda^n)
\tag2-3$$
yields $B_k^n(r_*,\lambda^n)=0$ and $\partial_r B_k^n(r_*,\lambda^n)\neq 0$.
$B_k^n(r,\lambda^n)$ is the order $k$ 
bifurcation function. 
Similarly by the Implicit Function Theorem applied to $B_k^n$, there is a local family of
limit 
cycles emerging from $\gamma (r_*)$, whereas there are at most $m$ such local families for a
root 
$r_*$ of multiplicity $m$ following from the Weierstrass Preparation Theorem \cite{7}.

In the case of an isochronous period annulus the isochronal assumption
is essential to our approach for determining the order $k$ bifurcation function $B_k^n$ under
the 
assumptions $B_j^n(r,\lambda^n) \equiv 0$ for $j<k$. It is well known (see, e.g., \cite{6})
that
the origin of the unperturbed system \thetag{$\Cal P_{0}$} is isochronous if and
only if there exists an analytic change of coordinates
$$(\Cal T_l):\ (u(x,y),v(x,y))=(x + o(|(x,y)|),y + o(|(x,y)|))
\tag2-4$$
in its neighborhood, reducing the system to the linear isochrone 
$\Cal I_0=-y\partial_x+x\partial_y$. 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 $B_k^n$. In fact through \thetag{$\Cal T_l$}, \thetag{$\Cal
P_\epsilon$} is simplified to the weakly linear system
$$\aligned
\dot u=&-v+\epsilon \bar p(u,v)\\
\dot v=&u+\epsilon \bar q(u,v),
\endaligned \tag{$\bar \Cal P_\epsilon$}
$$
whose orbits are in correspondence with the solutions of the one-parameter family 
of differential one-forms on the plane
$$\bar \omega_{\epsilon}=dH+\epsilon\bar \omega,
\tag2-5$$
with $H(u,v)=\frac{1}{2}(u^2+v^2)$, and $\bar \omega (u,v)=\bar q(u,v) du
-\bar p(u,v) dv$.

The expression of the first order bifurcation function is recalled in 
the following theorem that we proved in \cite{10}.

\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 first order branch point of periodic orbits of \thetag{$\bar \Cal P_\epsilon$} is a 
simple zero of the function
$$B_1^n(r,\lambda^n):=\int_0^{2\pi}{\left (\bar p(r\cos t,r\sin t)\cos t+\bar q(r\cos t,r\sin
t)
\sin t \right)dt}, 
\tag2-6$$
where $r$ is taken in an interval of $(0,\infty)$.
\endproclaim
At this point we must note that the resulting perturbation one-form $\bar \omega$ is not necessarily 
polynomial.  Actually formula \thetag{2-6} is equivalent to a 
classic Poincar\'e formula (See \cite{3})
$$B_1^n(r,\lambda^n)=d_{1}(r,0,\lambda^n)=-\int_{\gamma (r)}{\bar \omega}
\tag2-7$$
also called the first Melnikov function $M_1(r)$, along the level line 
$\gamma (r):\quad H=r$.
\subheading{2.1 Higher order Perturbations of the isochrone}

In general such as in \cite{10}, under the linearizing transformation, the resulting perturbation 
$\bar \omega (u,v)=\bar q(u,v) du-\bar p(u,v) dv$ is not necessarily a polynomial. We consider here the 
particuliar case where $\bar \omega (u,v)$ is a polynomial of $u$ and $v$ of degree $n$, and make use of the relative cohomology 
decomposition of polynomial one-forms in the plane to analyze the higher order perturbations. It goes 
as follows.  

$\lambda^n$ denotes the set of $n^2+3n$ coefficients of the polynomial one-form $\bar \omega$. Assume the first order 
bifurcation function $B_1^n(r,\lambda^n)$ vanishes identically as a function of $r$, for a value $\lambda_1^n$ of 
$\lambda^n$. We then need to compute $B_2^n(r,\lambda_1^n):=d_2(r,\lambda_1^n)$, whose positive 
roots give the branch points at second order. The relative cohomology decomposition (See for instance 
\cite{3,9} ) states that 
for such a function $H(u,v)$ in \thetag{2-5}, and if 
$\int_{\gamma (r)}{\bar \omega}\equiv 0$,
then there are 
polynomials $g_1^n(u,v)$ and $R_1^n(u,v)$ such that
$$\bar\omega (u,v)=g_1^n(u,v)dH+dR_1^n(u,v).
\tag2-8$$
This leads to
$$B_2^n(r,\lambda_1^n)=\int_{\gamma (r)}{(g_1^n\bar \omega)}\quad \text{(modulo
$B_1^n(r,\lambda^n)\equiv 0$).}
\tag2-9$$
Thus similarly to formula \thetag{2-6}, it entails
$$B_2^n(r,\lambda_1^n)=\int_0^{2\pi}{\left [(g_1^n\cdot\bar p)(r\cos t,r\sin t)\cos
t+(g_1^n\cdot\bar q)(r\cos t,r\sin t)
\sin t \right]dt}.
\tag2-10$$
We recall  briefly the construction \cite{3}. Let $\gamma_{\epsilon}(r)$ be solution of 
$0=\bar\omega_{\epsilon}=dH+\epsilon \bar \omega$. From the definitions of $B_1^n$ and the displacement function, integrating over
$\gamma_{\epsilon}$ yields
$$\epsilon B_1^n(r,\lambda^n)+\epsilon\int_{\gamma (r)}\bar \omega=0\quad \text{(modulo $\epsilon^2$)}.
\tag2-11$$
That is 
$$B_1^n(r,\lambda^n)=-\int_{\gamma (r)}\bar \omega.
\tag2-12$$
Assume $B_1^n(r,\lambda^n)\equiv 0$ and \thetag{2-8}. Integrating over $\gamma_{\epsilon}$ the equality
$$(1-\epsilon g_1^n)(dH+\epsilon \bar \omega)=d(H+\epsilon R_1^n)-\epsilon^2 g_1^n \bar \omega 
\tag2-13$$
gives 
$$\epsilon^2 B_2^n(r,\lambda^n)+\epsilon^2 \int_{\gamma (r)} g_1^n \bar \omega=0\quad \text{(modulo $\epsilon^3$)}.
\tag2-14$$
Hence formula \thetag{2-9}. Inductively, given 
$$\aligned
B_k^n(r,\lambda_{k-1}^n)&=(-1)^k \int_{\gamma(r)}{(g_{k-1}^n\bar\omega)}\\
&=(-1)^k\int_0^{2\pi}{\left (g_{k-1}^n\cdot\left[\bar p(r\cos t,r\sin t)\cos t+\bar q(r\cos t,r\sin t)
\sin t\right ] \right)dt}
\endaligned
\tag2-15$$
if $B_k^n(r,\lambda_{k-1}^n)\equiv 0$ (as a function of $r$), there exist polynomials $g_k^n$ and $R_k^n$ such that 
$$g_{k-1}^n\bar\omega (u,v)=g_k^n(u,v)dH+dR_k^n(u,v),
\tag2-16$$
and therefore the $(k+1)th$ bifurcation function is given by
$$B_{k+1}^n(r,\lambda^n)=(-1)^{k+1} \int_{\gamma (r)}{(g_{k}^n\bar\omega)}.
\tag2-17$$
Consequently constructing the sequence of polynomials $g_i^n\in{\Bbb R}[u,v],i=1,\cdots,k$ yields the 
computation of the first nonzero identically bifurcation function $B_k^n(r,\lambda^n)$ whose
positive 
roots give the branch points of order $k$.
\subsubhead{2.1.1 Computing the relative cohomology decomposition}\endsubsubhead
\smallskip
The polynomial $g_1^n(u,v)$ in \thetag{2-8} is determined by the following.
\proclaim{Proposition 2.2}
Assume that $\bar p(u,v)$ and $\bar q(u,v)$ in \thetag{2-5} are polynomials in $u$ and $v$. 
If $B_1^n(r,\lambda^n)$ vanishes identically then the polynomial $g_1^n(u,v)$ such that
$$\bar \omega (u,v)=g_1^n(u,v)dH+dR_1^n(u,v)$$
is given by the partial differential equation
$$u\frac{\partial g_1^n(u,v)}{\partial v}-v\frac{\partial g_1^n(u,v)}{\partial u}= 
Div(\bar p,\bar q)(u,v),
\tag2-18$$
where $Div(\bar p,\bar q)$ is the divergence of $\bar p$ and $\bar q$.
\endproclaim
\demo{Proof}
The cohomology decomposition  
$$
\bar \omega (u,v)=g_1^n(u,v)dH+dR_1^n(u,v)=\bar q(u,v) du-\bar p(u,v) dv
\tag2-19$$
yields
$$d\bar\omega=dg_1^n\wedge dH=d\bar q \wedge du-d\bar p \wedge dv.
\tag2-20$$
This entails
$$\left(v\frac{\partial g_1^n}{\partial u}-u\frac{\partial g_1^n}{\partial v}\right)du\wedge dv=
-\left(\frac{\partial \bar p}{\partial u}+\frac{\partial \bar q}{\partial v}\right)du\wedge dv.
\tag2-21$$
Hence the claim.
The $kth, k \geq 1$ cohomology decomposition factor $g_k^n$ is computed the same way using the polynomials 
$g_{k-1}^n \bar p$ and $g_{k-1}^n \bar q$. 
\qed\enddemo
Consequently $g_1^n(u,v)$ is a polynomial of maximum degree $d=n-1$. The second  order bifurcation 
function is then  
$$ 
B_2^n(r,\lambda_1^n)=\int_0^{2\pi}{\left [g_1^n(r\cos t,r\sin t)(\bar p(r\cos t,r\sin t)\cos
t+
\bar q(r\cos t,r\sin t)\sin t) \right]dt}.
\tag2-22$$
This yields an algorithmic construction of the bifurcation function $B_k^n(r,\lambda^n)$
modulo 
$B_j^n(r,\lambda^n)\equiv 0$ for $j<k$. 
\remark{Remarks 2.3}
\roster 
\item As a consequence of the explicit decomposition \thetag{2-19} and the algorithmic 
construction,  we have $B_k^n(r,\lambda^n)\in {\Bbb R}[\lambda^n]$, i.e., the bifurcation function 
$B_k^n(r,\lambda^n)$ depends polynomially on the system coefficients $\lambda^n$.
\item This construction yields an increasing sequence of ideals generated by the polynomials 
$B_k^n$ in the Noetherian ring ${\Bbb R}[\lambda^n]$ of polynomials in $\lambda^n$.
\item By Hilbert's basis theorem the ideal $I_{\bar
\omega}=<B_1^n,B_2^n,\cdots,B_k^n,\cdots>$ of all the bifurcation 
polynomials is finitely generated, i.e., there 
exists a positive integer $\tau=\tau(n)$ such that $I_{\bar \omega}=I_{\tau(n)}=
<B_1^n,\cdots,B_{\tau (n)}^n>$. We call $I_{\tau(n)}$ the 
Bautin-like ideal associated to the polynomial perturbation $\bar \omega$.
\item Therefore whenever the resulting perturbation $\bar \omega$ is polynomial under the 
linearizing transformation, the relative cohomology decomposition allows to compute explicitly 
the Bautin-like ideal \cite{1} which contains all the informations for finding the bound $\Cal M^{\tau(n)}(n)$ 
to the number of limit cycles to be born to the origin in a perturbation of the isochrone.
\endroster
\endremark
For the sake of illustration, first we address the case of the linear isochrone. Next as an example of 
a nonlinear isochrone we discuss the cubic Hamiltonian isochrone. This isochronous system admits a 
linearization that preserves the polynomial perturbation allowing the use of 
the relative cohomology decomposition-based approach. 
\smallskip
\heading{3. Higher order Perturbations of the linear isochrone}\endheading

Consider a perturbation of degree $n$ of the linear isochrone in the form
$$
\Cal I_\epsilon :=(-y+\epsilon p(x,y))\frac{\partial}{\partial x}
+(x+\epsilon q(x,y))\frac{\partial}{\partial y},
\tag3-1$$
with $p(x,y)$ and $q(x,y)$ given in \thetag{1-1}, and the set of system coefficients 
$\lambda^n=(p_{ij},q_{ij}, 1\leq i+j \leq n)$. Computing the first 
order bifurcation function from \thetag{2-6} yields
$$
B_1^n(r,\lambda^n)=\sum_{i=1}^{n}{r^iC_i(\lambda^n)},
\tag3-2$$
where (terms of negative subindex assumed zero)
$$C_i(\lambda^n)=\sum_{k=0}^{i+1}{(p_{i-k,k}+q_{i-k+1,k-1})\int_0^{2\pi}{\cos t^{i-k+1}\sin t^{k}}dt}.
\tag3-3$$
Simplifying through the well-known rules $\int_0^{2\pi}\cos t^m \sin t^l dt=0$ for $m$ or $l$
odd we get
$$
C_i(\lambda^n)\equiv 0 \quad\text{(resp. $C_i(\lambda^n)\not\equiv 0$) for $i$ even (resp. odd).}
\tag3-4$$
Note that the coefficients $C_i(\lambda^n)$ are of degree one in the component of $\lambda^n$. They are also 
linearly independent. For instance
$$
C_1(\lambda^n)=\pi (p_{10}+q_{01});\quad
C_3(\lambda^n)=\frac{\pi}{4}(3p_{30}+p_{12}+q_{21}+3q_{03}).
\tag3-5$$
From \thetag{3-2} the branch points are the real positive roots $\rho=r^2$ of
$$\bar B_1^n(\rho,\lambda^n)=C_1(\lambda^n)+C_3(\lambda^n)\rho+\cdots+C_{2N+1}(\lambda^n)\rho^{N},
\tag3-6$$
where $N=\frac{n-2}{2}$ (resp. $\frac{n-1}{2}$) for $n$ even (resp. $n$ odd).
Hence the following theorems we proved in \cite{10}.
\proclaim{Theorem 3.1} 
To first order, no more than $\Cal M^1(n)=(n-1)/2$, (resp.  $(n-2)/2$) continuous families of limit 
cycles can bifurcate from the linear isochrone in the direction of any autonomous polynomial 
perturbation of degree $n$, for $n$ odd (resp.  even).  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
For $n=2$, (resp. $n=3$) we have
\proclaim{Corollary 3.2} 
No (resp. at most one)  continuous family of limit cycles bifurcates from the linear isochrone in the 
direction of the quadratic (resp. cubic )
autonomous perturbation $(p,q)$. In the cubic case the maximum number one is attained if and only if the
coefficients satisfy the condition $C_1(\lambda^3)\cdot C_3(\lambda^3)<0$, where $C_1(\lambda^3)$ and 
$C_3(\lambda^3)$ are given in \thetag{3-5}. In this instance, this family emerges from the
real positive simple 
roots of the function
$$\Delta (\rho,\lambda^3):=C_1(\lambda^3)+C_3(\lambda^3)\rho.
\tag3-7$$
\endproclaim
We now proceed to the higher orders and prove the following.
\proclaim{Theorem 3.3}
From the linear isochrone, to second order, no more than
$\Cal M^2(n)=n-2$ continuous families of limit cycles can bifurcate
in the direction of any autonomous polynomial perturbation of degree $n$ independently of the
parity of $n$.
These families emerge from the real positive simple
roots of the $(n-2)th$ degree polynomial equation  
$$\overline B_2^n(\rho,\lambda_1^n):=C_3(\lambda_1^n)+C_5(\lambda_1^n)\rho+\cdots+
C_{2n-1}(\lambda_1^n)\rho^{n-2}.
\tag3-8$$
Moreover we can construct small perturbations with the maximum number of limit cycles as
below.
\endproclaim
\demo{Proof}
First note that in \thetag{3-8} there are $\frac{n+1}{2}$ (resp. 
$\frac{n}{2}$) $C_i(\lambda^n)$ for $n$ odd (resp. $n$ even.) Let 
$\lambda_1^n=\lambda^n|_{C_i(\lambda^n)=0}$ 
the set of system coefficients $(p_{ij},q_{ij})$ such 
that, from \thetag{3-6}  
$$C_1(\lambda_1^n)=C_3(\lambda_1^n)=\cdots=C_{i}(\lambda_1^n)=\cdots=C_{2N+1}(\lambda_1^n)=0. 
\tag3-9$$
That is $B_1^n(r,\lambda_1^n)\equiv 0$. Important to our analysis is the fact that every equation 
$C_i(\lambda_1^n)=0$ allows to derive one system coefficient in terms of the remaining in its expression. 
Therefore we have
$$
\operatorname{card}(\lambda_1^n)=\cases n^2+3n-\frac{n+1}{2}=\frac{2n^2+5n-1}{2},&\text{for $n$ odd}\\ 
n^2+3n-\frac{n}{2}=\frac{2n^2+5n}{2},&\text{for $n$ even},\endcases
\tag3-10$$
where $\operatorname{card}(\lambda_1^n)$ is the number of components $p_{ij},q_{ij}$ in $\lambda_1^n$.
Using the relative cohomology decomposition we compute the $(n-1)th$ degree polynomial $g_1^n(x,y)$ 
by solving equation \thetag{2-18}. Take $g_1^n(x,y)$ as
$$g_1^n(x,y)=\sum_{i=1}^{n-1}{\sum_{k=0}^{i}{g^1_{i-k,k}x^{i-k}y^{k}}}.
\tag3-11$$
The coefficients $g^1_{i-k,k}=g^1_{i-k,k}(\lambda_1^n)$ are determined by the relation
$$(k+1)g^1_{i-k-1,k+1}-(i-k+1)g^1_{i-k+1,k-1}=(i-k+1)p_{i-k+1,k}+(k+1)q_{i-k,k+1}.
\tag3-12
$$
Set
$$\aligned
G_i(\lambda_1^n,t)&=\sum_{k=0}^i{g^1_{i-k,k}\cos^{i-k}t\sin^kt}\\
F_{i+1}(\lambda_1^n,t)&=\sum_{k=0}^{i+1}{(p_{i-k,k}+q_{i-k+1,k-1})\cos t^{i-k+1}\sin t^k},
\endaligned 
\tag3-13$$
and compute the second order bifurcation function using \thetag{2-10}. It entails
$$
B_2^n(r,\lambda_1^n)=\sum_{i=2}^{2n-1}{r^i C_i(\lambda_1^n)},
\tag3-14$$
with 
$$C_i(\lambda_1^n)=\sum_{k=1}^{i-1}{\int_{0}^{2\pi}{G_{i-k}(\lambda_1^n,t)
F_{k+1}(\lambda_1^n,t)dt}},
\tag3-15$$
terms of negative subindex are assumed zero, $G_j(\lambda_1^n,t)=0$ for $j>n-1$, and $F_j(\lambda_1^n,t)=0$ 
for $j>n+1$. Through the rules $\int_0^{2\pi}\cos t^m \sin t^l dt=0$ for $m$ or $l$ odd it
results
$$
C_i(\lambda_1^n)\equiv 0 \quad \text{(resp. $C_i(\lambda_1^n)\not\equiv 0)$, for $i$ even (resp. $i$ odd)}.
\tag3-16$$
In particular $C_2(\lambda_1^n)=0$, and $C_{2n-1}(\lambda_1^n)\not\equiv 0$, independently of the parity 
of $n$. Hence the claim. 
\qed \enddemo
We repeat the above outlined process in the following $S_j,j=1,\cdots,M_n$ steps after which we obtain the first non 
identically zero $B_{\tau}^n$ and derive the overall upper bound $\Cal M^{\tau}(n)$. This procedure is called 
the {\it Step Reduction Process.} We prove
\proclaim{Theorem 3.4}
\roster   
\item For $n$ odd (resp. $n$ even), the first odd (resp. even) integer $\tau=\tau(n)=M_n$
determined by  \thetag{3-20} (resp. \thetag{3-22}) yields
$B_{\tau-1}^n\not\equiv 0$ (resp. $B_{\tau}^n\not\equiv 0$).
\item At most
$$\Cal M^{\tau}(n)=\cases \frac{\tau n-(\tau+2)}{2},&\text{for $n$ odd}\\
\frac{\tau n-(\tau+3)}{2},&\text{for $n$ even}\endcases
$$
branch points of limit cycles bifurcate from the linear isochrone in a $n-$degree polynomial
perturbation.
\item At any arbitrary order $1\leq k\leq \tau$ the $kth$ order upper bound of limit cycles is
given by \thetag{3-18}.
\endroster
\endproclaim
\demo{Proof}
At every step $S_j$ we compute the relative cohomology decomposition factor $g_{k}^n$ which 
is a polynomial of degree $k(n-1)th$ for $k=j+1$. At the corresponding coefficients 
$\lambda_k^n|_{C_i(\lambda_{k-1}^n)=0}$, the number of bifurcation 
coefficients $C_i(\lambda_{k-1}^n)$ is
$$\operatorname{card}(C_i(\lambda_{k-1}^n))=\cases \frac{kn-k}{2},&\text{for $k$ odd, $n$ odd}\\
\frac{kn-(k+1)}{2},&\text{for $k$ odd, $n$ even}.\endcases
\tag3-17$$
we determine the $kth$ order bifurcation function $B_{k}^n(r,\lambda_{k-1}^n)$ that yields a
$kth$ order 
upper bound of branch points
$$\Cal M^k(n)=\cases \frac{kn-(k+2)}{2},&\text{for $k$ even and every $n;$ $k$ odd and $n$ odd.}\\
 \frac{kn-(k+3)}{2},&\text{for $k$ odd and $n$ even.}\endcases
\tag3-18$$
As above we derive some system coefficients in function of others in solving 
$C_i(\lambda_{k-1}^n)=0$. Finally, we know from remark \thetag{2.3} that the process must stop giving the 
overall upper bound. Recall that the coefficients $C_i(\lambda_{k-1}^n)$ are linearly
independent and polynomials of degree $k$ in the components of $\lambda_{k-1}^n$. After the
last $M_n$ step the number of remaining system coefficients is less or 
equal to the number of bifurcation coefficients $C_i(\lambda_{M_n}^n)$. Thus at least the last $C_i$ is 
necessarily nonzero yielding $B_{M_n}^n\not\equiv 0$, as illustrated in the quadratic and
cubic cases below. 
We next determine $M_n$. 
\roster
\item For $n$ odd, after $M_n$ steps, from \thetag{3-10}, we have 
$$\frac{2n^2+5n-1}{2}\leq \sum_{k=2}^{M_n}{\frac{kn-k}{2}}
\tag3-19$$
This leads to $M_n$ satisfying
$$M_n(M_n+1)\geq 4\frac{n^2+3n-1}{n-1}
\tag3-20$$
\item For $n$ even, it amounts to determining $\overline M_n=M_n/2$ such that
$$\frac{2n^2+5n}{2}\leq \sum_{k=2}^{\overline M_n}{\left(\frac{kn-k}{2}+\frac{(k+1)n-(k+2)}{2}\right)}.
\tag3-21$$
We get 
$$\overline M_n(\overline M_n+1)\geq \frac{2n^2+9n-6}{n-1}
\tag3-22$$
\endroster
Hence the result.\qed 
\enddemo

For example, for $n=2$ we have 
\proclaim{Corollary 3.5}
In a quadratic perturbation of the linear isochrone
\roster
\item The maximum number of continuous families of limit cycles which can bifurcate is three.
\item To first order, second order, and third order no limit cycles can bifurcate.
\item The number of continuous families of limit cycles which can bifurcate is at most one to fourth order
and fifth order, at most two to sixth order and seventh order, at most three to eighth order.
\endroster
\endproclaim
\demo{Proof}
The result is straightforward by taking $n=2$ in formulas \thetag{3-18} and \thetag{3-22}. We 
obtain $M_2\geq 8$. Thus $B_{8}^2\not\equiv 0$.
\qed \enddemo 
Item one in the above corollary confirms results in \cite{2, section 3.1, and Theorem 4.8} whereas 
items $2,3$ correct and improve concluding remarks in \cite{4}.
The case $n=3$ yields 
\proclaim{Corollary 3.6}
In a cubic perturbation of the linear isochrone
\roster
\item The maximum number of continuous families of limit cycles which can bifurcate is five.
\item The number of continuous families of limit cycles which can bifurcate is at most one to first order 
and second order,  at most two to third order, at most three to fourth order, at most four to fifth order, 
at most five to sixth order.
\endroster
\endproclaim
\demo{Proof}
The result follows from $n=3$ in formulas \thetag{3-18} and \thetag{3-20}. We get $M_3\geq 5.3$. 
Thus $B_{6}^3\not\equiv 0$. \qed \enddemo

Similar corollaries can be formulated for fourth, fifth, $\cdots$, nth order perturbation of the linear isochrone.
We now discuss the nonlinear isochrone case of the cubic polynomial Hamiltonian isochrones. Unlike the Kukles 
isochrone \cite{10}, it admits a polynomial linearizing transformation that preserves the polynomial nature of the 
perturbation one-form, allowing the use of the relative cohomology decomposition.
\smallskip
\heading 4. Cubic Hamiltonian Isochrones\endheading
\smallskip
Assuming the degenerate singularity on the $y-$axis without loss of generality, a cubic 
Hamiltonian system may be written as
$$
\aligned
\dot x=& -y-a_1x^2-2a_2xy-3a_3y^2-a_4x^3-2a_5x^2y\\
\dot y=&x+3a_6x^2+2a_1xy+a_2y^2+4a_7x^3+3a_4x^2y+2b_5xy^2,
\endaligned 
\tag{$\Cal H_3$}$$
with Hamiltonian function
$$
H(x,y)=\frac{x^2+y^2}{2}+a_6x^3+a_1x^2y+a_2xy^2+a_3y^3+a_7x^4+a_4x^3y+a_5x^2y^2.
\tag4-1$$
Marde\v si\'c et al have established the following characterization in 
\cite{8}.
\proclaim{Theorem 4.1}
The Hamiltonian cubic system \thetag{$\Cal H_3$} is Darboux linearizable if and only if 
it is of the form
$$
\aligned
\dot x=& -y-Cx^2\\
\dot y=& x+2Cxy+2C^2x^3.
\endaligned 
\tag{$\Cal H_i$}$$
This system is linearizable through the canonical change of coordinates
$$(u(x,y),v(x,y))=(x,y+Cx^2).
\tag{$\Cal T_l$}$$
\endproclaim
\subheading{4.1 First Order Perturbation}
\smallskip
Consider a cubic autonomous perturbation $(\Cal H_{\epsilon})$ of system 
\thetag{$\Cal H_i$}
$$
\aligned
\dot x =&-y - C x^2+\epsilon p(x,y)\\
\dot y =&x + 2C xy +2 C^2 x^3+\epsilon q(x,y),
\endaligned \tag{$\Cal H_{\epsilon}$}
$$
where, along with small values of the parameter $\epsilon \in {\Bbb R}$, and $C\neq 0$ we take
$$
p(x,y)=\sum_{i=1}^3{\sum_{k=0}^i{p_{i-k,k} x^{i-k}y^k}},\quad
q(x,y)=\sum_{i=1}^3{\sum_{k=0}^i{q_{i-k,k} x^{i-k}y^k}}.
\tag4-2
$$
The system coefficients set is $\lambda^3=(C,p_{ij},q_{ij},1\leq i+j\leq 3)$ with 
$\operatorname{card}(\lambda^3)=19$. The 
linearizing change of coordinates \thetag{$\Cal T_l$} transforms 
\thetag{$\Cal H_\epsilon$} into system 
$$
\aligned
\dot u=& -v+\epsilon \bar p(u,v)\\
\dot v=& u+\epsilon \bar q(u,v),
\endaligned \tag{$\bar\Cal H_\epsilon$}
$$
with
$$
\aligned
\bar p(u,v)=& \sum_{i=1}^3{\sum_{k=0}^i{p_{i-k,k}u^{i-k}
(v-Cu^2)^k}}=\sum_{i=1}^6{\sum_{k=0}^i{\bar p_{i-k,k}u^{i-k}v^k}}\\
=&p_{10}u+p_{01}v+(p_{20}-Cp_{01})u^2+p_{11}uv+p_{02}v^2+(p_{30}-Cp_{11})u^3+\\
&(p_{21}-2Cp_{02})u^2v+p_{12}uv^2+p_{03}v^3+(c^2p_{02}-Cp_{21})u^4-2Cp_{12}u^3v-\\
&3Cp_{03}u^2v^2+C^2p_{12}u^5+3C^2p_{03}u^4v-C^3p_{03}u^6,\\
\bar q(u,v)=& 2 C u \bar p(u,v)+ \sum_{i=1}^3{\sum_{k=0}^i{q_{i-k,k}
u^{i-k}(v-Cu^2)^k}}=\sum_{i=1}^7{\sum_{k=0}^i{\bar q_{i-k,k}u^{i-k}v^k}}\\
=&q_{10}u+q_{01}v+(2Cp_{01}+q_{20}-Cq_{01})u^2+(2Cp_{01}+q_{11})uv+q_{02}v^2+\\
&(2C(p_{20}-Cp_{01})+q_{30}-Cq_{11})u^3+(2Cp_{11}+q_{21}-2Cq_{02})u^2v+(2Cp_{02}+\\
&q_{12})uv^2+q_{03}v^3+(2C(p_{30}-Cp_{11})+C^2q_{02}-Cq_{21})u^4+(2Cp_{21}-\\
&4C^2p_{02}-2Cq_{12})u^3v+(2Cp_{12}-3Cq_{03})u^2v^2+2Cp_{03}uv^3+C^2(2Cp_{02}-2q_{21}+\\
&q_{12})u^5+C^2(-4p_{12}+3q_{03})u^4v-6C^2p_{03}u^3v^2+C^3(2p_{12}-q_{03})u^6+\\
&6C^3p_{03}u^5v-2C^4p_{03}u^7.
\endaligned \tag4-3
$$
Therefore the resulting one-form  $\bar \omega=\bar q du -\bar p dv$ is polynomial of degree 
$deg(\bar \omega):=max(deg(\bar p),deg(\bar q))=7$. Denoting $\overline \lambda^7$ the system coefficients 
set after linearization 
$\operatorname{card}(\bar \lambda^7)=\operatorname{card}(\lambda^3)=19$. We then prove the following.
\proclaim{Theorem 4.2}

From a periodic trajectory in the period annulus $\Bbb A$ of the
nonlinear isochrone $(\Cal H_i)$, at most two local families of limit cycles
bifurcate to first order in the direction of the cubic perturbation $(p,q)$.
Moreover there are autonomous perturbations with exactly $0\leq N_+\leq 2$ families of
limit cycles.
These families emerge from the real positive simple roots of the quadratic function
$$\Delta (\rho,\lambda^3):=C_1(\lambda^3)+C_3(\lambda^3)\rho+C_5(\lambda^3)\rho^2,
\tag4-4$$
with the coefficients $C_i(\lambda^3),i=1,3,5$ given below.
\endproclaim
\demo{Proof}
Computation of the first order bifurcation function 
$$
B_1^n(r,\lambda^3)=\int_0^{2\pi}{\left (\bar p(r\cos t,r\sin t)\cos t+\bar q(r\cos t,r\sin t)\sin t\right)}dt
\tag4-5
$$
gives
$$
B_1^n(r,\lambda^3)=r\left(C_1(\lambda^3)+C_3(\lambda^3)\rho+C_5(\lambda^3)\rho^2\right),
\tag4-6$$
with $\rho=r^2$,
$$\aligned
C_1(\lambda^3)&=\pi (p_{10}+q_{01});\quad
C_3(\lambda^3)=\frac{\pi}{4}\left (3(p_{30}+q_{03})+p_{12}+q_{21}-C(p_{11}+
2q_{02})\right);\\\
C_5(\lambda^3)&=\frac{\pi}{8}(p_{12}+3q_{03})C^2.
\endaligned
\tag4-7
$$
The upper bound $\Cal M^1(3)$ is clearly two, more accurate than $\Cal M^1(n)=\frac{n-1}{2}=3$ for 
$n=7$ one might predict from the previous section. 

A construction of small perturbations with an indicated 
number $N_{+}$ of families of limit cycles may be done using for instance Descartes rule of 
signs. We outline the technique, not really necessary for this quadratic case but effective for higher orders. Indeed 
denoting $\nu$ the number of sign changes in the sequence of coefficients of 
$\Delta (\rho)=C_1(\lambda^3)+C_3(\lambda^3)\rho+C_5(\lambda^3)\rho^2$, the 
number $N_{+}$ of positive zeros is such that $N_{+}-\nu =2k,\quad k \in {\Bbb N}$.
Therefore 
$$ \gathered
C_1(\lambda^3) \cdot C_3(\lambda^3)<0 
\text{ and }C_3(\lambda^3)\cdot C_5(\lambda^3) <0,\text{ we get }N_{+}=2
\text{ or }0\,,\\
C_1(\lambda^3) \cdot C_3(\lambda^3)<0 
\text{ and }C_3(\lambda^3)\cdot C_5(\lambda^3) >0, \text{ gives }N_{+}=1
\text{ or }0\,.\\
C_1(\lambda^3),\quad C_3(\lambda^3),\quad C_5(\lambda^3) \text{ of same sign, there
is no positive zeros.}
\endgathered \tag4-8
$$
The analysis is completed by the following lemma.
\proclaim{Lemma 4.3}

Let $s(x)$ be a real polynomial, $s\neq 0$, and let ${s_0(x),s_1(x),\dots,s_m(x)}$ be 
the sequence of polynomials generated by the Euclidean algorithm started with 
$s_0:=s(x);$ $s_1:=s'(x)$. Then for any real interval $[\alpha,\beta]$ such that 
$s(\alpha)\cdots s(\beta) \neq 0$, $s(x)$ has exactly $\nu (\alpha)-\nu (\beta)$ 
distinct zeros in $[\alpha,\beta]$ where $\nu (x)$ denotes the number of changes of 
sign in the numerical sequence $(s_0(x),s_1(x),\dots,s_m(x))$. Moreover all zeros of 
$s(x)$ in $[\alpha,\beta]$ are simple if and only if $s_m$ has no zeros in 
$[\alpha,\beta]$.
\endproclaim
For a detailed proof, see \cite{5, Theorem 6.3d}. Assume $C_5(\lambda^3) \neq 0$ for 
a more general treatment, and set
$$\Delta (\rho)=\rho^2+\alpha_2 \rho +\alpha_0,\quad \text{with 
$\alpha_0:=\frac{C_1(\lambda^3)}{C_5(\lambda^3)};\quad \alpha_2:=\frac{C_3(\lambda^3)}{C_5(\lambda^3)}$.}
\tag4-9$$
We derive the following Euclidean sequence (up to constant factors):
$$ \gathered
s_0(x)=\Delta (r), \text{ and } s_1(x)=\Delta '(r) \\
s_2(x)=-\frac{\alpha_2}{2}r^2-\alpha_0,\text{ and }s_3(x)=\beta r\\
s_4(x)=\alpha_0,
\endgathered \tag4-10
$$
with $\beta=\frac{-2\alpha_2^2+8\alpha_0}{\alpha_2}$. We further assume $\alpha_0 \neq 0$ 
and $\alpha_2 \neq 0$, i.e., $C_1(\lambda^3)$ and $C_3(\lambda^3)$ nonzero. At $x=0$ we obtain the 
sequence $(\alpha_0,0,-\alpha_0,0,\alpha_0);$ hence $\nu (0)=2$. At $\infty$, where the 
leading terms dominate, we get $(1,4,-\frac{\alpha_2}{2},\beta,\alpha_0)$. As a result, 
to make $N_{+}=2$, (resp. $1$) we must have $\nu (\infty)=0$ (resp. $1$). It amounts to 
taking all the terms $-\frac{\alpha_2}{2}$, $\beta$, and $\alpha_0$ positive. Then it 
suffices to realize  $C_1(\lambda^3) \cdot C_5(\lambda^3) >0$, $C_3(\lambda^3) \cdot C_5(\lambda^3) <0$ and 
$4 C_1(\lambda^3) 
\cdot C_5(\lambda^3) < C_3^2(\lambda^3)$. And respectively $C_1(\lambda^3)\cdot C_3(\lambda^3)<0$ and 
$C_3(\lambda^3) \cdot C_5(\lambda^3) <0$. 
Moreover for $\alpha_0 \neq 0$, $s_4(x)$ is constant; therefore all zeros made to 
appear by the previous construction are simple. 
\qed\enddemo
\remark{Remarks 4.4}
One may see the resulting system \thetag{$\bar \Cal H_\epsilon$} as a $7th$ degree perturbation of the
linear isochrone and use the formulas in the previous section to predict the successive upper
bound $\Cal M^k(7),k=1,2,3...$. Although the results are not incorrect, the bound obtained is not the 
best one. To obtain the most accurate upper bound one must consider the explicit expression of
each perturbation polynomial in the building up of the combined cohomology decomposition-step 
reduction process.

Indeed \thetag{$\bar\Cal H_\epsilon$} is not a typical $7th$ degree polynomial perturbation of 
the linear isochrone so as to literally apply the previous section. For such a perturbation 
$\operatorname{card}(\lambda^7)=70$, which yields a more complicated step-reduction procedure than do the actual 
$19$ coefficients.
\endremark
\bigskip
\subheading{4.2 Higher Order Perturbations}
\smallskip
Set $\lambda_1^3=\lambda^3|_{C_i(\lambda^3)=0,i=1,3,5}$ that is 
$$p_{10}+q_{01}=p_{12}+3q_{03}=3p_{30}+q_{21}-C(p_{11}+2q_{02})=0.
\tag4-11$$
Thus $B_1^3(r,\lambda_1^3)\equiv 0$. We then analyze the second order perturbation and obtain 
the following result.
\proclaim{Theorem 4.5}

At second order there is a choice of the relative cohomology
decomposition first factor leading to a maximum of three, and four continuous families of 
limit cycles bifurcating in the direction of the cubic perturbation $(p,q)$ of the nonlinear 
isochrone $(\Cal H_i)$.
\endproclaim
\demo{Proof}

The particular expression of the resulting polynomial perturbation 
$\bar\omega$ impose the search of a $5th$ degree first relative cohomology 
decomposition polynomial $g_1^3(u,v)$. From formula \thetag{3-12} we obtain
$$\aligned 
g_1^3(u,v)&=g^1_{10}u+g^1_{01}v+g^1_{20}u^2+g^1_{02}v^2+g^1_{21}u^2v+g^1_{03}v^3+g^1_{40}u^4\\
&+g^1_{22}u^2v^2+g^1_{04}v^4+g^1_{50}u^5+g^1_{05}v^5,
\endaligned
\tag4-12$$
with
$$\aligned
&g^1_{10}=-(p_{11}+2q_{02});\quad g^1_{01}=2p_{20}+q_{11};\quad g^1_{02}-g^1_{20}=p_{21}+q_{12}\\
&g^1_{21}=-2C(p_{21}+q_{12});\quad g^1_{03}=-4C(p_{21}+q_{12})=2g^1_{21}\\
&g^1_{22}=2g^1_{40}=2g^1_{04};\quad g^1_{50}=g^1_{05}.
\endaligned
\tag4-13$$
This expression of $g_1^3(u,v)$ is particularly interesting. It shows the non-uniqueness of the cohomology 
decomposition in this case. Indeed, whereas in \thetag{4-13} the coefficients $g^1_{10}$, $g^1_{01}$, $g^1_{21}$, $g^1_{03}$ are fixed in 
terms of the components of $\lambda^3$ we have multiple choices for $g^1_{02}$ and 
$g^1_{20}$. Moreover $g^1_{22}$, $g^1_{40}$, $g^1_{04}$, $g^1_{50}$, $g^1_{05}$ are arbitrary. Consequently 
we may consider the following possibilities for $g_1^3$.
\roster
\item A cubic polynomial $\bar g_1^3$ by making $g^1_{20}=g^1_{04}=g^1_{05}=0$.
\item A $4th$ degree $\tilde g_1^3$ with $g^1_{04}\neq 0;\quad g^1_{05}=0$.
\item A $5th$ degree $\hat g_1^3$ for $g^1_{05}\neq 0$.
\endroster 
Of course the upper bounds $\Cal M^k(3), k\geq 2$ vary accordingly. Indeed following the process outlined 
previously the second bifurcation function $B_2^3(r,\lambda_1^3)$ reduces to
$$
B_2^3(r,\lambda_1^3)=\sum_{i=3,i odd}^N{r^iC_i(\lambda_1^3)},
\tag4-14$$
where the bifurcation coefficients $C_i(\lambda_1^3)$ are computed as in \thetag{3-15}. We get respectively 
$N=11$, for the $5th$ and $4th$ degree polynomial $\bar g_1^3$ and  $\widetilde g_1^3$ yielding 
a $2nd$ order upper bound $\Cal M^2(3)=(N-3)/2=4$. Whereas for the cubic polynomial $\hat g_1^3$ we get 
$N=9$ leading to $\Cal M^2(3)=(N-3)/2=3$. 
\qed \enddemo
 
In the sequel we choose the "best" relative cohomology decomposition first factor $\hat g_1^3$ which 
we denote again $g_1^3$ for convenience, by assuming zero the arbitrary coefficients in \thetag{4-13}. 
We follow the step procedure of the previous section to analyze the higher orders. We obtain 
\proclaim{Theorem 4.5}
In a cubic perturbation of the nonlinear cubic Hamiltonian isochrone
\roster
\item To third order (resp. fourth order) at most six (resp. nine) continuous families of 
limit cycles can bifurcate.
\item The maximum number of branch points of limit cycles is nine.
\endroster
\endproclaim
\demo{Proof}
For $\lambda_2^3=\lambda_1^3|_{C_i(\lambda_1^3)=0,i=3,5,7,9}$, $\operatorname{card}(\lambda_2^3)=12$, and  
$B_2^3(r,\lambda_2^3)\equiv 0$. It yields the determination of a $8th$ degree relative cohomology 
decomposition second factor $g_2^3$. We then compute the third order bifurcation function $B_3^3(r,\lambda_2^3)$ 
and the bifurcation coefficients $C_i(\lambda_2^3),i=3,5,7,8,9,11,13,15$ as in \thetag{3-15}. This 
entails the third order upper bound $\Cal M^3(3)=6$.

The equations $C_i(\lambda_2^3)=0,i=3,5,7,8,9,11,13,15$ yield a coefficient set 
$\lambda_3^3=\lambda_2^3|_{C_i(r,\lambda_2^3)=0,i=3,5,7,8,9,11,13,15}$ such that 
$B_3^3(r,\lambda_3^3)\equiv 0$, and $\operatorname{card}(\lambda_3^3)=6$. This leads to compute a $14th$ degree cohomology 
decomposition factor $g_3^3$, and ten bifurcation coefficients 
$C_i(\lambda_3^3),i=3,\cdots,21; odd$. It entails a $4th$ order bifurcation function non identically zero. We 
obtain the $4th$ order upper bound $\Cal M^4(3)=9$ as claimed.
\qed \enddemo
\bigskip
\head 5. Concluding Remarks \endhead
\smallskip
The relative cohomology decomposition of polynomial one-forms complemented with the step reduction procedure 
described above provides a useful technique for the investigation of higher order branching of periodic 
orbits of polynomial isochrones when the linearization preserves the polynomial characteristic of the 
perturbation. It yields a complete analysis of an arbitrary $n-$degree polynomial perturbation of the 
linear isochrone, and the nonlinear cubic Hamiltonian isochrone, by providing an explicit formula for any 
order bifurcation function, as well as for the overall upper bound $\Cal M(n)$ of the branch points of 
limit cycles, i.e, the finite number of the generators of the corresponding Bautin-like ideals. 

A similar technique might be obtained when the resulting perturbation after linearization is rational. 

\bigskip
\head Acknowledgment \endhead
\smallskip
We are very much grateful to J.P. Fran\c coise for fruitful discussions, as well as to I.D. Iliev whose review 
pointed out a gap in the proof of Theorem 3.2 in \cite{10}, and allowed the following corrections therein: There exist perturbations of the nonlinear 
Kukles isochrone with exactly three branch points of limit cycles at first order.

We are also very much grateful to the referee. His insightful comments help improve substantially the 
exposition of the paper.
\bigskip
\Refs
\widestnumber\key{1}
\ref \no1 \by N.N. Bautin
\paper On the number of limit cycles which appear with the variation of the 
coefficients from an equilibrium point of focus or center type
\jour Amer. Math. Transl. series 1
\vol 5 \yr 1962 \pages 396-413
\endref
\widestnumber\key{2}
\ref \key 2 \by C. Chicone, M. Jacobs
\paper Bifurcation of Limit Cycles from Quadratic Isochrones
\jour J. of Differential Equations \vol 91 \yr 1991 \pages 268-327
\endref

\widestnumber\key{3}
\ref \key 3 \by J.P. Fran\c coise
\paper Successive derivatives of a first return map, application to the study of quadratic 
vector fields
\jour Ergod. Th. \& Dynam. Sys. \vol 16 \yr 1996 \pages 87-96
\endref

\widestnumber\key{4}
\ref \key 4 \by H. Giacomini, L. Llibre, M. Viano
\paper On the nonexistence, existence and uniqueness of limit cycles
\jour Nonlinearity \vol 9 \yr 1996 \pages 501-516
\endref
\widestnumber\key{5}
\ref \key 5 \by P. Henrici
\book Applied and Computational Analysis
\publ John Wiley and Sons 
\yr 1974
\endref

\widestnumber\key{6}
\ref \key 6 \by P. Marde\v si\'c, C. Rousseau, B. Toni 
\paper Linearization of Isochronous centers 
\jour J. Differential Equations 
\vol 121\yr 1995\pages 67-108
\endref

\widestnumber\key{7}
\ref \key 7 \by J. Poenaru
\book Analyse Differentielle
\publ Springer Lectures Notes in math \vol 371
\yr 1974
\endref
\widestnumber\key{8}
\ref \key 8 \by P. Marde\v si\'c, L. Moser-Jauslin, and C. Rousseau,  
\paper Darboux Linearization and Isochronous centers with rational first integral
\jour J. of Differential Equations
\vol 134 \yr 1997 \pages 216-268
\endref
\widestnumber\key{9}
\ref \key 9 \by M. Sebastiani
\paper Preuve d'une conjecture de Brieskorn
\jour Manuscripta math.
\vol 2 \yr 1970 \pages 301-308
\endref
\widestnumber\key{10}
\ref \key 10 \by B. Toni
\paper Branching of periodic orbits from Kukles Isochrones
\jour Electronic Journal of Differential Equations
\vol 1998 \yr 1998 \pages No. 13 pp. 1-10
\endref

\endRefs

\enddocument