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\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 $j$ 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)}=
$. 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
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\enddocument