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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 86, pp. 1--5.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}
\begin{document}
\title[\hfilneg EJDE-2017/86\hfil Limit cycles for BiLi\'enard systems]
{Center conditions and limit cycles for BiLi\'enard systems}
\author[J. Gin\'e \hfil EJDE-2017/86\hfilneg]
{Jaume Gin\'e}
\address{Jaume Gin\'e \newline
Departament de Matem\`atica, Inspires Research Centre,
Universitat de Lleida,
Avda. Jaume II, 69,
25001 Lleida, Catalonia, Spain}
\email{gine@matematica.udl.cat}
\thanks{Submitted January 18, 2016. Published March 27, 2017.}
\subjclass[2010]{34C05, 37C10}
\keywords{Center problem; analytic integrability; Gr\"obner bases;
\hfill\break\indent polynomial BiLi\'enard differential systems;
decomposition in prime ideals}
\begin{abstract}
In this article we study the center problem for polynomial BiLi\'enard systems
of degree $n$. Computing the focal values and using Gr\"obner bases we
find the center conditions for such systems for $n=6$. We also establish
a conjecture about the center conditions for polynomial BiLi\'enard systems
of arbitrary degree.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}
\allowdisplaybreaks
\section{Introduction and statement of main results}\label{s1}
The so-called Li\'enard equation $\ddot{x}+f(x) \dot{x} + g(x)=0$ with
where $f(x)$ and $g(x)$ are polynomials, which we rewrite as a differential
system in the plane
\begin{equation}\label{lie}
\dot{x}= y, \quad \dot{y}= -g(x) - y f(x),
\end{equation}
arises frequently in the study of various mathematical models of physical,
chemical, biology and other areas.
We assume that the singular point is at the origin $g(0)=0$ and which is
nondegenerate $g'(0)>0$.
By means of the Li\'enard transformation $y \mapsto y + F(x)$, where
$F(x)=\int_0^x f(x) dx$, system \eqref{lie} becomes
\begin{equation}\label{lie2}
\dot{x}= y-F(x), \quad \dot{y}= -g(x).
\end{equation}
The centers of system \eqref{lie2} are orbitally reversible, that is,
are symmetric with respect to an analytic invertible transformation and
a scaling of time followed by a reversion of time, see \cite{C,CLP,G}.
We recall that system \eqref{lie2} has a center at the origin if all its
solutions in a neighborhood of the origin are closed. The center
problem consists in finding necessary and sufficient conditions
over $F$ and $g$ to have a center at the origin.
In fact the original system studied by Li\'enard was with $g(x)=x$,
see \cite{L}. Li\'enard equations were intensely studied as they
can be used to model oscillating circuits in vacuum tube technology,
see for instance \cite{G}. Moreover other equations may be reduced to
Li\'enard equations, see \cite{GL}.
In this work we study a family of polynomial systems
which is a generalization of the original Li\'enard system, and
corresponds to systems of the form
\begin{equation}\label{BL}
\dot{x}=-y+F(x), \quad \dot{y}=x+G(y),
\end{equation}
where $F(x)$ and $G(y)$ are polynomials without constant and
linear terms. These systems are called {\it BiLi\'enard systems},
see \cite{GT}. In \cite{GS} the center problem has been studied
when $F(x)$ and $G(y)$ are polynomials of fourth degree and
it was shown that all the centers are time-reversible.
We recall that a system is time-reversible if it is invariant under
the symmetry $(x,y,t) \mapsto (-x,y,-t)$ or
$(x,y,t) \mapsto (x,-y,-t)$.
Furthermore, there are families of centers for $F(x)$ and $G(y)$ of arbitrary degree,
see \cite{GT}. In \cite{GP} the authors classify all centers of the
family of the BiLi\'enard systems of degree five and find the maximum
number of limit cycles which can bifurcate from a fine focus for such systems.
In the following theorem we classify all centers of system \eqref{BL}
when $F(x)$ and $G(y)$ are polynomials of degree six.
\begin{theorem}\label{thm1}
Consider the differential system
\begin{equation}\label{sys1}
\begin{gathered}
\dot{x}=-y+F(x)=-y+a_2x^2+a_3x^3+a_4x^4+a_5x^5+a_6 x^6, \\
\dot{y}= \ x+G(y)=x+b_2y^2+b_3y^3+b_4y^4+b_5y^5+ b_6 y^6,
\end{gathered}
\end{equation}
where $a_i$ and $b_i$ are real numbers. The origin is a center if,
and only if, one of the following cases holds:
\begin{itemize}
\item[(a)] $a_2=a_3=a_4=a_5=a_6=b_3=b_5=0$;
\item [(b)] $b_3=-a_3$, $b_2=\pm a_2$, $b_4= \pm a_4$, $b_5=-a_5$ and $b_6=\pm a_6$;
\item [(c)] $a_3=a_5=b_2=b_3=b_4=b_5=b_6=0$.
\end{itemize}
Moreover, all centers at the origin are time-reversible.
\end{theorem}
The determination of the center conditions allows to study the small-amplitude
limit cycles which can bifurcate from the origin of perturbations of such systems,
see for instance \cite{DHZ,G1} and references therein.
For system \eqref{sys1} we have the following result.
\begin{proposition}\label{prop1}
The maximum number of small--amplitude limit cycles which can
bifurcate from the origin of system \eqref{sys1} is at least eight.
\end{proposition}
Theorem \ref{thm1} and Proposition \ref{prop1} are proved in section \ref{s.2}
and \ref{s.3} respectively.
From the results presented in this work we can establish the following conjecture
\begin{conjecture}\label{c1}
All the centers of system \eqref{BL} are time-reversible and given by the
following families
\begin{itemize}
\item[(i)] $F \equiv 0$ and $G(x)=G(-x)$;
\item [(ii)] $G \equiv 0$ and $F(x)=F(-x)$;
\item [(iii)] $F(x)=-G(x)$;
\item [(iv)] $F(x)=G(-x)$.
\end{itemize}
\end{conjecture}
Moreover the result should carry over to the case where $F$ and $G$ are
analytic functions. In the first case system \eqref{BL} is invariant
by the symmetry $(x, y, t) \mapsto (x, -y,-t)$.
In the second case system \eqref{BL} is invariant by the symmetry
$(x, y, t) \mapsto (-x, y,-t)$.
In fact these first two cases are classical Li\'enard families with a center. The
last two cases are centers because they are invariant by the symmetry
$(x, y, t) \mapsto (y, x,-t)$.
Cases (a) and (c) of Theorem \ref{thm1} correspond to case (i) and (ii)
of Conjecture \ref{c1}, respectively.
Case (b) of Theorem \ref{thm1} corresponds to the cases (iii) and (iv)
of Conjecture \ref{c1}.
\section{Proof of Theorem \ref{thm1}}\label{s.2}
First we determine the necessary conditions for having a center.
These necessary conditions can be determined by different methods,
see \cite{GS,RS}. We use here the method developed by Poincar\'e of
construction of a formal first integral.
To construct this first integral we will use polar coordinates
$x=r \cos \theta$ and $y= r \sin \theta$.
So we transform system \eqref{sys1} through this change of variables
and we propose the Poincar\'e series
\[
H(r, \theta)= \sum_{m=2}^{\infty} H_m(\theta) r^m,
\]
where $H_2(\theta)=1/2$ and $H_m(\theta)$ are homogeneous
trigonometric polynomials in $\theta$ of degree $m$.
We suppose that the transformed system \eqref{sys1} has
this power series as a formal first integral, i.e.,
\[
\dot{H}(r,\theta)= \frac{\partial H}{\partial r} \dot{r}
+ \frac{\partial H}{\partial \theta} \dot{\theta}
= \sum_{k=2}^{\infty} V_{2k} r^{2k}.
\]
Here $V_{2k}$ are the \emph{focal values} which are polynomials in the parameters
of system \eqref{sys1}.
The first nonzero focal value is $V_4=a_3 + b_3$. The next nonzero focal value is
\begin{align*}
V_6 =& -195 a_2^2 a_3 + 30 a_5 + 12 a_2^3 b_2 + 44 a_4 b_2 - 133 a_3 b_2^2 \\
&- 12 a_2 b_2^3 - 205 a_2^2 b_3 - 123 b_2^2 b_3 - 44 a_2 b_4 + 30 b_5.
\end{align*}
The size of the next focal values increases greatly hence we do not present
them explicitly here. The reader can easily compute these next focal values.
The Hilbert Basis theorem assures that the ideal $J=\langle V_4,V_6,\dots \rangle$
generated by the
focal values is finitely generated. This implies the existence of
$v_1,v_2,\dots ,v_k$ such that $J=\langle v_1,v_2,\dots ,v_k\rangle$. This
set of generators is a basis of $J$ and the conditions $v_j=0$
for $j=1,\ldots,k$ provide a finite set
of necessary conditions to have a center for system \eqref{sys1}.
In practice we compute a certain number of focal values thinking that inside this
number there is the set of generators. Let $J_i$ be the ideal generated only by the first $i-1$ focal values,
i.e., $J_{i}=\langle V_4, \ldots,V_{2i}\rangle$.
Next we decompose this algebraic set into its
irreducible components using the computer algebra system Singular \cite{Sh8}.
The computational tool used is the routine minAssGTZ \cite{primdec_lib}
which is based on the Gianni-Trager-Zacharias algorithm \cite{Gianni}.
Note that if for system \eqref{sys1} $a_6 \ne 0$, then by a linear
transformation we can take $a_6=1$.
Using this observation and in order to simplify calculations, we split
system \eqref{sys1} into two system considering separately the cases:
\[
(\alpha): a_6=1, \quad (\beta): a_6=0\,.
\]
For the case $(\alpha)$ the decomposition of the ideal $J_9$
given by $J_9=\langle V_4,V_6,\dots ,V_{18}\rangle$
consist of 3 components defined by the following prime ideals:
\begin{itemize}
\item[(1)] $ \langle a_3,a_5,b_2,b_3,b_4,b_5,b_6 \rangle $,
\item[(2)] $ \langle a_2+b_2,a_3+b_3,a_4+b_4,a_5+b_5,1+b_6 \rangle$,
\item[(3)] $\langle a_2-b_2,a_3+b_3,a_4-b_4,a_5+b_5,1-b_6 \rangle$,
\end{itemize}
We were not able to compute the decomposition over the field of rational
numbers because of the complexity of the computations.
Hence we use modular arithmetics. In fact the decomposition is obtained
over the field of characteristic $32003$. We have chosen this prime
number because the computations are relatively fast using this prime.
As we have used modular arithmetics we must check if the decomposition is
complete and no component is lost.
To do that we use the algorithm developed in \cite{RP}.
Let $P_i$ denote the polynomials defining each component.
Using the instruction \emph{intersect} of Singular we compute the
intersection $P=\cap_i P_i= \langle p_1, \dots, p_m \rangle$.
By the Strong Hilbert Nullstellensatz (see for instance \cite{RS})
to check whether $V(J_j)=V(P)$ it is sufficient to check if the
radicals of the ideals are the same, that is, if $\sqrt{J_j}=\sqrt{P}$.
Computing over characteristic $0$ reducing Gr\"obner bases of ideals
$\langle 1-w V_{2k},P:V_{2k}\in J_j\rangle$ we find that each of them is $\{1\}$.
By the Radical Membership Test this implies that $\sqrt{J_j}\subseteq \sqrt{P}$.
To check the opposite inclusion, $\sqrt{P}\subseteq \sqrt{J_j}$
it is sufficient to check that
\begin{equation} \label{rm2}
\langle 1-wp_k, J_j: p_k \text{ for } k=1,\dots, m \rangle= \langle 1 \rangle.
\end{equation}
Using the Radical Membership Test to check if \eqref{rm2} is true, we were
able to complete computations working in the field of characteristic zero
so we know that the decomposition of the center variety is complete.
For the case $(\beta)$ the obtained decomposition of the ideal $J_9$ consist
of 4 components defined by the following prime ideals:
\begin{itemize}
\item[(1)] $ \langle a_3,a_5,b_2,b_3,b_4,b_5,b_6 \rangle $,
\item[(2)] $ \langle a_2+b_2,a_3+b_3,a_4+b_4,a_5+b_5,b_6 \rangle$,
\item[(3)] $\langle a_2-b_2,a_3+b_3,a_4-b_4,a_5+b_5,b_6 \rangle$,
\item[(4)] $ \langle a_2,a_3,a_4,a_5,b_3,b_5 \rangle $,
\end{itemize}
This decomposition is also obtained using modular arithmetics so proceeding
as in the previous case we can check that this decomposition is complete.
In this case this is also true.
The sufficiency is derived from the results presented in the previous section.
\section{Proof of Proposition \ref{prop1}} \label{s.3}
To find the maximum number of small-amplitude limit cycles which can bifurcate
from the origin we use the method of finding a fine focus of maximum order,
see for instance \cite{GS}. From our calculations it is easy to see that if
$a_2=b_2=a_3+b_3=a_5+b_5=a_6+b_6=0$ then $V_4=V_6=V_8=0$ and $V_{10}$
takes the form
\[
V_{10}=(a_4 + b_4) (379 a_3 a_4 + 398 a_6 - 379 a_3 b_4).
\]
We vanish this focal value taking $a_6 -379a_3(a_4 - b_4)/398$ and
$V_{12}$ becomes
\[
V_{12}=(a_4 - b_4) (a_4 + b_4)(445561 a_5-3104010 a_3^2).
\]
Taking $a_5=3104010 a_3^2/445561$ we have $V_{12}=0$ and $V_{14}$ reads for
\[
V_{14}=(a_4 - b_4) (a_4 + b_4) (10770211123227 a_3^3 - 775833091250 a_4 b_4).
\]
Now we made the reparametrization $a_3 = z^{1/3}$ and we can vanish $V_{14}$
taking $z=775833091250 a_4 b_4 /$ $ 10770211123227$.
In this case $V_{16}$ and $V_{18}$ take the form
\begin{gather*}
\begin{aligned}
V_{16}&= (a_4 - b_4) (a_4 b_4)^{1/3} (a_4 + b_4) (68732087591790148677 a_4^2 \\
&\quad - 298114693011794424032 a_4 b_4 + 68732087591790148677 b_4^2),
\end{aligned} \\
\begin{aligned}
V_{18}&= (a_4 - b_4) (a_4 b_4)^{2/3} (a_4 + b_4)
(7226530034982884356352004477 a_4^2 \\
&\quad + 13348721106142735246693837622 a_4 b_4 \\
&\quad + 7226530034982884356352004477 b_4^2).
\end{aligned}
\end{gather*}
We can vanish $V_{16}$ taking one of the two reals roots of the quadratic polynomial
and under this assumption $V_{18}$ is different from zero if
$a_4b_4\neq 0$ and $a_4\neq \pm b_4$, and therefore we obtain a fine focus
of order eight for the BiLi\'enard system \eqref{sys1}.
\subsection*{Acknowledgments}
The author is partially supported by a MINECO/FEDER grant
number MTM2014-53703-P and an AGAUR (Generalitat de
Catalunya) grant number 2014SGR 1204.
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\end{document}