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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 140, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/140\hfil Existence of periodic solution]
{Existence of periodic solution for perturbed
generalized Li\'enard equations}

\author[I. Boussaada, A. R. Chouikha\hfil EJDE-2006/140\hfilneg]
{Islam Boussaada, A. Raouf Chouikha}  % in alphabetical order

\address{Islam Boussaada \newline
LMRS, UMR 6085, Universite de Rouen,
Avenue de l'universit\'e, BP.12,
76801 Saint Etienne du Rouvray, France}
\email{islam.boussaada@etu.univ-rouen.fr}

\address{A. Raouf Chouikha \newline
Universite Paris 13 LAGA,
Villetaneuse 93430, France}
\email{chouikha@math.univ-paris13.fr}

\date{}
\thanks{Submitted May 15, 2006. Published November 1, 2006.}
\subjclass[2000]{34C25}
\keywords{Perturbed systems; Li\'enard equation; periodic solution}

\begin{abstract}
 Under conditions of Levinson-Smith type, we prove the existence
 of a $\tau$-periodic solution for the perturbed generalized Li\'enard equation
 $$
 u''+\varphi(u,u')u'+\psi(u)=\epsilon\omega(\frac{t}{\tau},u,u')
 $$
 with periodic forcing term. Also we deduce sufficient condition for existence
 of a periodic solution for the equation
 $$ u''+\sum_{k=0}^{2s+1} p_k(u){u'}^k=\epsilon\omega(\frac{t}{\tau},u,u').
 $$
 Our method can be applied also to the equation
 $$
 u''+[u^2+(u+u')^2-1]u'+u=\epsilon\omega(\frac{t}{\tau},u,u').
 $$
 The results obtained are illustrated with numerical examples.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem*{remark}{Remark}
\newtheorem{proposition}[theorem]{Proposition}

\section{Introduction}

 Consider Li\'enard equation
\[
  u''+\varphi(u)u'+\psi(u)=0
\]
where $u'=\frac{du}{dt}$, $u''=\frac{d^2u}{dt^2}$, $\varphi$  and
$\psi$ are $C^{1}$.
Studying the existence of periodic solution of period $\tau_0$ has been
purpose of many authors:  Farkas \cite{F1} presents some typical
works on this subject, where the Poincar\'e-Bendixson theory plays
a crucial role.
In general, a periodic perturbation of the Li\'enard equation does
not possess a periodic solution as described by  Moser; see for example \cite{C}.

Let us consider the perturbed Li\'enard equation
\[
u''+\varphi(u)u'+\psi(u)=\epsilon\omega(\frac{t}{\tau},u,u')
\]
where $\omega$ is a \emph{controllably periodic perturbation} in the
Farkas sense; i.e., it is periodic with a period $\tau$ which can be choosen
appropriately. The existence of a non trivial periodic solution for (2) was
studied by Chouikha \cite{C}. Under very mild conditions it is proved that
to each small enough amplitude of the perturbation there belongs a one
 parameter family of periods $\tau$ such that the perturbed system has a
unique  periodic solution with this period.

 Let us consider now the following generalized
 Li\'enard equation, which is ``a more realistic assumption
in modelling many real world phenomena'' as stated in \cite[page 105]{F1}
 \begin{equation}\label{e1}
 u''+\varphi(u,u')u'+\psi(u)=0.
 \end{equation}
Where  $\varphi$  and  $\psi$ are $C^{1} $ and satisfy some assumptions
that will  be specified below.
 The leading work of investigation for the existence of periodic
solution of generalized Li\'enard systems
was established by Levinson-Smith \cite{L-S}.
 Let us define conditions $C_{LS}$.
\smallskip

\noindent {\bf Definition.}
The functions $\varphi$ and  $\psi$ satisfy the condition $C_{LS}$ if:
 $x\psi(x)>0$  for $|x|>0$,
 $$
\int_0^{x} \psi(s) ds =\Psi(x) \quad \text{and} \quad
\lim_{x\to +\infty} \Psi(x) = + \infty , \quad \varphi(0,0) < 0.
$$
Moreover, there exist some numbers $0 < x_0 < x_1$ and $M>0$ such that:
\begin{gather*}
 \varphi(x,y) \geq 0 \quad \text{for } |x|\geq x_0,\\
\varphi(x,y)\geq -M \quad \text{for } |x|\leq  x_0\\
x_1 > x_0,\quad \int_{x_0}^{x_1} \varphi(x,y(x))dx\geq
 10 M x_0
\end{gather*}
for every decreasing function $y(x)>0$.




\begin{proposition}[Levinson-Smith \cite{L-S}] \label{prop1}
When the functions $\varphi$ and  $\psi$ are of class $C^1$ and satisfy
condition $C_{LS}$ then the generalized Lienard equation \eqref{e1}
 has at least one non-constant $\tau_0$-periodic solution.
\end{proposition}

A non trivial solution will be denoted $u_0(t)$, and its period $\tau_0$.
This proposition has many improvements (under weaker hypotheses)
due to Zheng and Wax Ponzo; see \cite{F1},  among other authors.

 This article is organized as follows:
At first, we prove the existence of a periodic solution for the
 perturbed generalized Li\'enard equation
\begin{equation}\label{l1}
u''+\varphi(u,u')u'+\psi(u)=\epsilon\omega(\frac{t}{\tau},u,u'),
\end{equation}
Where  $t,\epsilon,\tau\in \mathbb{R}$ are such that
$|\tau-\tau_0|<\tau_1<\tau_0$, $|\epsilon|<\epsilon_0$ with
$\epsilon_0\in \mathbb{R}$ sufficient small and $\tau_1$ is a
fixed real scalar.
We will use the Farkas method which
was effective for perturbed Li\'enard equation.
In the third section, we will propose a criteria for the existence
 of periodic solution for
\begin{equation}\label{l2}
u''+\sum_{k=0}^{2s+1} p_k(u){u'}^k=\epsilon\omega(\frac{t}{\tau},u,u'),
\end{equation}
with $ s\in \mathbb{N} $ and $p_k$ are $C^1$ functions, for all $k\leq2s+1$.
In the second part of the section,  using a result of De Castro \cite{De}
 we will prove uniqueness of a periodic solution for the equation
\begin{equation}\label{l4}
u''+[u^2+(u+u')^2-1]u'+u=0.
\end{equation}
Sufficient condition for the existence of periodic solution to
\begin{equation}\label{l3}
u''+[u^2+(u+u')^2-1]u'+u=\epsilon\omega(\frac{t}{\tau},u,u').
\end{equation}
will be found.
At the end of the paper, some phase plane examples are given in order
to illustrate the above results. In particular, we describe
uniqueness of a solution for equation \eqref{l4} and the existence of
 a solution of equation \eqref{l3} for
 $\omega(\frac{t}{\tau},u,u') = (\sin 2t) \ u'$.

\section{Periodic solution of perturbed generalized Lienard equation}

In this part of this paper we  prove the existence of
periodic solution of the perturbed generalized Lienard equation \eqref{l1}
 such that  the unperturbed one \eqref{e1} has at least one periodic solution.
The method of proof that we will employ was described in \cite{C,F1}.

Consider the equation \eqref{e1}
We assume that $\varphi$ and $\psi$ are $C^1$ and satisfy $C_{LS}$.
Then by Proposition \ref{prop1} there exists at least a non trivial periodic
solution denoted $u_0(t)$.

Let the least positive period of the solution $u_0(t)$ be denoted
by $\tau_0$ and $U$ be an open subset of $\mathbb{R}^2$ containing $(0,0)$.
These notation will be used in the rest of the paper.

\begin{theorem}\label{monpre}
Let $\varphi$ and $\psi$ be $C^1$ and satisfy $C_{LS}$.
Suppose $1$ is a simple characteristic multiplier of the variational system
associated to \eqref{e1}.
 Then there are two real functions
$\tau,h$ defined on $U\subset\mathbb{R}^2$ and constants $\tau_1<\tau_0 $
 such that the periodic solution
$\nu(t,\alpha,a+h(\epsilon,\alpha),\epsilon,\tau(\epsilon,\alpha))$
  of the equation
 $$
u''+\varphi(u,u')u'+\psi(u)=\epsilon\omega(\frac{t}{\tau},u,u'),
$$
 exists for $(\epsilon,\alpha)\in U $, $|\tau-\tau_0|<\tau_1$,
 $\tau(0,0)=\tau_0$ and $h(0,0)=0$.
\end{theorem}

We point out that the characteristic multipliers are the eigenvalues of
the characteristic  matrix which is the fundamental matrix in the
time $\tau_0$.


\begin{proof}[Proof of Theorem \ref{monpre}]
Following the method used in \cite{F1}, we set
$x_2=u$ , $x_1=\frac{du}{dt}=u'$
and note $x=\mathop{\rm col}(x_1,x_2)= \mathop{\rm col}(u',u)$.
The plane equivalent system of \eqref{e1} is
\begin{equation}\label{l5}
x'=f(x)\Longleftrightarrow \left\{\begin{array}{l}
x'_1=-\varphi(x_2,x_1)x_1-\psi(x_2)\\
x'_2=x_1   \end{array}\right.
 \end{equation}
with
$$
f(x)=\mathop{\rm col}(-\varphi(x_2,x_1)x_1-\psi(x_2),x_1).
$$
Then the system  (\ref{l5}) has the periodic solution $q(t)$ with period
$\tau_0$.
We define
$$
q(t)=\mathop{\rm col}({u_0}'(t),u_0(t))
$$
and therefore
 $$
q'(t)=\mathop{\rm col}(-\varphi(u_0(t),{u_0}'(t)){u_0}'(t)-\psi(u_0(t)),{u_0}'(t)) .
$$
The variational system  associated with (\ref{l5}) is
\begin{equation}\label{l6}
y'={f_x}'(q(t))y,
\end{equation}
Without loss of generality, we take the initial conditions
 $$
t=0 ,\quad u_0(0)=a <0 \quad\text{and}\quad
u_0'(0)=0
$$
Hence ${f_x}'(q(t))$ is the matrix
\[\begin{pmatrix}
 -{\varphi'}_{x_1}(u_0(t),{u_0}'(t)){u_0}'(t)
 -\varphi(u_0(t),{u_0}'(t))
&-{\varphi'}_{x_2}(u_0(t),{u_0}'(t)){u_0}'(t)-\psi'(u_0(t))\\
1&0\end{pmatrix}
\]
Notice that
$q'(t)=\mathop{\rm col}(-\varphi(u_0(t),{u_0}'(t)){u_0}'(t)-\psi(u_0(t),
{u_0}'(t))$
 is the first solution of the variational system. Now
 we calculate the second one, denoted by
$\widehat{y}(t)=\mathop{\rm col}(\widehat{y}_1(t),\widehat{y}_2(t))$
and linearly independent with $q'(t)=y(t)$, in order to write the
fundamental matrix.
 Consider
$$
I(s)=\exp\Big[-\int_0^s({\varphi'}_{x_1}(u_0(\rho),{u_0}'(\rho)){u_0}'(\rho)+
 \varphi(u_0(\rho),{u_0}'(\rho)))d\rho\Big]
$$
and
\begin{align*}
\pi(t)&=-\int_0^t \big(\varphi(u_0(\rho),{u_0}'(\rho)){u_0}'(\rho)
+\psi(u_0(\rho))\big)^{-2} \big({\varphi'}_{x_2}(u_0(t),{u_0}'(t)){u_0}'(t)\\
&\quad +\psi'(u_0(t))\big)I(\rho)d\rho
\end{align*}
 We then obtain
\begin{gather*}
\widehat{y}_1(t)=-[\varphi(u_0(t),{u_0}'(t)){u_0}'(t)+\psi(u_0(t)]\pi(t),\\
\widehat{y}_2(t)={u_0}'(t)\pi(t)+\pi'(t)
 \frac{\varphi(u_0(t),{u_0}'(t)){u_0}'(t)+\psi(u_0(t)}
 {{\varphi'}_{x_2}(u_0(t),{u_0}'(t)){u_0}'(t)+\psi'(u_0(t))}
\end{gather*}
 It is known, \cite{C,F1}, that the fundamental matrix satisfying
$\Phi(0)=Id_2 $ is
$\Phi(t)$ equals to
\[
\begin{pmatrix}
\frac{\varphi(u_0(t),{u_0}'(t)){u_0}'(t)+\psi(u_0(t))}{\psi(a)}
&\psi(a)\pi(t)[\varphi(u_0(t),{u_0}'(t)){u_0}'(t)+\psi(u_0(t)]
     \\
-\frac{{u_0}'(t)}{\psi(a)}&-\psi(a)u'_0(t)\pi(t)-\psi(a)\pi'(t)
     \frac{\varphi(u_0(t),{u_0}'(t)){u_0}'(t)+\psi(u_0(t))}
{{\varphi'}_{x_2}(u_0(t),{u_0}'(t)){u_0}'(t)+\psi'(u_0(t))}
\end{pmatrix}
\]
Thus,
$$
\Phi(\tau_0)= \begin{pmatrix}1&{\psi(a)}^2
\pi(\tau_0)\\
0&\rho_2\end{pmatrix}.
$$
We use the Liouville's formula
 $$
\det\Phi(t)=\det\Phi(0)\exp\int_0^t \mathop{\rm Tr} ({f_x}'(q(\tau)))d\tau.
$$
 Since $\det(\Phi(0))=1$,
 we deduce the characteristic multipliers associated with (\ref{l6}):
$\rho_1=1$ and
 $\rho_2=I(\tau_0)=\exp\Big[-\int_0^{\tau_0}({\varphi'}_{x_1}(u_0(\rho),
{u_0}'(\rho)){u_0}'(\rho)+
 \varphi(u_0(\rho),{u_0}'(\rho)))d\rho\Big]$.

 From \cite{F1}, we have:
$$
J(\tau_0)=-Id_2+\begin{bmatrix}-\psi(a)&0
\\0&0\end{bmatrix}+\Phi(\tau_0)
$$
Hence we obtain the jacobian matrix
$$
J(\tau_0)=\left(\begin{array}{cc}-\psi(a)&{\psi(a)}^2 \pi(\tau_0)\\
0&\rho_2-1\end{array}\right),
$$
Since $1$ is a simple characteristic multiplier $(\rho_2\neq1)$,
$\det J(0,0,0,\tau_0)\neq 0 $.
We define the periodicity condition
\begin{equation}\label{lz} z(\alpha,h,\epsilon,\tau)
:=\nu(\alpha+\tau,a+h,\epsilon,\tau)-(a+h)=0.
 \end{equation}
By the Implicit Function Theorem there are ${\epsilon}_0>0$ and
${\alpha}_0>0$ and uniquely
determined functions $\tau$ and $h$ defined on
 $U=\{(\alpha,\epsilon)\in{\mathbb{R}}^{2}:|\epsilon|< \epsilon_0,|\alpha|<{\alpha}_0\}$
 such that: $\tau,h\in C^1$, $\tau(0,0)=T_0,h(0,0)=0$ and
 $z(\alpha,h,\epsilon,\tau)\equiv0$.
 Because of (\ref{lz}), the periodic solution of \eqref{l1} has  period
$\tau(\epsilon,\alpha)$   near $T_0$ and has  path near the path of the
unperturbed solution.
 \end{proof}

 In particular if $\rho_2<1$, the periodic solution is orbitally
asymptotically stable i.e. stable
 in the Liapunov sense and it is attractive see  \cite[page 346]{F1}.
 Thus, the following inequality is a criteria of the existence of orbital
  asymptotical stable periodic solution of the equation \eqref{l1}.
\begin{equation}\label{l7}
\rho_2<1\Longleftrightarrow
  \int_0^{\tau_0}({\varphi'}_{x_1}(u_0(\rho),{u_0}'(\rho)){u_0}'(\rho)+
 \varphi(u_0(\rho),{u_0}'(\rho)))d\rho>0.
 \end{equation}
Using Proposition \ref{prop1}, we conclude the existence of
non trivial periodic solution for perturbed generalized Li\'enard equation.

 \section{Results on the periodic solutions}

\subsection*{Special case}
Let us now consider the equation
\begin{equation}\label{l8}
u''+\sum_{k=0}^{2s+1} p_k(u){u'}^k=0.
\end{equation}
Let $p_k$ be $C^1$ function, for all$ k\leq2s+1$ for $ s\in \mathbb{N} $.
This is a special case of Li\'enard equation with
$p_0(u)=\psi(u)$  and
$$
\varphi(u,u')=\sum_{k=1}^{2s+1} p_k(u){u'}^{k-1} .
$$
We will suppose $\varphi$ and $\psi$ verify $C_{LS}$ conditions. Let
 $U $ be an open subset of $ \mathbb{R}^2$ containing $(0,0)$.
The associated perturbed equation, as denoted previously (\ref{l2}),
is equation
$$ u''+\sum_{k=0}^{2s+1}
p_k(u){u'}^k=\epsilon\omega(\frac{t}{\tau},u,u').
$$

\begin{remark} \rm
The last non-zero term of the finite sum $\sum_{k=0}^{2s+1} p_k(u){u'}^k$
has an odd index.
Then it is necessary to have the element $x_0 \neq 0$ in the $C_{LS}$
conditions.
\end{remark}

\begin{theorem}\label{monsec}
Let $\varphi$ and $\psi$ be  $C^1$ and satisfy $C_{LS}$.
If $1$ is a simple characteristic multiplier of the variational system
associated to (\ref{l8})  then there are two functions
$\tau,h :U\to R$ and constants $\tau_1<\tau_0 $
 such that the periodic solution
$\nu(t,\alpha,a+h(\epsilon,\alpha),\epsilon,\tau(\epsilon,\alpha))$
  of the equation
 $$
u''+\sum_{k=0}^{n} p_k(u){u'}^k=\epsilon\omega(\frac{t}{\tau},u,u')
$$
 exists for $(\epsilon,\alpha)\in U $ with $|\tau-\tau_0|<\tau_1$,
$\tau(0,0)=\tau_0$ and $h(0,0)=0$.
\end{theorem}

\begin{proof}
We will use the same method  as  in the existence theorem
 for non-trivial periodic solution of the perturbed system.
Consider the unperturbed equation to compute some useful elements.
First we assume that $2s+1=n$, to simplify  notation.
Let $x_2=u$ and $x_1=\frac{du}{dt}=u'$.
The equivalent plane system of (\ref{l8}) is
\begin{equation}\label{l9}
x'=f(x)\Longleftrightarrow\left\{\begin{array}
{l}x'_1=-\sum_{k=0}^{n} p_k(x_2){x_1}^k\\
x'_2=x_1        \end{array}\right.
 \end{equation}
with
$$
f(x)=\mathop{\rm col}(-\sum_{k=0}^{n} p_k(x_2){x_1}^k,x_1).
$$
Let $q(t)=\mathop{\rm col}(u'_0(t),u_0(t))$ the periodic solution
of (\ref{l9}).
The variational system associated to (\ref{l9}) is
$$
y'=f'_x(q(t))y
$$
with the periodic solution
 $$
q'(t)=\mathop{\rm col}(-\sum_{k=0}^{n} p_k(u_0)(t){u_0'}^k(t) , u'_0(t)),
$$
hence
$$
f'_x(q(t))=\begin{pmatrix}
-\sum_{k=1}^{n} k p_k(u_0(t)){u'_0(t)}^{k-1}
&-\sum_{k=0}^{n} p'_k(u_0(t)){u_0'(t)}^k\\
1&0\end{pmatrix}.
$$
We assume the initial values:
$$
t=0 ,\quad u_0(0)=a <0\quad\text{and}\quad {u_0}'(0)=0.
$$
 Then $q(0)=\mathop{\rm col}(0,a)$ and
$q'(0)=\mathop{\rm col}(-\psi(a),0)$.

In same way as the previous section we compute the fundamental
matrix associated with (\ref{l9}), denoted $ \Phi(t)$.
Determine the second vector solution (linearly
independent with $q'(t)=y(t)$).
A trivial calculation described in \cite{C,F1} gives us the second solution
denoted $\widehat{y}(t)$,
hence $\Phi(t)=(\frac{y(t)}{y(0)},y(0)\widehat{y}(t))$.
For that consider
$$
I(s)=\exp\Big[-\int_0^s(\sum_{k=1}^{n} k
p_k(u_0(\rho)){u'_0(\rho)}^{k-1})d\rho\Big],
$$
and denote as in the previous section
$$
\pi(t)=-\int_0^t(\sum_{k=0}^{n}
p_k(u_0)(\rho){u_0'}(\rho)^k)^{-2}(\sum_{k=0}^{n}
p'_k(u(t)){u'}^k(t))I(\rho)d\rho.
$$
Sine $\widehat{y}(t)=\mathop{\rm col}(\widehat{y}_1(t),\widehat{y}_2(t))$,
where
\begin{gather*}
\widehat{y}_1(t)=-(\sum_{k=0}^{n} p_k(u_0)(t){u_0'}(t)^k)\pi(t),\\
\widehat{y}_2(t)=u'_0(t)\pi(t)+\pi'(t)\frac{\sum_{k=0}^{n}
p_k(u_0)(t){u_0'}^k(t)}
{\sum_{k=0}^{n} p'_k(u_0(t)){u'_0(t)}^k}.
\end{gather*}
Hence the fundamental matrix associated with our variational system is
$$
\Phi(t)=\begin{pmatrix}
\frac{\sum_{k=0}^{n}p_k(u_0)(t){u_0'}^k(t)}{\psi(a)}
&\psi(a)(\sum_{k=0}^{n} p_k(u_0)(t){u_0'}(t)^k)\pi(t)
     \\
-\frac{{u_0}'(t)}{\psi(a)}&-\psi(a)u'_0(t)\pi(t)-\psi(a)\pi'(t)
     \frac{\sum_{k=0}^{n} p_k(u_0)(t){u_0'(t)}^k}
{\sum_{k=0}^{n} p'_k(u_0(t)){u_0'(t)}^k}
\end{pmatrix}.
$$
We deduce the principal matrix (the fundamental one with
$t=\tau_0$).
  $$
\Phi(\tau_0)=
\begin{pmatrix}
1&{\psi(a)}^2 \pi(\tau_0)\\
0&\rho_2\end{pmatrix}.
$$
By the Liouville's formula, we have the characteristic multipliers
 $\rho_1=1$ and
\begin{align*}
\rho_2&=\det(\Phi(\tau_0))\\
&=\exp\Big(\int_{0}^{\tau_0} (Tr{f_x}'(q(\tau))d\tau\Big)\\
&= \exp\Big(-\int_{0}^{\tau_0}\sum_{k=1}^{n} k
p_k(u_0(\tau)){u'_0(\tau)}^{k-1} )d\tau\Big)
\end{align*}
  Then we define the  equivalence (\ref{l7}):
\begin{equation}\label{la}
\rho_2 <1\Longleftrightarrow
  \int_{0}^{\tau_0}\Big(\sum_{k=1}^{n} k p_k(u_0(\tau)){u'_0(\tau)}^{k-1} \Big)
d\tau>0
\end{equation}
and the associated Jacobian matrix is
$$
J(\tau_0)=\begin{pmatrix}-\psi(a)&{\psi(a)}^2 \pi(\tau_0)\\
0&\rho_2-1\end{pmatrix}.
$$
\end{proof}

\subsection*{Uniqueness of the periodic solution for an unperturbed equation}
Let us consider now equation \eqref{l4}:
$$
u''+[u^2+(u+u')^2-1]u'+u=0,
$$
which is a special case of generalized Li\'enard equation with
$$
\varphi(u,u')=(u^2+(u'+u)^2-1)\ and \ \psi(u)=u.
$$
We will prove existence and uniqueness of non trivial periodic solution
 for equation \eqref{l4}. Existence will be ensured by $C_{LS}$ conditions
and for proving  uniqueness we use a De Castro's result \cite{R-S-C}
(see also \cite{De}).

\begin{proposition}[De Castro \cite{C}] \label{prop2}
Suppose the following system has at least one periodic orbit
\begin{gather*}
y'=-\varphi(x,y)y-\psi(x)\\
x'=y.
 \end{gather*}
Then under the following two assumptions:
\begin{itemize}
 \item[(a)] $\psi(x)=x$;
\item[(b)] $\varphi(x,y)$ increases, when $|x|$ or $|y|$ or the both
increase
\end{itemize}
this periodic orbit is unique.
\end{proposition}

Let us verify that \eqref{l4} satisfies the above assumptions:
Equation \eqref{l4} is satisfied if and only if
\begin{equation}\label{lb}
\begin{gathered}
 u''+\sum_{k=0}^{3} p_k(u){u'}^k=0,\\
p_0(u)=\psi(u)=u, \quad p_1(u)=2u^2-1, \quad p_2(u)=2u, \quad
p_3(u)=1.
\end{gathered}
\end{equation}
Also if and only if
 \begin{equation}\label{lc}
\begin{gathered}
u''+\varphi(u,u')u'+\psi(u)=0,\\
\varphi(u,u')=(u^2+(u'+u)^2-1), \quad \psi(u)=u.
\end{gathered}
\end{equation}
Clearly, the assumptions of Proposition \ref{prop2} are satisfied.
In the following, we firstly verify conditions $C_{LS}$.
In that case the equation
 $$
u''+\varphi(u,u')u'+\psi(u)=0$$
has at least a non trivial periodic
solution.
It is easy to see that $\psi(u)=u$ satisfies
\begin{gather*}
x \psi(x)>0\quad \text{for } |x|>0,\\
\int_0^{x} \psi(s) ds =\Psi(x) ,\quad
 lim_{x\to +\infty} \Psi(x) = + \infty
\end{gather*}
Now we have $\varphi(0,0)=-1<0$.
By taking $x_0=1$, $M=1$, we have
\begin{gather*}
 \varphi(x,y)\geq 0 \quad\text{for } |x|\geq x_0,\\
\varphi(x,y)\geq -M \quad \text{for } |x|\leq  x_0\,.
\end{gather*}
The following calculation gives us the optimal value of $x_1>x_0 $.
Let
\begin{align*}
 H&=\int_{x_0}^{x_1} \varphi(x,y) dx \\
&= \int_{1}^{x_1}[x^2+(x+y)^2-1]dx\\
&=\int_{1}^{x_1}[2x^2+2xy+y^2-1]dx \\
&=\Big[\frac{2}{3}x^3+x^2y+x(y^2-1)\Big]_{1}^{x_1}\\
&=(x_1-1)(\frac{{x_1}^2-2x_1+1}{6}
+2(\frac{x_1+1}{2})^2+2y(\frac{x_1+1}{2})+(y^2-1))\\
&=(x_1-1)\Big(\frac{{x_1}^2-2x_1+1}{6} +\varphi(\frac{x_1+1}{2},y)\Big)
\end{align*}
Since  $\frac{x_1+1}{2}\geqslant x_0=1$, using the inequality
$\varphi(x,y)\geq 0$ for $|x|\geq x_0$,
 we  obtain $H\geqslant\frac{(x_1-1)^3}{6}$.
Hence, if $\frac{(x_1-1)^3}{6}=10Mx_0=10$, then
$x_1=1+(60)^\frac{1}{3}$ which satisfies
$$
x_1 > x_0 ,\quad \int_{x_0}^{x_1} \varphi(x,y) \, dx
 \geq 10 M x_0 ,
$$
for every decreasing function $y(x)>0$.


\subsection*{Existence of periodic solution for perturbed equation
satisfying $C_{LS}$}
In the following we are dealing with the existence of periodic
solution for the equation (\ref{l3}).
We assume the initial values:
$$
t=0 ,\quad u_0(0)=a <0, \quad {u_0}'(0)=0.
$$

\begin{theorem}\label{montro} \label{thm3}
Suppose $1$ is a simple characteristic multiplier of the variational
system associated to \eqref{l4}.
 Then there are two functions $\tau,h :U\to R$ and constants $\tau_1<\tau_0 $
 such that the periodic solution
$\nu(t,\alpha,a+h(\epsilon,\alpha),\epsilon,\tau(\epsilon,\alpha))$
  of the equation
 $$
u''+{u'}^3+2u{u'}^2+(2u^2-1)u'+u=\epsilon\omega(\frac{t}{\tau},u,u'),
$$
 exists for $(\epsilon,\alpha)\in U $ with $|\tau-\tau_0|<\tau_1$,
$\tau(0,0)=\tau_0$ and $h(0,0)=0$.
 \end{theorem}

\begin{proof}
We proceed similarly as in the proof of Theorem \ref{monsec}.
We substitute the fundamental matrix,
 the second characteristic multiplier is $ \rho_2$.
The following holds for equation \eqref{l4},
$$
\rho_2<1\Longleftrightarrow
  \int_{0}^{\tau_0}(\sum_{k=1}^{3} k p_k(u_0(\tau)){u'_0(\tau)}^{k-1}
)d\tau>0,
$$
 then
$$ \rho_2 <1\Longleftrightarrow
 \int_{0}^{\tau_0}[2{u_0}^2(\tau)+4u_0(\tau)u'_0(\tau)
+3{u'_0(\tau)}^{2}-1]d\tau>0.
$$
 It ensures that  $1$ is a simple characteristic multiplier of the variational
system associated to \eqref{l4} it implies $J(\tau_0)\neq 0$.
Then a periodic solution for
the perturbed equation (\ref{l3}) exists.
\end{proof}

   Using Scilab we will describe the phase plane of equation \eqref{l4}
$u''+[u^2+(u+u')^2-1]u'+u=0$.
We take
 $x_0=u_0(0)=a=-0.7548829$, $y_0={u_0}'(0)=0$ and the step time
of integration $(step=.0001)$.
 Recall that the periodic orbit is unique.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.45\textwidth]{fig1a}
\includegraphics[width=0.45\textwidth]{fig1b}
\end{center}
\caption{ (A) The unique periodic orbit for $u''+[u^2+(u+u')^2-1]u'+u=0$.
 (B) Zoom on the periodic orbit ($\times20$)}
\end{figure}


We take $\epsilon\omega(\frac{t}{\tau},u,u')=\epsilon sin(2t)u'$.
 Some illustrations of the phase portrait for the perturbed equation (\ref{l3}),
 those can explain existence of a bound $\epsilon_0$, from which periodicity of the
 orbit will be not insured.
 In order to localize $\epsilon_0$, we have taken several values of $\epsilon$.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.45\textwidth]{fig2c}
\includegraphics[width=0.45\textwidth]{fig2d}
\end{center}
\caption{(C) The periodic orbit for 
$u''+[u^2+(u+u')^2-1]u'+u=\epsilon\omega(\frac{t}{\tau},u,u')$,
 $\epsilon=0.001$.
(D) Zoom on the periodic orbit ($\times20$)}
\end{figure}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.45\textwidth]{fig3e}
\includegraphics[width=0.45\textwidth]{fig3f}
\end{center}
\caption{(E) Orbit for  $u''+[u^2+(u+u')^2-1]u'+u
=\epsilon\omega(\frac{t}{\tau},u,u')$, $\epsilon=0.01$.
(F) Zoom on the orbit ($\times10$) and loss of periodicity.}
\end{figure}

 We see that from the range of $\epsilon=0.01$ the orbit loses
the periodicity.

\begin{table}[ht]
\caption{Period $\tau$ for some values of $\epsilon$}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$\epsilon$ & 0 & 1/1000 & 1/900 & 1/800 & 1/700\\
\hline
$\tau$ & 5.4296 & 5.4287 & 5.4286 &5.4285 &5.4283 \\
\hline\hline
$\epsilon$ & 1/600 & 1/500 & 1/400 & 1/300 & 1/200 \\
\hline
$\tau$ & 5.4281 & 5.4278 & 5.4274 & 5.4267 & 5.4252\\
\hline
\end{tabular}
\end{center}
\end{table}

\subsection*{Acknowledgements}
We thank Professors Miklos Farkas and Jean Marie Strelcyn for their
helpful discussions; also the referees for their suggestions.

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\end{document}