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\markboth{\hfil Solutions series \hfil EJDE/Conf/10}
{EJDE/Conf/10 \hfil A. Raouf  Chouikha \hfil}

\begin{document}
\setcounter{page}{115}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
Fifth Mississippi State Conference on Differential Equations and
Computational Simulations, \newline
Electronic Journal of Differential Equations,
Conference 10, 2003, pp 115--122. \newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu (login: ftp)}
 \vspace{\bigskipamount} \\
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Solutions series for some non-harmonic motion equations
%
\thanks{ {\em Mathematics Subject Classifications:} 34A20.
\hfil\break\indent
{\em Key words:} power series solution, trigonometric series.
\hfil\break\indent
\copyright 2003 Southwest Texas State University. \hfil\break\indent
Published February 28, 2003. } }

\date{}
\author{A. Raouf  Chouikha}
\maketitle

\begin{abstract}
 We consider the class of nonlinear oscillators of the form
 \begin{gather*}
 {d^2 u\over{dt^2}} + f(u)  = \epsilon g(t)  \\
 u(0) = a_0, \quad u'(0) = 0,
 \end{gather*}
 where $g(t)$ is a $2T$-periodic function, $f$ is a function only dependent
 on $u$, and $\epsilon $ is a small parameter.
 We are interested in the periodic solutions with minimal period $2T$, when
 the restoring term $f$ is such that $f(u)=\omega ^2u+u^2$ and $g$ is a 
 trigonometric polynomial with period $2T=\frac{\pi}{\omega}$. 
 By using method based on expanding the solution as a sine power series, 
 we prove the existence of periodic solutions for this perturbed equation.
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\numberwithin{equation}{section}

\section{Introduction}

Consider the second order differential equation
\begin{equation}
 \begin{gathered} \label{e2}
 x'' + g(t,x,x',\epsilon) = 0 \\
u(0) = a_0, \quad u'(0) = 0\,,
\end{gathered}
\end{equation}
where $\epsilon > 0$ is a small parameter,  $g$ is a $2T$-periodic function
in $t$ with\\ $g(t,x,0,0) = g(x)$ is independent of $t$.

The existence of solutions for this equation in the case where $g$ is
independent on $x'$ and  continuously differentiable has been studied by
 many authors.  In the latter case, this proved the existence of
solutions to
 $$ x'' + g(t,x,\epsilon ) = 0. $$
For a detailed review, we refer the reader to the book by Chow and Hale \cite{C-H}.
Furthermore, the example given by Hartman proved  non existence cases
 of (1).  So, we cannot expect to generalize their results.
On the other hand,  Loud, Willem and others considered the case where $T$
is independent on $\epsilon $.
Concerning the case $T = T(\epsilon )$,  Fonda and Zanolin  studied the
system
$$
 u'' + f(u)  = \epsilon e(t,u,\epsilon )
$$
where $f$ is semilinear with $xf(x) > 0$.  Here $e$ is a
$T_\epsilon $-periodic function such that
$| e(t,x,\epsilon )| < K$, $\lim_{\epsilon \to 0} T_\epsilon
= +\infty $ and  $\lim_{\epsilon \to 0} \epsilon T_\epsilon = 0$.
They proved the existence of  periodic solutions.

In the present paper we are interested in the class of non linear oscillators
of the form
\begin{equation} \label{e3}
\begin{gathered}
{d^2 u\over{dt^2}} + f(u)  = \epsilon g(t) \\
u(0) = a_0, \quad u'(0) = 0\,,
\end{gathered}
\end{equation}
where the restoring term $f \equiv f(u) $ depends on $u$ and the function
$g\equiv g(t)$ is  $2T$-periodic. We look for periodic
solutions with minimal period $2T$, when $f$ and $g$ satisfy suitable
conditions.


When $f$ is a quadratic polynomial function, we are able to solve \eqref{e2} , in
some situations and without  $\epsilon $  to be small, or any restricted
condition on the period $T$. We will prove for that  the existence of
a trigonometric expansions of (periodic) solutions in Sinus powers.

Note that, some times under certain conditions, (1) cannot have a periodic
solution, as described in the  example  below. However, under other suitable
conditions of controllability of the period, Equation (1)
has a periodic solution.

\subsection*{A non existence result}

According to Hartman \cite[p.39]{H}, Equation (1) in general does not have a
non constant periodic solution, even if $x g(t,x,x') > 0$.
The following example, given by Moser, proves the non existence of a non
constant periodic solution of
$$ x'' + \phi (t,x,x') = 0. $$
Let
$$
\phi (t,x,y) = x + x^3 + \epsilon f(t,x,y), \qquad \epsilon > 0
$$
satisfying the following conditions:
$\phi \in C^1(R^3)$;  $f(t+1,x,y) = f(t,x,y)$  with
$f(0,0,0) = 0$ and $f(t,x,y) = 0$ if $xy = 0$;
$$
\frac{\phi }{x} \to \infty \quad \mbox{as } x \to \infty
$$
uniformly in $(t,y) \in R^2$;
$$
\frac{\delta f }{\delta y} > 0  \quad\mbox{if } xy >0,
\quad \mbox{and}\quad  \frac{\delta f   }{\delta y} = 0 \quad\mbox{otherwise}.
$$
and $x,y$ satisfying  $ | x | < \epsilon , | y | < \epsilon $.
In fact, we have  $x f(t,x,y) $  and $ y f(t,x,y)) > 0 $
if $ x y > 0, | x| < \epsilon$, $y$  arbitrary  and $ \phi  = 0 $ otherwise.

We remark that $\frac{\delta f }{\delta y} $  is small.
The function  $V = 2 x^2 + x^4 + 2 {x'}^2$  satisfies
$V' = -4 \epsilon x' f(t,x,x')$, so that  $ V' < 0 $ if
$ x x' > 0$, $| x | < \epsilon $ and $V' =0$ otherwise.
 Thus  $x$ cannot be periodic unless  $V' = 0$.

\subsection*{A controllably periodic perturbation}

In this part, we give brief explanation  of a method due to Farkas \cite{F}
inspired of the one of Poincar\'e.  The determination of controllably
periodic perturbed solution. This method appears to be efficient particularly
for the perturbations of various autonomous systems.
Since we know it for example a good application for perturbed Van der Pol
equations type as well as Duffing equations type.

The perturbation is supposed to be `controllably periodic', i.e., it is
periodic with a period which can be chosen appropriately. Let $u_0(t)$ be
the periodic solution of the (unperturbed) equation. Under very mild
conditions it is proved that to each small enough amplitude of the
perturbation
there belongs a one parameter family of periods such that the perturbed system has a unique periodic solution with this
period.

It is assumed the existence of the fundamental matrix solution of the first
variational 2-dimensional system of  $\dot x = h(x)$ and the unique periodic
solution $p(t)$ corresponding to  $u_0(t)$.
For more details concerning applications of this method we may refer to \cite{F}.


\section{Periodic perturbations of Duffing  non linear equations}

In this section, we will apply the methods of trigonometric series in sinus
powers to resolve in an explicit way non linear oscillators of Duffing type.
This approach already was used previously in an effective way, see \cite{S-S} and
\cite{Ch}.  One can also apply it for periodic perturbations of various
differential equations.  Our results generalize and improve those of \cite{S-S}
and \cite{Ch}. Their also contribute in an interesting way to the general problem
of the periodic perturbations roughly exposed in the introduction.

\subsection*{Case of finite trigonometric polynomial}

Firstly, let us consider the restoring function
$$
f(u) = \omega ^2 u + u^2,
$$
where $\omega $ is a constant and $g(t)$ a finite trigonometric polynomial
with period $2T = \frac {\pi }{\omega }$, which  has a finite Fourier series
 expansion. So, it can be written
$$
g(t) = b_0 + b_2 \sin^{2}({\pi t\over{2T}}) + \dots + b_{2p_0}
 \sin^{2p_0}({\pi t\over{2T}}) = \sum_{n \leq
 p_0} b_{2n} \sin^{2n}({\pi t\over{2T}}), \quad   p_0 \in  \mathbb{N} .
$$
Consider the following system
\begin{equation} \label{e4}
 \begin{gathered}
{d^2 u\over{dt^2}} + \omega ^2 u + u^2  = \epsilon  g(t) \\
u(0) = a_0, \quad u'(0) = 0\,.
\end{gathered}
\end{equation}

\begin{theorem} \label{thm1}
The $T$-periodic solutions of  \eqref{e3} may be expressed in the form
$$
u(t)  = c_0 + c_2\sin ^2({\pi t\over{2T}}) + c_4\sin^4 ({\pi t\over{2T}})
+ c_6\sin^6 ({\pi t\over{2T}}) + \dots
= \sum_{n \in IN} c_{2n} \sin^{2n}({\pi t\over{2T}})
$$
The coefficients $c_{2n}$ satisfy  $| a_0 | < C$, $a_0 = c_0$,
  $2 \omega^2 c_2 = - \omega ^2 c_0 - c_0^2 +  \epsilon b_0$,
($\omega = {\pi \over{2T}}$) and
the recursion formula
\begin{multline} \label{e6} % and 5
(2n+1)(2n+2) c_{2n+2}\\
 =\begin{cases}
(4n^2 - 1) c_{2 n} - {1 \over \omega^2} \sum_{r=0}^{n} c_{2r} c_{2n-2r}
+ \frac {\epsilon }{\omega ^2}b_{2n}, &\mbox{for }  n\leq p_0,\\
 (4n^2 - 1) c_{2 n} - {1 \over \omega^2} \sum_{r=0}^{n} c_{2r} c_{2n-2r},
 &\mbox{for } n > p_0\,.
\end{cases}
\end{multline}
\end{theorem}

\paragraph{Proof}
Suppose the solution  $u$ of equation \eqref{e3} can be written in the form
$$
u(t) = \sum_{n \in IN} c_{2n} \sin^{2n}(\omega t).
$$
We shall apply the previous method in \cite{Ch} to:
(i) obtain recursion formula of the coefficients  $c_{2n}$, and
(ii) prove the convergence of the series expansion of the solution.

This method finds a solution to \eqref{e3}in the form
\begin{equation} \label{e7}
u(t) = c_0 + c_1 \sin\omega t + c_2 \sin^2 \omega t + c_3 \sin^3 \omega t
+ \dots
\end{equation}
where  $c_i$, $i = 0, 1, 2, \dots$, are coefficients to be determined by the
substitution of \eqref{e6} in \eqref{e3}.
In fact, we get  $\omega = {\pi \over 2T}$  where $T$ is the  period of the
solution.
So, a trivial computation gives
$$  2 \omega^2 c_2 = - \omega ^2 c_0 - c_0^2 +  \epsilon b_0.
$$
We get also
$$
12 c_4 = 3 c_2 - \frac {2}{\omega ^2}c_0 c_2 + \frac {\epsilon }{\omega ^2}b_2
$$
By identification for $ n  \geq 1$, we obtain  the recursion formula of
these coefficients
\begin{equation} \label{e8}
(2n+1)(2n+2) c_{2n+2} = (4n^2 - 1) c_{2 n} - {1 \over \omega^2}
\sum_{r=0}^{n} c_{2r} c_{2n-2r} + \frac {\epsilon }{\omega ^2}b_{2n},
\quad \mbox{for } n\leq p_0.
\end{equation}
and
\begin{equation}
(2n+1)(2n+2) c_{2n+2} = (4n^2 - 1) c_{2 n} - {1 \over \omega^2}
\sum_{r=0}^{n} c_{2r} c_{2n-2r} , \quad\mbox{for } n  >  p_0.
\end{equation}
Equation \eqref{e3} implies  $c_1 = 0$. Relations \eqref{e7} yields
$c_3 = 0$, $c_5 = 0$, \dots.
However, the even order coefficients  $c_{2n}$  do not vanish.
The solution to \eqref{e3} can now be written as
\begin{equation}
u(t) = c_0  - [c_0-\frac {1}{ \omega ^2}c_0^2
+\frac { \epsilon}{ \omega ^2} b_0] \sin^2 \omega t
+ [\frac {1}{4}c_2-\frac {1}{6 \omega ^2}c_0 c_2
+\frac {\epsilon }{12 \omega ^2}b_2] \sin^2 \omega t + \dots
\end{equation}
Relations of the coefficients and further induction show that $c_{2i}$,
$i=1,2,\dots$ all vanish for \quad $a_0 =  0$  and  $b_ {2i} = 0$.
So, the trivial solution  $u \equiv 0$  is included in \eqref{e6} as a special case.
The following lemma is strictly analogous of \cite[Lemma 2]{Ch}, corresponding
to the case  $\epsilon =0$. It implies that
$$ \sum_{n\geq 1} | c_{2n}| < + \infty ,
$$
which implies the convergence of the expansion. \hfill$\square$


\begin{lemma} \label{lm1}
There exist a positive constants $k$ and  $\alpha $ verifying
 $1 <\alpha < \frac {3}{2}$
such that the coefficients $c_{2n}$ of the series expansion \eqref{e6} solution of
the differential equation  \eqref{e3} satisfy the inequality
\begin{equation}
| c_{2n} | < {k \over{(2n)^\alpha }}.
\end{equation}
\end{lemma}

\paragraph{Remark}
Following \cite{S-S}, it is interesting to write the power series solution for
the system \eqref{e3} in the form
$$ u(t) = v(\sin \omega t).
$$
Let $\tilde g$ defined by  $g(t) = \tilde g(\sin \omega t)$.
Then  $v$  is a solution of the differential equation
\begin{equation} \label{e12}
\begin{gathered}
(1 - x^2){d^2 v\over{dx^2}} - x {d v\over{dx}}+ v + {1\over{\omega ^2}}v^2
= \frac {\epsilon}{\omega ^2} \tilde g(x) \\
v(0) = a_0, \quad \frac {dv}{dx}(0) = 0
\end{gathered}
\end{equation}
Recall that the latter method permits  to compare approximate solutions
of the non-harmonic motion of the oscillator.

\subsection*{A more general case}

Now, consider a more general situation where the function $g(t)$ in \eqref{e2} may
be written as an infinite expansion in Sinus power
$$
g(t) = b_0 + b_2 \sin^{2}({\pi t\over{2T}}) + \dots + b_{2n}
\sin^{2n}({\pi t\over{2T}}) + \dots  = \sum_{n \in IN} b_{2n}
\sin^{2n}({\pi t\over{2T}}) .
$$
It is the case in particular, when the function  $g(t)$ has a finite Fourier
series expansion. When  $g$  has an (infinite) Fourier series expansion,
 $g(t)$  may be expressed as an infinite expansion in Sinus power. But
we have to prove its convergence.

Now, we are interested in solving
\begin{equation} \label{e13}
\begin{gathered}
{d^2 u\over{dt^2}} + \omega ^2 u + u^2
= \epsilon  \sum_{n \in IN} b_{2n} \sin^{2n}({\pi t\over{2T}})  \\
u(0) = a_0, \quad u'(0) = 0\,.
\end{gathered}
\end{equation}
We prove the following theorem which extends Theorem \ref{thm1}.

\begin{theorem} \label{thm2}
Suppose that the coefficients of the expansions of the function  $g(t)$
satisfies
 $$ | b_{2n}| < {1\over{(2n)^\beta}} ,\ {\it with }\ \beta \geq 1,
$$
 then the solutions of \eqref{e12} may be expressed in the form
$$
u(t)  = a_0 + c_2\sin ^2({\pi t\over{2T}}) + c_4\sin^4 ({\pi t\over{2T}})
+ c_6\sin^6 ({\pi t\over{2T}}) + \dots
= \sum_{n \in IN} c_{2n} \sin^{2n}({\pi t\over{2T}}).
$$
The coefficients $c_{2n}$ satisfy the conditions
$| a_0 | < C$, $a_0 = c_0$,
$2 \omega^2 c_2 = - \omega ^2 c_0 - c_0^2 +  \epsilon b_0$,
$(\omega = {\pi \over{2T}})$  and the recursion formula, for $n > 0$:
\begin{equation}
(2n+1)(2n+2) c_{2n+2}
= (4n^2 - 1) c_{2 n} - {1 \over \omega^2} \sum_{r=0}^{n} c_{2r} c_{2n-2r}
+ \frac {\epsilon }{\omega ^2}b_{2n}.
\end{equation}
\end{theorem}

\paragraph{Proof}
The proof starts in the same manner as that of Theorem \ref{thm1}. In order to
establish the recursion relation between the coefficients we may proceed
as previously. The difference is that one obtains a less good estimation
of the $c_{2n}$.  The crucial point is to state an analogous one of
Lemma \ref{lm2},
ensuring thus the convergence of the series solution. It may be deduced from
$$ \sum_{n\geq 1} | c_{2n}| < + \infty
$$
For that, we prove the following lemma.

\begin{lemma} \label{lm2}
For any positive number $\alpha $  such that
 $1 < \alpha  < 3/2$, there exists a positive constant $k$
satisfying
 $$ k < {1 \over \omega^2}{3\over 4} ({3\over 2} - \alpha ) 4^{1-\alpha }
 $$
such that the coefficients $c_n$ of the series expansion  solution of the
differential equation  \eqref{e12} satisfies the inequality
\begin{equation}
| c_n | < {k \over{n^\alpha }}.
\end{equation}
\end{lemma}

\paragraph{Proof}
We first notice that Lemma \ref{lm2} gives an optimal result, because our method
does not work for  $\alpha = {3\over 2}$.
The coefficients  $c_n$ of the power series  solution, satisfy the recursion
formula \eqref{e13}. We shall prove there exist two positive constants $k > 0$, and
$\alpha > 1$, such that  the following inequality holds
$$
| c_n | < {k\over{n^\alpha }}
$$
for any integer $n \geq 1$.
Suppose for any $n \leq p$, we get $| c_n | < {k\over{n^\alpha }}$.
In particular, it implies that
$$
\sum_{0<r<p} c_r c_{p-r} < \sum_{0<r<p} {k^2\over{r^\alpha (r - p)^\alpha }}
\leq {k^2\over{(p-1)^{\alpha -1}}}.
$$
Equality \eqref{e13} gives
$$
c_{p+2} = {{p^2 - 1}\over{(p+1)(p+2)}} c_p - {1 \over \omega^2 (p+1)(p+2)}
\sum_{r=0}^{p-1} c_r c_{p-r} + {\epsilon \over\omega ^2(p+1)(p+2)} b_p.
$$
Thus, if we prove the inequality
$$
{{p^2-1} \over{(p+1)(p+2)}}{k\over{p^\alpha }}
+{1 \over \omega^2 (p+1)(p+2)}{k^2\over{(p-1)^{\alpha -1}}}
+{\epsilon | b_p| \over\omega ^2(p+1)(p+2)} \leq {k\over{(p+2)^\alpha }},
$$
we can conclude
\begin{equation}
| c_{p+2} | < {k\over{(p+2)^\alpha }}.
\end{equation}
Since the coefficients $b_p$ satisfy the hypothesis
 $ | b_{2n}| < {1\over{(2n)^\beta}}$, with $\beta \geq 1$, it is not difficult
to  exhibit an integer $p_0$ depending on $\beta $ such that the following
inequality holds for  $p \geq p_0$:
\begin{equation}
{{p^2-1} \over{(p+1)(p+2)}}{k\over{p^\alpha }}
+ {1 \over \omega^2 (p+1)(p+2)}{k^2\over{(p-1)^{\alpha -1}}}
\leq {k\over{(p+2)^\alpha }}
\end{equation}
This inequality  is equivalent to
\begin{equation}
k < {\beta \over \omega^2} p f(p) g(p)
\end{equation}
where
\begin{gather*}
f(p) = {p+1\over{p}} ({p-1\over{p+2}})^{\alpha -1}\\
g(p) = 1 - {(p^2 - {\beta \over \omega^2}c_0)(p+2)^{\alpha -1}
\over{(p+1)p^\alpha }}
\end{gather*}
Using MAPLE, we  prove that $f(p)$ is an increasing positive function in $p$.
Moreover, for any $p \geq 1$, $f(p)$ is bounded below as
$$
f(p) \geq ({3\over 2}) 4^{1-\alpha }.
$$
The function $g(p)$ is such that
$$
p g(p) = p - {(p^2 - {\beta \over \omega^2}c_0)({p+2\over p})^{\alpha -1}
\over{(p+1)}}
$$
is a  strictly decreasing and bounded function. More exactly,
we may calculate the lower bound
$$
g(p) >{ (3 - 2\alpha )\over p}.
$$
Thus, if \ $(3 - 2\alpha ) = \epsilon > 0$, it suffices to chose
$$ k \leq  ({3\over 2}) 4^{1-\alpha }(3 - 2\alpha )
$$
for inequality (17) to hold.

\paragraph{Remark for the case $\epsilon = 0$:}
Note that the choice of $k$ depends on $\alpha $ value.
For $\alpha  = 3/2$, using MAPLE we can prove that
 $$
 p g(p) = p - {(p^2 - { 3\over 2\omega^2}c_0)({p+2\over p})^{1/2}
 \over{(p+1)}}
$$
is positive and strictly decreasing  to $0$. While $p^2 g(p)$ is a bounded
function.  Moreover, it appears that  $p f(p) g(p)$ is a decreasing function
which tends to $0$ as $p$ tends to infinity.
Thus, our method fails since it does not permit to determine a non negative
constant $k$.

\begin{thebibliography}{00} \frenchspacing

\bibitem{C-H}  S. N. Chow,  J. Hale,
{\it Methods of bifurcation theory},  Springer, Berlin (1982).

\bibitem{Ch}  A. R. Chouikha,
{\it Remark on Periodic Solutions of Non-linear Oscillators},
 Appl. Math. Lett.,vol. 14 (8), p.963-968, (2001).

\bibitem{Ch1}  A. R. Chouikha,
{\it Fonctions elliptiques et bifurcations d'equations differentielles},
 Canad.  Math. Bull.,  vol. 40 (3), p. 276-284, (1997).

\bibitem{F}   M. Farkas,
{\it Estimates on the existence regions of perturbed periodic solutions},
SIAM J. Math. Anal., vol 9, p. 867-890, (1978).

\bibitem{H}  P. Hartman,
{\it On boundary value problems for superlinear second order differential
equation}, J. of Diff Eq. , vol 26, p. 37-53, (1977).

\bibitem{S-S}  A. Shidfar and A. Sadeghi,
{\it Some Series Solutions of the anharmonic Motion Equation},
 J. Math. Anal. Appl.  120, p. 488-493 (1986).

\bibitem{S-S1}  A. Shidfar and A. Sadeghi,
{\it The Periodic Solutions of Certain Non-linear Oscillators},
Appl. Math. Lett., vol 3, n 4, p. 21-24, (1990).

\end{thebibliography}

\noindent\textsc{A. Raouf  Chouikha}\\
University of Paris-Nord, Laga,\\
CNRS UMR 7539  Villetaneuse F-93430,\\
e-mail: chouikha@math.univ-paris13.fr

\end{document}