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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 06, pp. 1--19.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/06\hfil Slow and fast systems]
{Slow and fast systems with Hamiltonian reduced problems}

\author[M. Benbachir, K. Yadi , R. Bebbouchi\hfil EJDE-2010/06\hfilneg]
{Maamar Benbachir, Karim Yadi, Rachid Bebbouchi}

\address{Maamar Benbachir \newline
Universit\'e de B\'echar, BP 417, Route de Kenadsa, 08000 B\'echar
Alg\'erie}
\email{mbenbachir2001@yahoo.fr}

\address{Karim Yadi \newline
Laboratoire des Syst\'emes Dynamiques et Applications\\
Universit\'e Aboubekr Belka\"{\i}d, BP 119, 13000 Tlemcen Alg\'erie}
\email{yadikdz@yahoo.fr, k\_yadi@mail.univ-tlemcen.dz}

\address{Rachid Bebbouchi \newline
Facult\'e de Math\'ematiques\\
Universit\'e des Sciences et de la Technologie \newline
Houari Boumedi\'ene, Alger Bab ezzouar, Alg\'erie}
\email{rbebbouchi@hotmail.com}

\thanks{Submitted June 1, 2009. Published January 13, 2010.}
\subjclass[2000]{34D15, 34E18, 70H09, 03H05}
\keywords{Singular perturbations; Hamiltonian system; stroboscopy lemma;
\hfil\break\indent nonstandard analysis}

\begin{abstract}
 Slow and fast systems are characterized by having some of the derivatives
 multiplied by a small parameter $\epsilon$. We study systems of reduced
 problems which are Hamiltonian equations, with or without a slowly varying
 parameter. Tikhonov's theorem gives approximate solutions for times of order 
 $1$. Using the stroboscopic method, we give approximations for time of order 
 $1/\epsilon$. More precisely, the variation of the total energy of the
 problem, and the eventual slow parameter, are approximated by a certain
 averaged differential equation. The results are illustrated by some
 numerical simulations. The results are formulated in classical mathematics
 and proved within internal set theory which is an axiomatic approach to
 nonstandard analysis.
\end{abstract}

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

\section{Introduction}

A slow and fast system (or two-time scale system) is a perturbed system of
the form 
\begin{equation}
\begin{gathered} 
\frac{dx}{dt}=F(x,z,\epsilon),\quad x(0) =\alpha_{\epsilon},\\ 
\epsilon\frac{dz}{dt}=G(x,z,\epsilon),\quad z(0)=\beta_{\epsilon},
\end{gathered}  \label{a}
\end{equation}
where $x\in \mathbb{R}^{n}$ and $z\in \mathbb{R}^{m}$ are the slow and fast
components and $\epsilon $ is a positive real number small enough. The
functions $F$ and $G$ are continuous and defined on an open subset of 
$\mathbb{R}^{n+m}$. The initial conditions depend continuously on 
$\epsilon $. The system \eqref{a} is more exactly a family of problems depending on the
parameter $\epsilon $ varying in a small interval $]0,\epsilon _{0}]$. The
fact that $\epsilon $ multiplies the derivative makes non valid the theory
of continuous dependence of the solutions with respect to parameters. We are
in presence of a singularly perturbed problem. The purpose of 
\textit{Singular Perturbation Theory} is to investigate the behavior of solutions of 
\eqref{a} as $\epsilon \rightarrow 0$ on a bounded, or eventually, unbounded
time interval. A recommended reference is the tenth chapter of the book of
Wasow \cite{wasow}. The change of the time scale $s=t/\epsilon $ transforms 
\eqref{a} into 
\begin{gather*}
\frac{dx}{ds}=\epsilon F(x,z,\epsilon ),\quad x(0)=\alpha _{\epsilon }, \\
\frac{dz}{ds}=G(x,z,\epsilon ),\quad z(0)=\beta _{\epsilon },
\end{gather*}
which is a one parameter deformation of the unperturbed system 
\begin{equation}
\begin{gathered} \frac{dx}{ds}=0,\quad x(0)=\alpha_0,\\
\frac{dz}{ds}=G(x,z,0),\quad z(0)=\beta_0. \label{a'} \end{gathered}
\end{equation}
The following system, where $x$ is considered as a parameter, is called the
\textit{\ fast equation} 
\begin{equation}
\frac{dz}{ds}=G(x,z,0).  \label{b}
\end{equation}
Hence, the $z$-component of the solution of \eqref{a} varies very quickly
according to the so-called \textit{boundary layer equation} 
\begin{equation}
\frac{dz}{ds}=G(\alpha _{0},z,0),\quad z(0)=\beta _{0},  \label{c}
\end{equation}
where $x$ has been frozen at its initial value. The set of the zeros of the
right member of \eqref{b} is called the \textit{slow manifold} of the
problem \eqref{a}. It is formed by the equilibrium points of the fast
dynamics described by \eqref{b}. A solution of \eqref{b} may be unbounded
when $s\rightarrow \infty $, or may tend to an equilibrium point or may
approach a more complex attractor and this asymptotic behavior depends on
the initial condition. Assume for instance that the solutions of the fast
equation \eqref{b} tend toward an equilibrium point $\xi (x)$ on the slow
manifold. The equation $z=\xi (x)$ defines a component $\mathcal{L}$ of the
slow manifold. A fast transition brings a solution of \eqref{a} close to the
slow manifold. Then a slow motion takes place near the slow manifold and is
approximated by the \textit{reduced problem} 
\begin{equation}
\frac{dx}{dt}=F(x,\xi (x),0),\quad x(0)=\alpha _{0}.  \label{d}
\end{equation}
When the solution of \eqref{a} is unique, the results of Tikhonov's Theorem 
\cite{tykhonov} (see \cite{lobry}), under suitable conditions (among the
others, the asymptotic stability of the equilibrium $\xi (x)$ uniformly with
respect to $x$ in a compact domain), are mainly as follows:

\begin{quote}
Let $\tilde{z}(s)$ and $\bar{x}(t)$ be the solutions, supposed to be unique,
of the boundary layer equation \eqref{c} and the reduced problem \eqref{d}.
Suppose that $\bar{x}(t)$ is defined on $[0,T]$. Then for $\epsilon$
sufficiently small, the solution $(x(t,\epsilon),z(t,\epsilon))$ of the
perturbed problem \eqref{a} is defined at least on $[0,T]$ and satisfies 
\begin{gather*}
\lim_{\epsilon\to 0^{+}} x(t,\epsilon) =\bar{x}(t)\quad \text{for all }t\in[
0,T], \\
\lim_{\epsilon\to 0^{+}} z(t,\epsilon) =\xi(\bar{x}(t))\quad \text{for all }
t\in]0,T], \\
\lim_{\epsilon\to 0^{+}} z(\epsilon s,\epsilon) =\tilde{z}(s\quad \text{for
all }s\geq0\,.
\end{gather*}
\end{quote}

Another tool, namely the \textit{stroboscopic method}, is needed in this
work to give approximations of the solutions for larger time of order 
$1/\epsilon$. It is a method of the nonstandard perturbation theory of
differential equations proposed by Callot and Reeb and improved by Lutz and
Sari (see \cite{callot-sari,lutz,sari2,sari,saristrobo}). The principle is
as follows (one should at this stage admit the intuitive meaning of the
symbol ``$\simeq$''; i.e., ``infinitely close to''): Let $\phi(t)$ be a
function. Suppose we are able to define a sequence $(t_{n}
,\phi_{n}=\phi(t_{n}))$ such that $t_{n+1}\simeq t_{n}$, $\phi(t)\simeq
\phi_{n}$ on $[t_{n},t_{n+1}]$ and 
\[
\frac{\phi_{n+1}-\phi_{n}}{t_{n+1}-t_{n}}\simeq F(t_{n},\phi_{n}),
\]
where $F$ is a continuous function. We can conclude that the function 
$\phi(t)$ is infinitely close to a solution of the differential equation 
$\frac{d\phi}{dt}=F(t,\phi)$. Actually, we use in this paper an
improved version established by T. Sari.

After the generalizations of the famous results of Tikhonov and
Pontryagin-Rodygin for slow and fast systems obtained in 
\cite{lobry,sariyadi}, it was quite natural to think about the case where the
trajectories approach an oscillating motion on the slow manifold. As far as
we know, this case has been described in \cite{remili}. Unfortunately, this
reference has not been diffused. The author considered among others the
scalar slow-fast system 
\[
\frac{dx}{dt}=f(x,y,z),\quad \frac{dy}{dt}=g(x,y,z),\quad
\epsilon\frac{dz} {dt}=h(x,y)-z,
\]
the slow equation of which presents periodic orbits and admits a first
integral. With the use of the first return map of Poincar\'e, he showed
that, after a fast transition, the considered trajectory fills the region of
oscillations lying on the slow manifold. Note that this study was entirely
qualitative. The starting point of the present work was the study of the
singular perturbation of the harmonic oscillator \cite{benbachir} 
\[
\epsilon \frac{d^{3}x}{dt^{3}}+\frac{d^{2}x}{dt^{2}}+x=0,
\]
or more exactly of the associated system 
\[
\frac{dx}{dt}=y,\quad \frac{dy}{dt}=z,\quad \epsilon\frac{dz}{dt}=-x-z.
\]
By the change of variable $\epsilon z_1=x+z,$ one has 
\[
\frac{dx}{dt}=y,\quad \frac{dy}{dt}=-x+\epsilon z_1,\quad \epsilon\frac{dz_1}{dt}=y-z_1.
\]
This system is a particular case of the general slow-fast problem 
\begin{equation}
\begin{gathered} \frac{dx}{dt}=\frac{\partial H}{\partial y}(x,y)+\epsilon
f(x,y,z,\epsilon),\\ \frac{dy}{dt}=-\frac{\partial H}{\partial
x}(x,y)+\epsilon g(x,y,z,\epsilon),\\
\epsilon\frac{dz}{dt}=h(x,y,z,\epsilon), \end{gathered}  \label{original}
\end{equation}
the slow equation of which is a Hamiltonian system. We also consider the
more general case where the Hamiltonian depends on a slowly varying
parameter, more exactly 
\begin{equation}
\begin{gathered} 
\frac{dx}{dt}=\frac{\partial H}{\partial
y}(x,y,\lambda)+\epsilon f(x,y,z,\lambda,\epsilon),\\
\frac{dy}{dt}=-\frac{\partial H}{\partial x}(x,y,\lambda)+\epsilon
g(x,y,z,\lambda,\epsilon),\\
\epsilon\frac{dz}{dt}=h(x,y,z,\lambda,\epsilon),\\
\frac{d\lambda}{dt}=\epsilon\alpha(x,y,z,\lambda,\epsilon). \end{gathered}
\label{1}
\end{equation}
We define an averaged system 
\begin{equation}
E'=M_1(E,\lambda),\quad \lambda'=M_{2}(E,\lambda),
\label{averaged}
\end{equation}
where the prime denotes the derivative with respect to $\tau=\epsilon t$ and
the functions $M_1$ and $M_{2}$ are the averages of the functions 
\[
\Omega(x,y,\lambda)=\omega(x,y,\xi(x,y,\lambda),\lambda),\quad
A(x,y,\lambda)=\alpha(x,y,\xi(x,y,\lambda),\lambda,0),
\]
on the closed orbits of the Hamiltonian system 
\begin{equation}
\frac{dx}{dt}=\frac{\partial H}{\partial y}(x,y,\lambda),\quad 
\frac{dy}{dt}
=-\frac{\partial H}{\partial x}(x,y,\lambda),  \label{H}
\end{equation}
where $\lambda$ is considered as a constant parameter. The function $\omega$
is given by 
\begin{align*}
\omega(x,y,z,\lambda) &=\frac{\partial H}{\partial x}(x,y,\lambda
)f(x,y,z,\lambda,0)+\frac{\partial H}{\partial y}(x,y,\lambda)
g(x,y,z,\lambda ,0) \\
&\quad +\frac{\partial H}{\partial\lambda}(x,y,\lambda)\alpha(x,y,z,\lambda,0),
\end{align*}
and the function $\xi$ defines the slow manifold $z=\xi(x,y,\lambda)$ of 
(\ref{1}). We prove in Theorem \ref{parametre} that for any solution 
$(x(\tau,\epsilon),y(\tau,\epsilon),z(\tau,\epsilon),\lambda (\tau,\epsilon))$
of (\ref{1}), written in the time scale $\tau$, the functions 
\[
E(\tau)=H(x(\tau,\epsilon),y(\tau,\epsilon))
\]
and $\lambda(\tau,\epsilon)$ are approximated by a solution of the averaged
system (\ref{averaged}). We must attract the attention that this result
could be deduced from a result published by T. Sari in \cite{sari3}.
Actually, this author considers the non Hamiltonian perturbation of a
Hamiltonian system with slowly varying parameters 
\begin{equation}  \label{sari}
\begin{gathered} \frac{dx}{dt} =\frac{\partial H}{\partial
y}(x,y,\lambda)+\epsilon f_1(x,y,\lambda,\epsilon),\\ \frac{dy}{dt}
=-\frac{\partial H}{\partial x}(x,y,\lambda)+\epsilon
g_1(x,y,\lambda,\epsilon),\\ \frac{d\lambda}{dt}
=\epsilon\alpha_1(x,y,\lambda,\epsilon ), \end{gathered}
\end{equation}
to show how to use the stroboscopic method to obtain adiabatic invariants.
He proves (see Theorem~2 in \cite{sari3} and references therein for
classical results) that its solutions are approximated by the solutions of
the averaged system(\ref{averaged}) where the functions $M_1$ and $M_{2}$
are the averages of the functions 
\[
\Omega_1(x,y,\lambda)=\omega_1(x,y,\lambda),\quad
A_1(x,y,\lambda)=\alpha_1(x,y,\lambda,0),
\]
on the closed orbits of the Hamiltonian system (\ref{H}). Here, the function 
$\omega_1$ is given by 
\begin{align*}
\omega_1(x,y,\lambda) & =\frac{\partial H}{\partial x}(x,y,\lambda
)f_1(x,y,\lambda,0)+\frac{\partial H}{\partial y}(x,y,\lambda)g_1
(x,y,\lambda,0) \\
&\quad +\frac{\partial H}{\partial\lambda}(x,y,\lambda)\alpha_1(x,y,\lambda,0).
\end{align*}

To show how the last result in \cite{sari3} can be used to prove our result
despite the fact that the slow-fast system we consider contains also the
fast variable $z$, we need to say something about the Geometric Singular
Perturbation Theory, namely the Fenichel invariant manifold Theorem (for
details and definitions one can see \cite{kaper}). This last statement
concerns systems of the form 
\begin{equation}
y'=\epsilon u(y,z,\epsilon),\quad z'=v(y,z,\epsilon),
\label{Fenichel}
\end{equation}
where $u$ and $v$ are $C^{\infty}$ in an open subset $U\times I$ of 
$\mathbb{R}^{m+n+1}$, $0\in I$. Suppose that the set 
$\mathcal{N}_0=\{(y,z), v(y,z,0)=0\} $ is a normally hyperbolic
 manifold given by the graph
of a $C^{\infty}$ function $z=\xi(y)$ defined on a compact subset $Y$. Under
these assumptions, Fenichel's Theorem ensures that ``$\mathcal{N}_0$ 
\textit{persists for small values of} $\epsilon$'', more precisely, \textit{for any
positive integer } $r$ \textit{and for any} $\epsilon>0$ \textit{small
enough, there exists a} $C^{r}$ \textit{function} $z=\mathcal{Z}(y,\epsilon)$
\textit{defined for} $y$ in $Y$ \textit{such that the manifold} 
$\mathcal{N}_{\epsilon}=\{(y,z)$, $z=\mathcal{Z}(y,\epsilon)\}$ is locally invariant
under (\ref{Fenichel}). Moreover $\mathcal{N}_{\epsilon}\to \mathcal{N}_0$ 
\textit{when} $\epsilon \to 0$. Hence, on the invariant Fenichel slow
manifold $\mathcal{N}_{\epsilon}$, the system (\ref{Fenichel}) is reduced to 
\[
y'=u(y,\mathcal{Z}(y,\epsilon),\epsilon).
\]

Let us start by saying that the assumptions we make in our work do not
require strong conditions of differentiability of functions appearing in the
problem. Hence the slow manifold is not supposed to be differentiable nor
normally hyperbolic, that is for what Fenichel's theory is not completely
satisfactory in this case. However, if we suppose that the problem (\ref{1})
admits an invariant manifold given by $z=\mathcal{Z}(x,y,\lambda,\epsilon)$,
it becomes simply 
\begin{gather*}
\frac{dx}{dt}=\frac{\partial H}{\partial y}(x,y,\lambda)+\epsilon f(x,y,
\mathcal{Z}(x,y,\lambda,\epsilon),\lambda,\epsilon), \\
\frac{dy}{dt}=-\frac{\partial H}{\partial x}(x,y,\lambda)+\epsilon g(x,y,
\mathcal{Z}(x,y,\lambda,\epsilon),\lambda,\epsilon), \\
\frac{d\lambda}{dt}=\epsilon\alpha(x,y,\mathcal{Z}(x,y,\lambda
,\epsilon),\lambda,\epsilon),
\end{gather*}
which is a perturbed Hamiltonian system with slowly varying parameters of
the form \eqref{sari} where 
\begin{gather*}
f_1(x,y,\lambda,\epsilon)=f(x,y,\mathcal{Z}(x,y,\lambda,\epsilon
),\lambda,\epsilon), \\
g_1(x,y,\lambda,\epsilon)=g(x,y,\mathcal{Z}(x,y,\lambda,\epsilon
),\lambda,\epsilon), \\
\alpha_1(x,y,\lambda,\epsilon)=\alpha(x,y,\mathcal{Z}(x,y,\lambda
,\epsilon),\lambda,\epsilon).
\end{gather*}
Moreover, since $\mathcal{Z}(x,y,\lambda,0)=\xi(x,y,\lambda)$, one has 
\[
\Omega_1(x,y,\lambda)=\Omega(x,y,\lambda), \quad A_1(x,y,\lambda
)=A(x,y,\lambda).
\]
According to \cite[Theorem 2]{sari3}, the solutions of (\ref{1}) are
approximated by the solutions of the averaged system (\ref{averaged}),
provided that the conditions of Fenichel's Theorem are fulfilled. It is
worth noting that our contribution consists of a direct proof based on both
Tikhonov's theory and the stroboscopic method.

For convenience, we prefer to detail our approach for the analysis of the
particular case of system \eqref{original}. In Section \ref{approximative},
we state two theorems, the first being just an application of Tikhonov's
Theorem giving the behavior of the solutions of \eqref{original} over time 
$1 $. Theorem \ref{main} provides an approximation of the total energy of 
\eqref{original} over time $1/\epsilon$. Section$~$\ref{nsa} is devoted to
the ``non standard version'' of the first results and the proof is given in
Section \ref{proof}. Internal Set Theory is an extension of ordinary
mathematics due to E.~Nelson \cite{nelson}. It axiomatizes Robinson's
nonstandard analysis (NSA) \cite{robinson}. For a short tutorial in NSA, one
can consult for instance \cite{lobry} or \cite{sariyadi}. Historically, the 
\textit{nonstandard perturbation theory of differential equations}, which is
today a well-established tool in asymptotic theory, has its roots in the
seventies, when the Reebian school 
(see \cite{diener-lobry,lutz-goze,lutz-sari,sari1}) introduced the use of nonstandard
analysis into the field of perturbed differential equations. For more
information on the subject, the interested reader is referred to texts such
as \cite{diener-diener} and to papers such as
 \cite{lakrib,lobry,sari2,sariyadi} among many others. In Section \ref{parameter},
we consider the system (\ref{1}) where the Hamiltonian system depends on a
slowly varying vectorial parameter. We choose to present this last result
(Theorem \ref{parametre}) only in a non standard form. In the last section,
we give some examples of application of Theorems \ref{main} and \ref{parametre}.

\section{Averaging on the slow manifold\label{approximative}}

Consider the Hamiltonian system 
\begin{equation}
\begin{gathered} \frac{dq}{dt}=\frac{\partial H}{\partial p}(p,q),\\
\frac{dp}{dt}=-\frac{\partial H}{\partial q}(p,q). \end{gathered}
\label{hamiltonian}
\end{equation}
The level curves $H(p,q)=E$, where $E$ is constant (energy), are integral
curves of (\ref{hamiltonian}). We call \textit{region of oscillations} of
the Hamiltonian function $H(p,q)$ an interval $I$ of $\mathbb{R}$ such that,
for all $E$ in $I$, $H(p,q)=E$ defines a non trivial closed curve $C(E)$ in
the $p,q$-plane which does not contain any singular point where both 
$\frac{\partial H}{\partial p}$ and $\frac{\partial H}{\partial q}$ vanish. A
periodic solution of (\ref{hamiltonian}) corresponding to the closed orbit 
$C(E)$ is denoted by $(q(t,E),p(t,E))$ with period $P(E)$ and is defined for
all $t$. Consider the system \eqref{original} with the initial conditions 
\begin{equation}
x(0)=q_0,\quad y(0)=p_0,\quad z(0)=z_0,  \label{initialdata}
\end{equation}
where $(x,y,z)\in\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{n}$. First, we
describe the solutions of the system \eqref{original} on a finite time
interval when $\epsilon\to 0$, by the use of the theory of singular
perturbations. We use the following assumptions:

\begin{itemize}
\item[(H1)] The functions $f$, $g$, $h$ and the partial derivatives of $H$
are continuous with respect to their arguments.
\end{itemize}

We assume that the fast equation 
\begin{equation}
\frac{dz}{ds}=h(x,y,z,0),  \label{fast}
\end{equation}
where $s=t/\epsilon$, has an asymptotically stable equilibrium point 
$z=\xi(x,y)$. More exactly

\begin{itemize}
\item[(H2)] There exists a compact domain $K$ in $\mathbb{R}^{2}$ and a
continuous function $\xi$ such that for all $(x,y)\in K$, $z=\xi(x,y)$ is an
isolated root of $h(x,y,z,0)=0$. The point $z=\xi(x,y)$ is an asymptotically
stable equilibrium of (\ref{fast}) uniformly over $K$.
\end{itemize}

The graph of $z=\xi(x,y)$ is an attractive component of the slow manifold 
$h(x,y,z,0)=0$. The slow equation is the Hamiltonian system 
(\ref{hamiltonian}). We refer to the boundary layer equation as 
\begin{equation}
\frac{dz}{ds}=h(x,y,z,0),~z(0)=z_{0,}  \label{layer}
\end{equation}
and to the reduced equation as 
\begin{equation}
\begin{gathered} \frac{dq}{dt}=\frac{\partial H}{\partial
q}(p,q),~q(0,E_0)=q_0,\\ \frac{dp}{dt}=-\frac{\partial H}{\partial
p}(p,q),\text{~}p(0,E_0)=p_0, \end{gathered}  \label{reduced}
\end{equation}
where $(q,p)$ is in the interior $\mathrm{int}K$ of $K$ and $E_0$ is the
energy level such that $H(p_0,q_0)=E_0$. We make the last assumptions:

\begin{itemize}
\item[(H3)] The fast equation (\ref{fast}) and the slow equation 
(\ref{hamiltonian}) have the property of uniqueness of solutions with prescribed
initial conditions, $(q_0,p_0)\in intK$ and $z_0$ is in the basin of
attraction of $\xi(q_0,p_0)$.
\end{itemize}

The theorem below is just an application of Tikhonov's theorem for slow and
fast systems and gives an approximation of the solutions of \eqref{original}, 
\eqref{initialdata} over time 1 for $\epsilon$ sufficiently small 
\cite{tykhonov,lobry}. We do not give its proof.

\begin{theorem}\label{tykhonov}
Suppose that {\rm (H1), (H3)} are satisfied. Let
$\tilde{z}(s)$ be the solution of the boundary layer equation
\eqref{layer} and $(q(t,E_0),p(t,E_0))$ the solution of the reduced
equation \eqref{reduced}. Let $T>0$ be in the positive interval
of definition of
$(q(t,E_0),p(t,E_0))$. For every $\eta>0$, there exists $\epsilon
^{\ast}>0$ such that, for all $0<\epsilon<\epsilon^{\ast}$, any solution
$\gamma(t,\epsilon)=((x(t,\epsilon),y(t,\epsilon),z(t,\epsilon))$
of \eqref{original}, \eqref{initialdata} is defined at least on
$[0,T]$ and there exists $\omega>0$ such that
\begin{gather*}
\epsilon\omega<\eta,  \\
|z(\epsilon s)-\tilde{z}(s)|<\eta \quad \text{for all }0\leq s\leq\omega,\\
|x(t,\epsilon)-q(t,E_0)|<\eta,\quad |y(t,\epsilon)-p(t,E_0)|<\eta \quad
\text{for all }0\leq t\leq T,\\
|z(t,\epsilon)-\xi(q(t,E_0),p(t,E_0))|<\eta \quad
\text{for all }\epsilon\omega\leq t\leq T.
\end{gather*}
\end{theorem}

According to what precedes, for $\epsilon$ small enough, a phase trajectory 
$\gamma(t,\epsilon)$ starting at the point $(q_0,p_0,z_0)$ jumps, after a
small time $t_0=\epsilon\omega$, to the neighborhood of the slow manifold 
$z=\xi(x,y)$, $(x,y)\in\mathrm{int}K$. Then it remains close to the curve 
$C(E_0)$ defined by $H(x,y)=E_0$ over time $1$. Now, the total energy 
$E(t,\epsilon)=H(x(t,\epsilon),y(t,\epsilon))$ of the system \eqref{original}
is slowly varying since its derivative is given by 
\begin{equation}
\frac{dE}{dt}=\epsilon\omega(x,y,z,\epsilon),  \label{slowlyvarying}
\end{equation}
where 
\begin{equation}
\omega(x,y,z,\epsilon)=\frac{\partial H}{\partial x} (x,y).f(x,y,z,\epsilon)+
\frac{\partial H}{\partial y} (x,y).g(x,y,z,\epsilon).  \label{omega}
\end{equation}
Over time $1$, the quantity $E(t,\epsilon)$ remains nearly constant and the
problem is to describe what happens over time $1/\epsilon$. It is more
natural to consider system \eqref{original} and equation 
\eqref{slowlyvarying} at the time scale $\tau=\epsilon t$. Let 
${}'=d/d\tau$ be the derivative with respect to $\tau$. 
System \eqref{original}, and equation \eqref{slowlyvarying} becomes 
\begin{equation}
\begin{gathered} x'=\frac{1}{\epsilon}\frac{\partial H}{\partial y}
(x,y)+f(x,y,z,\epsilon),\\ y'=-\frac{1}{\epsilon}\frac{\partial H}{\partial
x} (x,y)+g(x,y,z,\epsilon),\\ z'=\frac{1}{\epsilon^{2}}h(x,y,z,\epsilon),
\end{gathered}  \label{original'}
\end{equation}
and 
\begin{equation}
E'=\omega(x,y,z,\epsilon).  \label{slowlyvarying'}
\end{equation}
Let us denote $\Omega(x,y):=\omega(x,y,\xi(x,y),0)$, and make another
assumption to avoid boundary problems:

\begin{itemize}
\item[(H4)] The region of oscillations $I$ of (\ref{hamiltonian}) is non
empty and there exists a compact subinterval $J$ of $I$ such that 
$K=\cup_{E\in J} C(E)$.
\end{itemize}

Consider the averaged equation 
\begin{equation}
\bar{E}'=\frac{M(\bar{E})}{P(\bar{E})}:=\frac{1}{P(\bar{E})}
\int_0^{P(\bar{E})} \Omega(q(v,\bar{E}),p(v,\bar{E}))dv,  \label{strobos}
\end{equation}
defined in $\mathrm{int}J$. We recall that $(q(t,E),p(t,E))$ is the periodic
solution of (\ref{hamiltonian}) of energy $E$ and period $P(E)$.

\begin{itemize}
\item[(H5)] Equation \eqref{strobos} has the property of uniqueness of
solutions with prescribed initial conditions.
\end{itemize}

\begin{theorem}\label{main}
Suppose that assumptions $H1$ to {\rm (H5)} are satisfied. Let
$\gamma(\tau,\epsilon)=((x(\tau,\epsilon),y(\tau,\epsilon
),z(\tau,\epsilon))$ be a solution of \eqref{original'} with initial
condition \eqref{initialdata}. Suppose that
$E_0=H(q_0,p_0 )\in\mathrm{int}J$. Let
$E(\tau)=H(x(\tau,\epsilon),y(\tau,\epsilon))$
be the total energy of $\gamma(\tau,\epsilon)$. Let $\bar{E}(\tau)$
be the solution of the averaged equation \eqref{strobos} with initial
condition $E_0$ and let $L$ be in its positive interval of definition.
Then, for every $\eta>0$, there exists $\epsilon^{\ast}>0$ such that
for all $0<\epsilon <\epsilon^{\ast}$ the function $E(\tau)$
satisfies $|E(\tau)-\bar{E} (\tau)|<\eta$ for all $\tau$ in $[0,L]$.
\end{theorem}

\section{External results}

\label{nsa}

Theorems \ref{tykhonov2} and \ref{main2} below are external results which
respectively imply Theorems \ref{tykhonov} and \ref{main}.

\begin{theorem}\label{tykhonov2}
Let $f$, $g$, $H$, $h$, $\xi$, $p_0$, $q_0$, $z_0$ and
$E_0$ be standard. Suppose that {\rm  (H1)--(H3)} are satisfied.
Let $\tilde{z}(s)$ be the solution of the boundary layer equation
\eqref{layer} and $(q(t,E_0),p(t,E_0))$ the solution of the reduced
equation \eqref{reduced}. Let $\epsilon>0$ be infinitesimal and $T$ be a
standard real number in the positive interval of definition of $(q(t,E_0
),p(t,E_0))$. Then, any solution $\gamma(t)=(x(t),y(t),z(t))$ of
\eqref{original} is defined at least on $[0,T]$ and there exists $\omega$ such
that $\epsilon\omega\simeq0$ and
\begin{gather*}
z(\epsilon s)\simeq\tilde{z}(s), \quad \text{for all }0\leq s\leq\omega,\\
x(t)\simeq q(t,E_0),\quad y(t)\simeq p(t,E_0) \quad
 \text{for all }0\leq t\leq T,\\
z(t)\simeq\xi(q(t,E_0),p(t,E_0)),\quad  \text{for all }
 \epsilon\omega\leq t\leq T.
\end{gather*}
\end{theorem}

\begin{theorem} \label{main2}
Suppose that {\rm (H1)--(H5)} are satisfied. Let $f$,
$g$, $H$, $h$, $\xi$, $p_0$, $q_0$, $z_0$ be standard. Let $\epsilon$
be positive infinitesimal. Let $\gamma(\tau)=((x(\tau),y(\tau),z(\tau))$ be a
solution of \eqref{original'} with initial condition \eqref{initialdata}.
Suppose that $E_0=H(q_0,p_0)\in\mathrm{int}J$. Let $E(\tau
)=H(x(\tau),y(\tau))$ be the total energy of $\gamma(\tau)$. Let $\bar{E}
(\tau)$ be the solution of the averaged equation \eqref{strobos} with initial
condition $E_0$ and let $L$ standard be in its positive interval of
definition. Then, the function $E(\tau)$ satisfies $E(\tau)\simeq\bar{E}
(\tau)$ for all $\tau\in[0,L]$.
\end{theorem}

The proof is postponed to Section \ref{proof}. Let us first show that
Theorem \ref{main2} reduces to Theorem \ref{main}. We will need the
following frequent reduction formula~due to Nelson \cite{nelson} 
\begin{equation}
\forall x~(\forall^{st}y~A\Rightarrow\forall^{st}z~B)\equiv\forall
z~\exists^{fin}y'~\forall x~(\forall y\in y'~A\Rightarrow
B),  \label{reduction}
\end{equation}
where $A$ (respectively $B$) is an internal formula with free variable $y$
(respectively $z$) and standard parameters. The notation $\forall^{st}$
means ``for all standard'' and $\exists^{fin}$ means ``there is a finite''.

\begin{proof}[Proof of Theorem \ref{main}]
Let $B$ be the formula occurring in Theorem \ref{main}: ``the function 
$E(\tau)$ satisfies $|E(\tau)-\bar{E}(\tau)|<\eta$ for all $\tau$ in 
$[0,L]$''. To say that ``the function $E(\tau)$ satisfies 
$E(\tau)\simeq \bar{E}(\tau)$ for all $\tau$ in $[0,L]$'' 
is the same as to say $\forall^{st}\eta$ 
$B$. To say that ``$\epsilon>0$ is infinitesimal'' is the same as to say 
$\forall^{st}\epsilon^{\ast}$ $0<\epsilon<\epsilon^{\ast}$. Hence, 
Theorem \ref{main2} asserts that 
\[
\forall\epsilon~(\forall^{st}\epsilon^{\ast}~0<\epsilon
<\epsilon^{\ast}\Rightarrow\forall^{st}\eta\text{~}B).
\]
In this formula, $f$, $g$, $H$, $h$, $p_0$, $q_0$, $E_0$ and $L$ are
standard parameters, $\epsilon$ and $\eta$ range over the strictly positive
real numbers. By \eqref{reduction}, the last formula is equivalent to 
\[
\forall\eta\;\exists^{fin}\epsilon^{\ast'}\;
\forall\epsilon\;(\forall\epsilon^{\ast}\in\epsilon^{\ast'}\; 
0<\epsilon<\epsilon^{\ast}\Rightarrow B).
\]
But for $\epsilon^{\ast'}$ finite set, to say $\forall\epsilon
^{\ast}\in\epsilon^{\ast'}$~$0<\epsilon<\epsilon^{\ast}$ is the
same as to say $0<\epsilon<\epsilon^{\ast}$ for $\epsilon^{\ast}
=\min\epsilon^{\ast'}$. Hence, the formula is equivalent to 
\[
\forall\eta\text{~}\exists\epsilon^{\ast}\text{~}\forall\epsilon
~(0<\epsilon<\epsilon^{\ast}\Rightarrow B).
\]
This means that for any standard $f$, $g$, $H$, $h$, $p_0$, $q_0$, $E_0$ and 
$L$, the statement of Theorem \ref{main} holds, thus by transfer, it holds
for all $f$, $g$, $H$, $h$, $p_0$, $q_0$, $E_0$ and $L>0$.
\end{proof}

\section{Case of a slowly varying parameter}

\label{parameter}

As explained in the introduction, we present the case where the slow motion
is described by a Hamiltonian system depending on a slowly varying parameter 
$\lambda\in D$ where $D$ is a compact of $\mathbb{R}^{k}$ such that 
$\mathrm{int}D\neq\emptyset$. More exactly, at the time scale 
$\tau =\epsilon t$ , we examine the problem\footnote{
Contrarily to system (\ref{1}) given in the introduction, we dropped the
parameter $\epsilon$ in the expressions of the functions without lost of
generality.} 
\begin{equation}
\begin{gathered} x'=\frac{1}{\epsilon}\frac{\partial H}{\partial
y}(x,y,\lambda )+f(x,y,z,\lambda),\\ y'=-\frac{1}{\epsilon}\frac{\partial
H}{\partial x} (x,y,\lambda)+g(x,y,z,\lambda),\\
z'=\frac{1}{\epsilon^{2}}h(x,y,z,\lambda),\\ \lambda'=\alpha(x,y,z,\lambda),
\end{gathered}  \label{slowlyvar}
\end{equation}
with initial condition 
\begin{equation}
x(0)=q_0,\quad y(0)=p_0,\quad z(0)=z_0,\quad \lambda(0)=\lambda_0.
\label{initial}
\end{equation}
We still denote by $J$ a compact region of oscillations of the Hamiltonian
function $H(p,q,\lambda)$. The total energy of a solution $\gamma
(\tau)=((x(\tau),y(\tau),z(\tau),\lambda(\tau))$ verifies 
\begin{equation}
E'=\omega(x,y,z,\lambda),  \label{Elambda}
\end{equation}
where 
\begin{equation}
\omega(x,y,z,\lambda)=\frac{\partial H}{\partial x}.f
+\frac{\partial H}{\partial y}.g+\frac{\partial H}{\partial\lambda}\alpha.  \label{omegalambda}
\end{equation}
Under Tikhonov's Theorem conditions, the trajectory is supposed to jump to
the neighborhood of the slow attractive manifold $\{z=\xi(x,y,\lambda),
\lambda=\lambda_0\}$ and is first approximated by the solution of the
reduced equation 
\begin{gather*}
\frac{dq}{dt}=\frac{\partial H}{\partial p}(p,q,\lambda_0),\quad q(0,E_0
,\lambda_0)=q_0, \\
\frac{dp}{dt}=-\frac{\partial H}{\partial q}(p,q,\lambda_0),\quad
p(0,E_0,\lambda_0)=p_0.
\end{gather*}
We want to give an approximation of the very slow drift of $E$ and 
$\lambda$. Let us denote by 
\begin{equation}  \label{oa}
\begin{gathered} \Omega(x,y,\lambda)
:=\omega(x,y,\xi(x,y,\lambda),\lambda),\\ A(x,y,\lambda)
:=\alpha(x,y,\xi(x,y,\lambda),\lambda), \end{gathered}
\end{equation}
and define the equations 
\begin{equation}
\bar{E}'=\frac{M(\bar{E},\bar{\lambda})}{P(\bar{E},\bar{\lambda} )}
:=\frac{1}{P(\bar{E},\bar{\lambda})} \int_0^{P(\bar{E},\bar{\lambda})}
\Omega(q(\nu,\bar{E},\bar{\lambda}),p(\nu,\bar{E},\bar{\lambda}),\bar{
\lambda })d\nu,  \label{strobosbis}
\end{equation}
and 
\begin{equation}
\bar{\lambda}'=\frac{N(\bar{E},\bar{\lambda})}{P(\bar{E},\bar{
\lambda })}:=\frac{1}{P(\bar{E},\bar{\lambda})} \int_0^{P(\bar{E},\bar{
\lambda})} A(q(\nu,\bar{E},\bar{\lambda}),p(\nu,\bar{E},\bar{\lambda}),\bar{
\lambda} )d\nu,  \label{stroboster}
\end{equation}
where $(q(t,E,\lambda),p(t,E,\lambda))$ is the periodic solution of 
\[
\frac{dq}{dt}=\frac{\partial H}{\partial p}(p,q,\lambda),~\frac{dp} {dt}=-
\frac{\partial H}{\partial q}(p,q,\lambda),
\]
of energy $E$ and period $P(E,\lambda)$. We claim what follows.

\begin{theorem}\label{parametre}
Let $f$, $g$, $H$, $h$, $\alpha$, $\xi$, $p_0$, $q_0$,
$z_0$, $\lambda_0$ be standard. Suppose that Tikhonov's conditions are
satisfied and that  \eqref{strobosbis} and \eqref{stroboster} have
the property of uniqueness. Let $\epsilon>0$ be infinitesimal. Let
$\gamma(\tau)=((x(\tau),y(\tau),z(\tau),\lambda(\tau))$ be a solution of
\eqref{slowlyvar} with initial condition \eqref{initial}. Suppose that
$E_0=H(q_0,p_0,\lambda_0)\in\mathrm{int}J$ and $\lambda_0
\in\mathrm{int}D$. Let $E(\tau)=H(x(\tau),y(\tau),\lambda(\tau))$ be
the total energy of $\gamma(\tau)$. Let $\bar{E}(\tau)$ and $\bar{\lambda}(\tau)$ be the
solutions of the equations \eqref{strobosbis} and \eqref{stroboster} with
initial conditions $E_0$ and $\lambda_0$ and let $L$ standard be in their
positive interval of definition. Then, the functions $E(\tau)$ and
$\lambda(\tau)$ satisfy $E(\tau)\simeq\bar{E}(\tau)$ and $\lambda(\tau)\simeq$
$\bar{\lambda}(\tau)$ for all $\tau\in[0,L]$.
\end{theorem}

The proof is given in the next Section.

\section{Proof of the main results}

\label{proof}

The key for proving the main results is the so called \textit{Stroboscopy
Lemma} (see \cite{sari,saristrobo}) which is an extension and an improvement
of the stroboscopic method outlined in the introduction. Let $\mathcal{O}$
be a standard open subset of $\mathbb{R}^{n}$, $F:\mathcal{O}\to \mathbb{R}
^{n}$ a standard continuous function. Let $\mathcal{I}$ be an interval of 
$\mathbb{R}$ containing $0$ and $\phi:\mathcal{I}\to \mathbb{R}^{n}$ a
function such that $\phi(0)$ is nearstandard in $\mathcal{O}$. Let 
$\mathcal{J}$ be a connected subset of $\mathcal{I}$, eventually an external
collection, such that $0\in\mathcal{J}$.

\begin{definition}[Stroboscopic property] \rm
Let $t$ and $t'$ be in $\mathcal{J}$. The
function $\phi$ is said to satisfy the stroboscopic property $\mathcal{S}
(t,t')$ if $[t,t']\subset\mathcal{J}$, $t'\simeq t$,
$\phi(s)\simeq\phi(t)$ for all $s$ in $[t,t']$ and
\[
\frac{\phi(t)-\phi(t')}{t-t'}\simeq F(\phi(t)).
\]
\end{definition}

Under suitable conditions, the Stroboscopy Lemma asserts that the function 
$\phi$ is approximated by the solution of the initial value problem 
\begin{equation}
\frac{dx}{dt}=F(x),\quad x(0)={}^\circ (\phi(0)),  \label{F}
\end{equation}
where ${}^\circ(\phi(0))$ denotes the standard part of $\phi(0)$.

\begin{theorem}[Stroboscopy Lemma]\label{stroboscopylemma}
Suppose that
\begin{itemize}
\item[(i)] There exists $\mu>0$ such that, whenever $t\in\mathcal{J}$
is limited and $\phi(t)$ is nearstandard in $\mathcal{O}$, there is
$t' \in\mathcal{J}$ such that $t'-t\geq\mu$ and the function $\phi$
satisfies the stroboscopic property $\mathcal{S}(t,t')$.

\item[(ii)] The initial value problem (\ref{F}) has a unique solution
$x(t)$.
\end{itemize}
Then, for any standard $L$ in the maximal positive interval of
definition of $x(t)$, we have $[0,L]\subset\mathcal{J}$ and
$\phi(t)\simeq x(t)$ for all $t\in[0,L]$.
\end{theorem}

\subsection*{Proof of Theorem \ref{main2}}

Consider $\tau_1\geq0$ such that $[0,\tau_1]\subset[0,L]$ and $E(\tau)$ is
nearstandard in $\mathrm{int}J$ for all $\tau\in[0,\tau _1]$. Let us
consider the external collection 
\[
\mathcal{J}=\{\tau\geq0:E(s)\text{ nearstandard in int$J$ for all }s\in[0,\tau]\}
\]
which contains the interval $[0,\tau_1]$. Let us show that $E(\tau)$
satisfies the hypothesis $(i)$ of the Stroboscopy Lemma 
(Theorem \ref{stroboscopylemma}).

Let $\mu=\epsilon\min_{E\in J} P(E)$. Since $P$ does not vanish and is
continuous and $J$ is a compact subset, $\mu$ is positive. Let 
$\tau'$ limited in $\mathcal{J}$, thus $E(\tau')$ is nearstandard in 
$\mathrm{int}J$. Let us make the change of variables 
\begin{equation}
r=\frac{\tau-\tau'}{\epsilon},\quad F(r)
=\frac{E(\tau'+\epsilon r)-E(\tau')}{\epsilon},  \label{change}
\end{equation}
which transforms the system formed by \eqref{original'} and 
\eqref{slowlyvarying'} with initial condition \\
$(x(\tau'),y(\tau'),z(\tau'),E(\tau'))$ into 
\begin{equation}
\begin{gathered} \frac{dx}{dr}=\frac{\partial H}{\partial y}(x,y)+\epsilon
f(x,y,z,\epsilon),\\ 
\frac{dy}{dr}=-\frac{\partial H}{\partial
x}(x,y)+\epsilon g(x,y,z,\epsilon),\\
\epsilon\frac{dz}{dr}=h(x,y,z,\epsilon),\quad \frac{dF}{dr}=\omega
(x,y,z,\epsilon), 
\end{gathered}  \label{dF}
\end{equation}
with initial condition $(x(\tau'),y(\tau'),z(\tau'),0)$. 
Moreover, according to Tikhonov's Theorem, the components $x(r)$, 
$y(r)$ and $F(r)$ of (\ref{dF}) are infinitely close, for all limited $r$, to
the solution of the standard system 
\begin{gather*}
\frac{dx}{dr}=\frac{\partial H}{\partial y}(x,y), \\
\frac{dy}{dr}=-\frac{\partial H}{\partial x}(x,y), \\
\frac{dF}{dr}=\Omega(x,y),
\end{gather*}
with initial condition $(^{o}x(\tau'^oy(\tau'),0)$, where 
$^{o}x(\tau')$ and $^{o}y(\tau')$ are the standard parts of 
$x(\tau')$ and $y(\tau')$. Hence, for all limited $r$, 
\begin{gather*}
x(r)\simeq q( r,E(\tau')) \simeq q( r,E') , \\
y(r)\simeq p( r,E(\tau')) \simeq p( r,E') ,
\end{gather*}
where $E'$ is the standard part of $E(\tau')$ and 
\[
F(r)\simeq \int_0^{r} \Omega(q(\nu,E'),p(\nu,E'))d\nu.
\]
In particular, by periodicity, we obtain 
\begin{equation}
F(P(E'))\simeq \int_0^{P(E')} \Omega(q(\nu,E'),p(\nu,E'))d\nu.  \label{C}
\end{equation}
We define now the successive instant of observation by 
$\tau''=\tau'+\epsilon P(E')$. We claim that 
$\tau''\in\mathcal{J}$. Indeed, since $\tau'$ is in $\mathcal{J}$, 
$E(s)$ is nearstandard in $\mathrm{int}J$ for all $s\in[0,\tau']$.
On the other hand, let $s\in[\tau',\tau'']$. Let 
$s=\tau'+\epsilon r$. By (\ref{change}) we have that $E(s)=E(\tau
')+\epsilon F(r)\simeq E(\tau')\simeq E'$ for all 
$r$ in $[0,P(E')]$. Thus, $E(s)$ is nearstandard in 
$\mathrm{int}J$. We proved that, for any $\tau'$ limited in $\mathcal{J}$ and 
$E(\tau')$ nearstandard in $\mathrm{int}J$, there exists 
$\tau''$ such that $0\simeq\tau''-\tau'\geq\mu$, 
$[\tau',\tau'']\subset\mathcal{J}$, 
$E(s)\simeq E(\tau')$ for all $s$ in $[\tau',\tau'']$. Moreover, by (\ref{C}), 
\[
\frac{E(\tau'')-E(\tau')}{\tau''-\tau'}=\frac{F(P(E'))}{P(E')}\simeq\frac{
M(E')}{P(E')}\simeq\frac{M(E(\tau'))}{
P(E(\tau'))}.
\]
By the Stroboscopy Lemma \ref{stroboscopylemma}, 
\begin{equation}
[0,L]\subset\mathcal{J}\quad\text{and}\quad E(\tau)\simeq\bar{E}(\tau)\text{
for all }\tau\in[0,L].  \label{strob}
\end{equation}

\subsection*{Proof of Theorem \ref{parametre}}

Consider $\tau_1\geq0$ such that $[0,\tau_1]\subset[0,L]$ and $E(\tau)$,
(resp. $\lambda(\tau)$) nearstandard in $\mathrm{int}J$ (in $\mathrm{int}D$
) for all $\tau\in[0,\tau_1]$. Let us consider the external collection 
\[
\mathcal{J}=\{\tau\geq0:E(s)\text{, (resp. } \lambda(s)\text{) nearstandard
in }\mathrm{int}J\text{ (in }\mathrm{int}D\text{ ) for all }s\in[ 0,\tau]\}.
\]
Let $\mu=\epsilon\min\{P(E,\lambda),\ E\in J,\ \lambda\in D\}$. Let 
$\tau'$ limited in $\mathcal{J}$. The change of variables 
\begin{equation}
r=\frac{\tau-\tau'}{\epsilon},\quad F(r)=\frac{E(\tau'+\epsilon r)-E(\tau')}{\epsilon},\quad \Lambda(r)=\frac{
\lambda(\tau'+\epsilon r)-\lambda(\tau')}{\epsilon},
\label{changevar}
\end{equation}
transforms the system formed by \eqref{slowlyvar} and (\ref{Elambda}) with
initial condition 
\[
(x(\tau'),y(\tau'),z(\tau'),\lambda(\tau'),E(\tau'))
\]
into 
\begin{equation}
\begin{gathered} \frac{dx}{dr}=\frac{\partial H}{\partial
y}(x,y,\lambda(\tau') +\epsilon\Lambda(r))+\epsilon
f(x,y,z,\lambda(\tau')+\epsilon\Lambda(r)),\\ \frac{dy}{dr}=-\frac{\partial
H}{\partial x}(x,y,\lambda(\tau') +\epsilon\Lambda(r))+\epsilon
g(x,y,z,\lambda(\tau')+\epsilon\Lambda(r)),\\
\epsilon\frac{dz}{dr}=h(x,y,z,\lambda(\tau')+\epsilon \Lambda(r)),\\
\frac{d\Lambda}{dr}=\alpha(x,y,z,\lambda(\tau')+\epsilon \Lambda(r)),\\
\frac{dF}{dr}=\omega(x,y,z,\lambda(\tau')+\epsilon\Lambda(r)), \end{gathered}
\label{dFL}
\end{equation}
with initial condition $(x(\tau'),y(\tau'),z(\tau'),0,0)$. Let $\lambda'$ be the standard part of $\lambda(\tau
')$. According to Tikhonov's\ theorem one can state that for all
limited $r$, the components $x(r)$, $y(r)$, $\lambda(r)$ and $F(r)$ of 
(\ref{dFL}) are infinitely close to the solution of the standard system 
\begin{gather*}
\frac{dx}{dr}=\frac{\partial H}{\partial y}(x,y,\lambda'), \\
\frac{dy}{dr}=-\frac{\partial H}{\partial x}(x,y,\lambda'), \\
\frac{d\Lambda}{dr}=A(x,y,\lambda'), \\
\frac{dF}{dr}=\Omega(x,y,\lambda'),
\end{gather*}
with initial condition $(^{o}x(\tau'^oy(\tau'),0,0)$, where 
$^{o}x(\tau')$ and $^{o}y(\tau')$ are the standard parts
of $x(\tau')$ and $y(\tau')$. That is, if $E'$ is
the standard part of $E(\tau')$, then for all limited $r$, 
\begin{gather*}
x(r)\simeq q( r,E',\lambda') , \\
y(r)\simeq p( r,E',\lambda')) , \\
F(r)\simeq \int_0^{r} \Omega(q(\nu,E',\lambda'),
p(\nu,E',\lambda'),\lambda')d\nu, \\
\Lambda(r)\simeq \int_0^{r} A(q(\nu,E',\lambda'),p(\nu,E',\lambda'),\lambda')d\nu.
\end{gather*}
By periodicity, we also have 
\begin{gather*}
F(P(E',\lambda'))\simeq \int_0^{P(E',\lambda')} \Omega(q(\nu,E',\lambda'),
p(\nu,E',\lambda'),\lambda')d\nu, \\
\Lambda(P(E',\lambda'))\simeq \int_0^{P(E',\lambda')} 
A(q(\nu,E',\lambda'),p(\nu,E',\lambda'),\lambda')d\nu.
\end{gather*}
Let $\tau''=\tau'+\epsilon P(E',\lambda
')\in\mathcal{J}$ be the successive instant. By (\ref{changevar}), 
\begin{gather*}
\frac{E(\tau'')-E(\tau')}{\tau''-\tau'}
=\frac{F(P(E',\lambda'))}{P(E',\lambda')}\simeq\frac{M(E',\lambda')}{
P(E',\lambda')}\simeq\frac{M(E(\tau'),\lambda(\tau'))}{P(E(\tau'),\lambda(\tau'))},
\\
\frac{\lambda(\tau'')-\lambda(\tau')}{\tau''-\tau'}=\frac{A(P(E',\lambda'))}{
P(E',\lambda')}\simeq\frac{N(E',\lambda')
}{P(E',\lambda')}\simeq\frac{N(E(\tau'),\lambda(\tau'))}{P(E(\tau'),\lambda(\tau'))}
\end{gather*}
By the Stroboscopy Lemma, $[0,L]\subset$ $\mathcal{J}$, $E(\tau)\simeq\bar {E
}(\tau)$ and $\lambda(\tau)\simeq\bar{\lambda}(\tau)$ for all $\tau$ in 
$[0,L]$.

\section{Applications}

\label{application}

The following examples should be viewed more as didactic examples to
illustrate the results, than as arising from practical problems.

\subsection*{Example 1}

The system associated to the following third order differential equation 
\[
\epsilon \dddot{x}=h(x,\ddot{x}),
\]
where $\epsilon>0$ is a small real parameter and $h$ a sufficiently smooth
function, is given by 
\begin{equation}
\dot{x}=y, \quad \dot{y}=z_1, \quad \epsilon\dot{z}_1=h(x,z_1),
\label{oscillator}
\end{equation}
where the dot denotes the derivative with respect to $t$. We suppose that 
$z_1=u(x)$ is an isolated root of $h(x,z_1)=0$ and that Tikhonov's Theorem
conditions are satisfied; in particular, $\frac{\partial h}{\partial z_1}
(x,z_1)<0$, which makes the slow manifold $z_1=u(x)$ attractive. To obtain
the general form \eqref{original}, we apply the change of variable 
\[
\epsilon z=z_1-u(x),
\]
which transforms (\ref{oscillator}) into the system 
\begin{equation}
\dot{x}=y, \quad \dot{y}=u(x)+\epsilon z, \quad \epsilon\dot{z}=\tilde {h}
(x,y,z_1,\epsilon),  \label{oscillator'}
\end{equation}
where 
\[
\tilde{h}(x,y,z,\epsilon)=\frac{\partial h}{\partial z_1} (x,u(x)).z-u^{
\prime }(x)y+o(\epsilon).
\]
The slow equation 
\begin{equation}
\dot{q}=p,\quad \dot{p}=u(q),  \label{slowmotion}
\end{equation}
is a Hamiltonian system with the Hamiltonian function 
\[
H(p,q)=-U(q)+\frac{p^{2}}{2},
\]
where $U'=u$. The formula (\ref{omega}) becomes 
\[
\omega(x,y,z,\epsilon)=\epsilon yz,
\]
and the averaged equation \eqref{strobos}, where $\tau=\epsilon t$, takes
the form 
\begin{equation}
\frac{d\bar{E}}{d\tau}=\frac{M(\bar{E})}{P(\bar{E})}=\frac{1}{P(\bar{E})}
\int_0^{P(\bar{E})} \Omega(q(v,\bar{E}),p(v,\bar{E}))dv,  \label{moy}
\end{equation}
where 
\begin{equation}
\Omega(q,p)=u'^2\big( \frac{\partial h}{\partial z_1 }(q,u(q))\big) 
^{-1},  \label{omegaex}
\end{equation}
and $(q(v,\bar{E}),p(v,\bar{E}))$ is a $P(\bar{E})$-periodic solution of 
(\ref{slowmotion}). One can see that 
\begin{equation}
P(\bar{E})=2\int_{q_1(\bar{E})}^{q_{2}(\bar{E})}\frac{dq}{\sqrt{2(\bar {E}
+U(q))}},  \label{periodex}
\end{equation}
where $q_1(\bar{E})$ and $q_{2}(\bar{E})$ are respectively the minimum and
the maximum of an oscillation on the closed orbit $C(\bar{E})$. According to
Theorem \ref{tykhonov}, the solution $(x(t,\epsilon),y(t,\epsilon),z(t,
\epsilon))$ of (\ref{oscillator'}) with initial condition $(p_0,q_0,z_0)$
mainly satisfies 
\begin{gather*}
\lim_{\epsilon\to 0} (x(t,\epsilon),y(t,\epsilon)) =(q(t,E_0),p(t,E_0))\quad 
\text{for all }t\in[0,kP(E_0)],\; k\in \mathbb{N}, \\
\lim_{\epsilon\to 0} z(t,\epsilon) =u(q(t,E_0))\quad \text{for all }
t\in]0,kP(E_0)],
\end{gather*}
where $E_0=\frac{p_0^{2}}{2}-U(q_0)$. Moreover, according to Theorem 
\ref{main}, the total energy $E(t,\epsilon)=H(x(t,\epsilon ),y(t,\epsilon))$ of
the system satisfies 
\[
\lim_{\epsilon\to 0} E(t,\epsilon)=\bar{E}(\epsilon t)\quad \text{for all }t
\in[0,L/\epsilon],
\]
where $\bar{E}(\tau)$ is the solution of (\ref{moy}) with initial condition 
$E_0$ and defined on $[0,L]$.

To illustrate the effectiveness of the stroboscopy method, we present a
numerical simulation of the example above where we chose $h(x,\ddot{x} )=-
\ddot{x}-x$, which gives a singularly perturbed harmonic oscillator. Hence, 
$u(x)=-x$ and system (\ref{oscillator'}) corresponds to 
\begin{equation}  \label{oscillator1}
\begin{gathered} \dot{x} =y,\\ \dot{y} =-x+\epsilon z,\\ \epsilon\dot{z}
=y-z. \end{gathered}
\end{equation}
The Hamiltonian function of the corresponding slow Hamiltonian equation is 
\[
H(q,p)=\frac{1}{2}q^{2}+\frac{1}{2}p^{2},
\]
and the period is exactly 
\[
P(\bar{E})=2\int_{-\sqrt{2\bar{E}}}^{\sqrt{2\bar{E}}} \frac{dq}{\sqrt{2(\bar{
E}-\frac{1}{2}q^{2})}}=2\pi.
\]
Moreover, according to (\ref{omegaex}) and to the first equation of 
\eqref{oscillator1}, 
\begin{align*}
M(\bar{E}) & = \oint_{C(\bar{E})} p^{2}(\nu,\bar{E})d\nu= \oint_{C(\bar{E})} 
\sqrt{2\bar{E}-q^{2}}dq \\
& =2\sqrt{2\bar{E}}\int_{-\sqrt{2\bar{E}}}^{\sqrt{2\bar{E}}}\sqrt{1-( \frac{q
}{\sqrt{2\bar{E}}}) ^{2}}dq.
\end{align*}
By the change of variable $X=q/\sqrt{2\bar{E}}$, one has 
\[
M(\bar{E})=4 \bar{E}\int_{-1}^{1}( \sqrt{1-X^{2}}) dX=2\pi\bar {E}.
\]
If we fix the initial conditions $(p_0,q_0,z_0)=(1,2,1)$, the averaged
equation (\ref{moy}) is simply 
\begin{equation}
\frac{d\bar{E}}{d\tau}=\bar{E},\quad \bar{E}(0)=\frac{5}{2},
\label{averagex1}
\end{equation}
with exact solution 
\[
\bar{E}(\tau)=\frac{5}{2}e^{\tau}.
\]

Figure \ref{jump} shows how the considered trajectory jumps to the
neighborhood of the slow manifold $z=y$ of the system \eqref{oscillator1}
before it evolves along the closed orbits of the slow equation drawn on this
slow manifold. Figure \ref{example1} is a comparison between the exact
variation of the total energy $E(\tau)$ and the solution $\bar {E}(\tau)$ of
the averaged equation (\ref{averagex1}) with the mentioned initial
conditions. Note that the oscillating curve corresponds to $E(\tau)$.

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig1} 
\end{center}
\caption{Numerical simulation of the trajectory of \eqref{oscillator1} with
initial condition $(1,2,10)$, $\epsilon=.01$, $t=0..100$}
\label{jump}
\end{figure}

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig2} 
\end{center}
\caption{Comparison between $E(\tau)$ and $\bar{E}(\tau)$
for the system \eqref{oscillator1} with $\epsilon=.01$}
\label{example1}
\end{figure}

\subsection*{Example~2}

Consider the system 
\begin{equation}  \label{slowlyvale}
\begin{gathered} \dot{x} =y,\\ \dot{y} =-\lambda x+\epsilon z,\\
\epsilon\dot{z} =-z+\lambda y,\\ \dot{\lambda} =\epsilon\lambda xy,
\end{gathered}
\end{equation}
In the same way as above, one can obtain 
\[
P(\bar{E},\bar{\lambda})=2\int_{-\sqrt{2\bar{E}/\bar{\lambda}}}^{\sqrt {2
\bar{E}/\bar{\lambda}}}\frac{dq}{\sqrt{2(\bar{E}-\frac{1}{2}\bar{\lambda }
q^{2})}}=\frac{2\pi}{\sqrt{\bar{\lambda}}}.
\]
According to (\ref{omegalambda}), (\ref{oa}), \eqref{strobosbis} and the
first equation of (\ref{slowlyvale}), we get 
\begin{align*}
M(\bar{E},\bar{\lambda}) & = \oint_{C(\bar{E},\bar{\lambda})} [\bar{\lambda}
p^{2}(\nu,\bar{E},\bar{\lambda})+\frac{\bar{\lambda}}{2} q^{3}(\nu,\bar{E},
\bar{\lambda})p(\nu,\bar{E},\bar{\lambda}]d\nu \\
&= \oint_{C(\bar{E},\bar{\lambda})} \bar{\lambda}pdq+ \oint_{C(\bar{E},\bar{
\lambda})} \frac{\bar{\lambda}}{2}q^{3}dq \\
& =2\sqrt{2\bar{E}}\int_{-\sqrt{2\bar{E}/\bar{\lambda}}}^{\sqrt{2\bar{E} /
\bar{\lambda}}}\bar{\lambda}\sqrt{2\bar{E}-\bar{\lambda}q^{2}}dq+0 \\
&=\frac{2\pi\bar{\lambda}\bar{E}}{\sqrt{\bar{\lambda}}}.
\end{align*}
According to (\ref{oa}), \eqref{stroboster} and the first equation of 
(\ref{slowlyvale}), we also get 
\[
N(\bar{E},\bar{\lambda})= \oint_{C(\bar{E},\bar{\lambda})} \bar{\lambda}
^{2}qpd\nu=2\int_{-\sqrt{2\bar{E}/\bar{\lambda}}}^{\sqrt{2\bar {E}/
\bar{\lambda}}}qdq=0.
\]
Hence, the averaged system describing the drift of $E$ and $\lambda$ is
given by the simple system 
\begin{equation}
\begin{gathered}
\bar{E}'=\frac{M(\bar{E},\bar{\lambda})}{P(\bar{E},\bar{\lambda}
)}:=\lambda\bar{E},\\
\bar{\lambda}'=\frac{N(\bar{E},\bar{\lambda})}{P(\bar{E},\bar
{\lambda})}:=0. \end{gathered}  \label{elambda}
\end{equation}

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig3} 
\end{center}
\caption{Comparison between $E(\tau)$ and $\bar{E}(\tau)$
for the system (\ref{slowlyvale}) with $\epsilon=.01$}
\label{energy}
\end{figure}

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig4} 
\end{center}
\caption{Comparison between $\lambda(\tau)$ and 
$\bar{\lambda}(\tau)$ for the system (\ref{slowlyvale})
with $\epsilon=.01$}
\label{variable}
\end{figure}

Figures \ref{energy} and \ref{variable} compare the exact solutions $E(\tau)$
and $\lambda(\tau)$ of (\ref{slowlyvale}) with initial condition $E_0=5/2$
and $\lambda_0=1$ at time scale $\tau=\epsilon t$, and the solutions 
$\bar{E}(\tau)=\frac{5}{2}e^{\tau}$ and $\bar{\lambda} (\tau)=1$ of (\ref{elambda}).
It is worth noting that in Figure \ref{variable} the difference between the
oscillating curve and the averaged one does not exceed 0.06 for 
$0\leq\tau\leq1$, that is for $0\leq t\leq100$.

\subsection*{Acknowledgements}
The authors wish to thank Prof. Tewfik Sari for his constructive and useful
comments, and the anonymous referees for their valuable advice and
recommendations.

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\end{document}