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\markboth{\hfil On Tykhonov's theorem for convergence \hfil EJDE--1998/19}%
{EJDE--1998/19\hfil Claude Lobry, Tewfik Sari, \&  Sefiane Touhami\hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol.\ {\bf 1998}(1998), No.~19, pp. 1--22. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113}
 \vspace{\bigskipamount} \\
  On Tykhonov's theorem for convergence of solutions of slow and fast systems 
\thanks{ 
{\em 1991 Mathematics Subject Classifications:} 34D15, 34E15, 03H05.
\hfil\break\indent
{\em Key words and phrases:} singular perturbations, deformations,
\hfil\break\indent asymptotic stability, nonstandard analysis.
\hfil\break\indent
\copyright 1998 Southwest Texas State University  and University of
North Texas. \hfil\break\indent
Submitted September 30, 1997. Published July 9, 1998.
\hfil\break\indent Supported by the GdR CNRS 1107.} }
\date{}
\author{Claude Lobry, Tewfik Sari, \&  Sefiane Touhami }

\maketitle

\begin{abstract} 
Slow and fast systems gain their special structure from the presence of two
time scales. Their analysis is achieved with the help of Singular
Perturbation Theory. The fundamental tool is Tykhonov's theorem which
describes the limiting behaviour, for compact interval of time, of solutions
of the perturbed system which is a one-parameter deformations of the
so-called unperturbed system. Our aim here is to extend this description to
the solutions of all systems that belong to a small neighbourhood of the
unperturbed system. We investigate also the behaviour of solutions on the
infinite time interval. Our results are formulated in classical mathematics.
They are proved within Internal Set Theory which is an axiomatic approach to
Nonstandard Analysis.
\end{abstract}

\section{Introduction}
Let us consider an initial value problem (IVP) of the form
\begin{eqnarray} 
&\varepsilon\dot x =  F(x,y,\varepsilon) \quad x(0)=\alpha_\varepsilon\,,&
\label{fs}\\
&\dot y =  G(x,y,\varepsilon)\quad y(0)=\beta_\varepsilon\,,& \nonumber 
\end{eqnarray}
where the dot ($\dot {}$) means $d/dt$, $x\in{\Bbb R}^n$, $y\in{\Bbb R}^m$ 
and parameter $\varepsilon$ is a positive real number. 
We consider the  behaviour of solutions when $\varepsilon$ is small. 
The small parameter
$\varepsilon$ multiplies the derivative so the usual theory of
continuous dependence of the solutions with respect to the parameters can not
be applied. The analysis of such systems is achieved with the help of the so
called {\em Singular Perturbation Theory}. The purpose of Singular Perturbation
Theory is to investigate the behaviour of solutions of (\ref{fs})
as $\varepsilon\rightarrow 0$ for $0\leq t\leq T$ and also for $0\leq
t<+\infty$. The vectors $x$ and $y$ are
the fast and slow components of the system. This system is called a 
{\em fast and slow system}. If we use the {\em fast time}, $\tau=t/\varepsilon$,
Problem (\ref{fs}) becomes
\begin{eqnarray}
&x' =  F(x,y,\varepsilon)\quad x(0)=\alpha_\varepsilon\,,&\label{fsl}\\
&y' =  \varepsilon G(x,y,\varepsilon)\quad y(0)=\beta_\varepsilon\,,&\nonumber
\end{eqnarray}
where $'=d/d\tau$. This problem is called the {\em perturbed problem}. 
It is a {\em regular perturbation} of the {\em unperturbed problem}
\begin{eqnarray} 
&x' =  F(x,y,0)\quad x(0)=\alpha_0\,,&\label{fs0}\\
&y' =  0 \quad y(0)=\beta_0\,.& \nonumber
\end{eqnarray}
Hence, first $x$ varies very quickly and is approximated by the solution of
the {\em boundary layer equation}
\begin{equation}\label{ble}
x'=F(x,\beta_0,0)\quad x(0)=\alpha_0\,,
\end{equation}
and $y$ remains close to its initial value $\beta_0$. The system of
differential equations
\begin{equation}\label{fe}
x'=F(x,y,0)\,,
\end{equation}
in which $y$ is a parameter, is called the {\em fast equation}. A solution of
(\ref{fe}) may behave in one of several ways: it may be unbounded as
$\tau\rightarrow\infty$,
it may tend toward an equilibrium point, or it may approach a more complex
attractor. Obviously, if the fast equation has multiple stable equilibria,
the asymptotic behaviour of a solution is determined by its initial value.
Assume the second case occurs, that is, the solutions of (\ref{fe}) tend
toward an equilibrium $\xi(y)$, where $x=\xi(y)$ is a root of equation
\begin{equation}\label{sm}
F(x,y,0)=0\,.
\end{equation}
The manifold of equation (\ref{sm}) is called the slow
manifold: it is
the set of equilibrium points of the fast equation (\ref{fe}); the
surface ${\cal{L}}$ of equation $x=\xi(y)$ is a component of the slow manifold.
The solution of (\ref{fs0}) is defined for all $\tau\geq0$ and tends
to $(\xi(\beta_0),\beta_0)$, namely to a point of
${\cal{L}}$. Hence a fast
transition brings the solution of problem (\ref{fs}) near the slow manifold.
Then, a slow motion takes place near the slow manifold, and is approximated
by the solution of the {\em reduced problem
}\begin{equation}\label{rp}
\dot y=G(\xi(y),y,0)\quad y(0)=\beta_0\,.
\end{equation}

The preceding description is definitely heuristic and imprecise. In a more
rigorous description we usually consider $\varepsilon$ as a parameter that
tends to 0 and we assume that Problem (\ref{fs}) has a unique solution
$x(t,\varepsilon)$,
$y(t,\varepsilon)$. Let $y_0(t)$ be the solution of the reduced Problem
(\ref{rp}),
which is assumed to be defined for $0\leq t\leq T$, then we have
$\lim_{\varepsilon\rightarrow0}y(t,\varepsilon)=y_0(t)$
for $0\leq t\leq T$.
We also have $\lim_{\varepsilon\rightarrow0}x(t,\varepsilon)=\xi(y_0(t))$,
but the limit holds only for $0<t\leq T$, since there is a boundary layer at 
$t=0$, for the $x$-component. Indeed, let $x_0(\tau)$ be the solution of the
boundary layer equation (\ref{ble}) then
$\lim_{\varepsilon\rightarrow0}x(\varepsilon\tau,\varepsilon)=x_0(\tau)$
for $0\leq\tau<+\infty$. This description of the solution of Problem
(\ref{fs}) was given
by Tykhonov \cite{T}, under the hypothesis that the equilibrium point $\xi(y)$
of equation (\ref{fe}) is asymptotically stable for all $y$ and that the
asymptotic stability is uniform with respect to $y$ (see Section~\ref{t}). A
highly recommended
classical reference for these matters is Chapter~X (especially Section~39) of
Wasow's book \cite{W}.

Hence, Singular Perturbation Theory describes the solutions of (2),
which is a one-parameter deformation of (3).
Actually, as noticed by Arnold (see \cite{A}, footnote page 157), the
behaviour of the perturbed problem solutions ``takes place in all
systems that are close to the original unperturbed system. Consequently, one
should simply study neighbourhoods of the unperturbed problem in a suitable
function space. However, here and in other problems of perturbation theory,
for the sake of mathematical convenience, in the statements of the results of
an investigation such as an asymptotic result, we introduce (more or less
artificially) a small parameter $\varepsilon$ and, instead of
neighborhoods, we
consider one-parameter deformations of the perturbed systems. The situation
here is as with variational concepts: the directional derivative (Gateaux
differential) historically preceded the derivative of a mapping (the
Fr\'echet differential)''.

The aim of this paper is to define a suitable function space of IVPs, and to
study small neighbourhoods of the unperturbed problem. This paper is organized
as follows. In Section~2 we describe a topology on the set of IVPs and we give
the asymptotic behaviour of the solutions of an IVP that lies in a small
neighbourhoods of an IVP which satisfies various hypotheses (Theorem~1). We
also investigate the solutions behaviour on the infinite time interval
(Theorem~2). For the proofs of Theorem~1 and 2, we use {\em Internal Set
Theory }(IST), which is an axiomatic approach of A. Robinson's {\em
Nonstandard Analysis }(NSA) \cite{R},
proposed by E. Nelson \cite{IST}. Section~3 starts with a short tutorial on
IST. Then we present
the nonstandard translates (Theorems~3 and 4) in the language of IST of
Theorems~1 and 2. This section ends with an external discussion of the
notion of uniform asymptotic stability, which is the crucial assumption for
the validity of the results.
In Section~4 we give the proofs of Theorems~3 and 4. We recall that IST is a
{\em conservative extension }of ordinary mathematics. This means that any
statement
of ordinary mathematics which is a theorem of IST was already a theorem of
ordinary mathematics, so there is no need to translate the proofs. 

We want to
emphasize that Theorems~3 and 4 were obtained directly from \cite{LST,S,Tou}.
Afterwards, we noticed
that the classical translations of these results are nothing more than
considering neighbourhoods, as suggested by Arnold. NSA allowed also 
the discovery and good understanding of new phenomena which are
not covered by Tykhonov's theory, namely the so called {\em canard }solutions.
These solutions are related to the important phenomenon of delayed loss of
stability in dynamical bifurcations \cite{L}. For more informations on the
applications of NSA to the asymptotic theory of differential equations, the
reader is referred to \cite{B,vdB,DD,DL,DW,LG,S}.


\section{Singular Perturbations}\label{SP}
\subsection{Slow and Fast Vectors Fields}
Our main problem is to study IVPs for fast and slow systems of the form
\begin{eqnarray}
&\varepsilon \dot x = f(x,y)\quad x(0)=\alpha & \label{sf}\\
&\dot y= g(x,y)\quad y(0)=\beta\,, & \nonumber
\end{eqnarray}
where $f:D\rightarrow{\Bbb R}^n$ and $g:D\rightarrow{\Bbb R}^m$
are continuous, $D$ is an open subset of ${\Bbb R}^{n+m}$, and
$(\alpha,\beta)\in D$. We denote by
\begin{eqnarray*}
{\cal{T}}&=&\big\{(D,f,g,\alpha,\beta):D\mbox{ open subset of }{\Bbb R}^{n+m},\\
&&\ (f,g):D\rightarrow{\Bbb R}^{n+m} \mbox{ continuous },~(\alpha,\beta)\in D
\big\}\,.
\end{eqnarray*}
Our aim is to study Problem (\ref{sf}) when $\varepsilon$ is small and
$(D,f,g,\alpha,\beta)$ is sufficiently close to an element
$(D_0,f_0,g_0,\alpha_0,\beta_0)$
satisfying various hypothesis. The hypothesis, which are denoted by the
letter H, are listed below.
The system of differential equations
\begin{equation}\label{f}
x'=f_0(x,y)\,,
\end{equation}
in which $y$ is a parameter, will be called the {\em fast equation}.

\proclaim (H1). For all $y$, the fast equation (\ref{f}) has the
uniqueness of the solutions with prescribed initial conditions.

We assume that the we are given an $n$-dimensional compact manifold
${\cal{L}}$,
which is contained in the set
\begin{equation}\label{sm1}
f_0(x,y)=0
\end{equation}
of equilibrium points of the fast equation (\ref{f}). The manifold
${\cal{L}}$ is
given as the graph of a function, that is, there is a
continuous mapping $\xi:Y\rightarrow{\Bbb R}^n$, $Y$ being a compact domain
in ${\Bbb R}^m$, such that
$(\xi(y),y)\in D_0$ for all $y\in Y$ and ${\cal{L}}=\{(x,y):x=\xi(y),~y\in
Y\}$.

\proclaim (H2).
The set $Y$ is a compact domain. The function $\xi$ is continuous.
For all $y\in Y$, $x=\xi(y)$ is an {\em isolated root }of equation (\ref{sm1}),
that is,
$f_0(\xi(y),y)=0$, and there exists a number $\delta>0$ such that the
relations $y\in Y$,
$\Vert x-\xi(y)\Vert<\delta$ and $x\neq\xi(y)$ imply $f_0(x,y)\neq0$.

It is not excluded that equation (\ref{sm1}) may have other roots beside
$\xi(y)$. The manifold defined by equation (\ref{sm1}) is called the {\em slow
manifold}.
We recall the concept of {\em uniform asymptotic stability }of equilibrium
points of equations depending on parameters.

\proclaim Definition 1. The equilibrium point $x=\xi(y)$ of the equation
(\ref{f}) is said to be\\
1. Stable (in the sense of Liapunov) if for every $\mu>0$ there exists a
$\eta$ with the property that any solution $x(\tau)$ of (\ref{f}) for
which $\Vert x(0)-\xi(y)\Vert<\eta$ can be continued for all $\tau>0$ and
satisfies the
inequality $\Vert x(\tau)-\xi(y)\Vert<\mu$.\\
Asymptotically stable if it is stable and, in addition,
$\lim_{\tau\rightarrow\infty}x(\tau)=\xi(y)$ for
all solutions such that $\Vert x(0)-\xi(y)\Vert<\eta$.\\
2. Attractive if it admits a basin of attraction, that is, a neighbourhood
${\cal{V}}$ with the property that any solution $x(\tau)$ of (\ref{f}) for
which
$x(0)\in{\cal{V}}$ can be continued for all $\tau>0$ and satisfies
$\lim_{\tau\rightarrow\infty}x(\tau)=\xi(y)$.\\
Moreover we say that the basin of attraction of the equilibrium point
$x=\xi(y)$ is uniform over $Y$ if there exists $a>0$ such that for all
$y\in Y$,
the ball ${\cal{B}}=\{x\in{\Bbb R}^n:\Vert x-\xi(y)\Vert\leq a\}$, of
center $\xi(y)$ and radius $a$, is a basin
of attraction of $\xi(y)$.

It is easy to see that an equilibrium point is asymptotically stable if and
only if it is stable and attractive. We must require that
\proclaim (H3). For each $y\in Y$, the point $x=\xi(y)$ is an
asymptotically stable equilibrium point of the fast equation (\ref{f}) and
the basin of attraction of $x=\xi(y)$ is uniform over $Y$.

The system of differential equations
\begin{equation}\label{s}
\dot y=g_0(\xi(y),y)\,,
\end{equation}
defined on the interior $Y_0$ of $Y$, will be called {\em the slow equation}.
Since the set $Y$ is compact, we restricted the slow equation
on $Y_0$ to avoid non essential technicalities with the maximal
interval of definition of a solution.

\proclaim (H4). The slow equation (\ref{s}), has the uniqueness of the
solutions with prescribed initial conditions.

\proclaim (H5). The point $\beta_0$ is in $Y_0$. The
point $\alpha_0$ is in the basin of attraction of the equilibrium point
$x=\xi(\beta_0)$.

We refer to the problem
\begin{equation}\label{ble1}
x'=f_0(x,\beta_0)\quad x(0)=\alpha_0\,,
\end{equation}
consisting of the fast equation (\ref{f}), where $y=\beta_0$, together with
the
initial condition $x(0)=\alpha_0$ as the boundary layer equation. Let
$x_0(\tau)$ be the
solution of the
boundary layer equation. According to the hypothesis~(H5), $x_0(\tau)$ is
defined
for all $\tau\geq0$ and $\lim_{\tau\rightarrow\infty}x_0(\tau)=\xi(\beta_0)$.
 We refer to the problem
\begin{equation}\label{rp1}
\dot y=g_0(\xi(y),y)\quad y(0)=\beta_0\,,
\end{equation}
consisting of the slow equation (\ref{s}) together with the
initial condition
$y(0)=\beta_0$, as the {\em reduced problem}.
 Let $y_0(t)$ be the solution of the
reduced problem. Let $I=[0,\omega)$, $0<\omega\leq+\infty$ be its maximal
positive interval of
definition.

Our result asserts that the curve $C$
consisting of two continuous arcs $C_1$ and $C_2$, where $C_1$ is the arc
$x=x_0(\tau)$, $y=\beta_0$, $0\leq\tau<+\infty$, and $C_2$ is the arc
$x=\xi(y_0(t))$, $y=y_0(t)$,
$t\in I$, gives
an approximation of the solutions $((x(t),y(t))$ of Problem (\ref{sf}),
when $\varepsilon$ is small enough and $(D,f,g,\alpha,\beta)$ is close to
$(D_0,f_0,g_0,\alpha_0,\beta_0)$ .
The closeness is measured by a topology on the set ${\cal{T}}$. To have a
convenient
definition of this topology, we introduce the notation
$\Vert h\Vert_\Delta=\sup_{x\in\Delta}\Vert h(x)\Vert$, where $h$ is a
function
defined on a set $\Delta$, with values in a normed space.

\proclaim Definition 2. The {\em topology of uniform convergence on
compacta} on
the set ${\cal{T}}$ is the topology for which the neighbourhood system of an
element
$(D_0,f_0,g_0,\alpha_0,\beta_0)$ is generated by the sets
\begin{eqnarray*}
V(\Delta,a)&=&\{(D,f,g,\alpha,\beta)\in{\cal{T}}:\Delta\subset D,~\Vert
f-f_0\Vert_\Delta<a,\\
&&\Vert
g-g_0\Vert_\Delta<a,~\Vert\alpha-\alpha_0\Vert<a,~\Vert\beta-\beta_0\Vert<a\}\end{eqnarray*}
where $\Delta$ is a compact subset of $D_0$ and $a$ is a real positive number.

We are now in a position to state our main result.

\proclaim Theorem~1. Let $f_0:D_0\rightarrow{\Bbb R}^n$,
$g_0:D_0\rightarrow{\Bbb R}^m$ and $\xi:Y\rightarrow{\Bbb R}^n$ be continuous
functions and let $(\alpha_0,\beta_0)$ be in $D_0$. Let hypotheses (H1) to
(H5) be satisfied. Let
$x_0(\tau)$ be the solution of the boundary layer equation (\ref{ble1}). Let
$y_0(t)$ be the
solution of the reduced problem (\ref{rp1}). Let $T$ be in $I$, $I$ being the
positive interval of
definition of $y_0$. For every $\eta>0$, there exists $\delta>0$ and a
neighbourhood
${\cal{V}}$ of $(D_0,f_0,g_0,\alpha_0,\beta_0)$ in ${\cal{T}}$ with the
properties that for all $\varepsilon<\delta$, and
all $(D,f,g,\alpha,\beta)\in{\cal{V}}$, any solution $(x(t),y(t))$ of the
problem (\ref{sf}) is
defined at least on $[0,T]$ and there exists $L>0$ such that $\varepsilon
L<\eta$, $\Vert x(\varepsilon\tau)-
x_0(\tau)\Vert<\eta$ for $0\leq\tau\leq L$, $\Vert x(t)-
\xi(y_0(t))\Vert<\eta$ for $\varepsilon L\leq t\leq T$ and $\Vert
y(t)-y_0(t)\Vert<\eta$
for $0\leq t\leq T$.

Let us discuss now the approximations for $t\in[0,+\infty)$. Let
$y_\infty\in Y_0$ be an
equilibrium point of the slow equation (\ref{s}), that is,
$g_0(\xi(y_\infty),y_\infty)=0$.

\proclaim (H6). The point $y=y_\infty$ is an asymptotically
stable equilibrium point of equation (\ref{s}) and $\beta_0$ lies in the
basin of
attraction of $y_\infty$.

When (H6) is satisfied, the solution $y_0(t)$ of the reduced
problem
is defined for all $t\geq0$ and satisfies the property
$\lim_{t\rightarrow\infty}y_0(t)=y_\infty$.
In this case the approximation given by
Theorem~1 holds for all $t\geq0$.

\proclaim Theorem~2. Let $f_0:D_0\rightarrow{\Bbb R}^n$,
$g_0:D_0\rightarrow{\Bbb R}^m$ and $\xi:Y\rightarrow{\Bbb R}^n$ be continuous
functions. Let $y_\infty$ be in $Y_0$ and $(\alpha_0,\beta_0)$ be in $D_0$.
Assume that
hypothesis (H1) to (H6)
hold. Let $x_0(\tau)$ be the solution of the boundary layer equation
(\ref{ble1}).
Let $y_0(t)$
be the solution of the reduced problem (\ref{rp1}). For every $\eta>0$, there
exists $\delta>0$
and a neighbourhood ${\cal{V}}$ of $(D_0,f_0,g_0,\alpha_0,\beta_0)$ in
${\cal{T}}$ with the properties that
for all $\varepsilon<\delta$, and all $(D,f,g,\alpha,\beta)\in{\cal{V}}$,
any solution $(x(t),y(t))$ of the
problem (\ref{sf}) is defined for all $t\geq0$ and there exists $L>0$ such
that
$\varepsilon L<\eta$, $\Vert x(\varepsilon\tau)- x_0(\tau)\Vert<\eta$ for
$0\leq\tau\leq L$, $\Vert x(t)- \xi(y_0(t))\Vert<\eta$ for
$t\geq\varepsilon L$ and
$\Vert y(t)-y_0(t)\Vert<\eta$ for $t\geq0$.

The proofs Theorems 1 and 2 are postponed to Section~3.2.

\subsection{Non-autonomous systems}
The problem (\ref{sf}) contains an apparently more general situation
$$\begin{array}{c}
 \varepsilon \dot x = f(x,y,t)\quad x(t_0)=\alpha\,, \\[5pt]
 \dot y= g(x,y,t)\quad y(t_0)=\beta\,, 
 \end{array} \eqno(8')
$$
where $f$ and $g$ are defined on an open set $D\subset{\Bbb R}^{n+m+1}$ and
$(\alpha,\beta,t_0)\in D$. To
see this, we consider $t$ as a dependent slow
variable and append the trivial equation $\dot t=1$. The fast equation is
$$
x'=f_0(x,y,t)\eqno{(9')}
$$
where $y$ and $t$ are parameters. The component ${\cal{L}}$
of the slow
manifold
$$
f_0(x,y,t)=0\eqno{(10')}
$$
is given as the graph of a function $x=\xi(y,t)$, where $(y,t)$ belongs to a
compact domain $Y\subset{\Bbb R}^m\times{\Bbb R}$. The slow equation,
considered on the interior $Y_0$
of $Y$, is
$$
\dot y=g_0(\xi(y,t),y,t).\eqno{(11')}$$
There is no change in the formulation of hypotheses (H1) to (H4), except that
equations (9), (10) and (11) are replaced by equations ($9'$), ($10'$) and
($11'$).
The formulation of hypothesis (H5) is: $(\beta_0,t_0)\in Y_0$ and
$\alpha_0$ lies in the
basin of attraction of $\xi(y_0,t_0)$. Thus Theorem~1 applies to problem
($8'$). Let $x_0(\tau)$ be the solution of the boundary layer equation
$x'=f_0(x,\beta_0,t_0),~x(0)=\alpha_0$. Let $y_0(t)$ be the solution,
defined on $[t_0,T]$,
of the reduced problem $\dot y=g_0(\xi(y,t),y,t),~y(t_0)=\beta_0$. For any
$\eta>0$ and
any solution $(x(t),y(t))$ of problem ($8'$),
there exists $L>0$ such that $\varepsilon L<\eta$,
$\Vert x(t_0+\varepsilon\tau)- x_0(\tau)\Vert<\eta$ for $0\leq\tau\leq L$,
$\Vert x(t)- \xi(y_0(t))\Vert<\eta$ for $t_0+\varepsilon L\leq t\leq T$ and
$\Vert y(t)-y_0(t)\Vert<\eta$
for $t_0\leq t\leq T$, as long as $\varepsilon$ is small enough and
$(D,f,g,\alpha,\beta)$ is in a small
neighbourhood of $(D_0,f_0,g_0,\alpha_0,\beta_0)$.

When the function $g_0(\xi(y,t),y,t)$ depends nontrivially on the variable
$t$,
Theorem~2 does not apply to Problem ($8'$), because Hypothesis (H6) does
never hold for the non-autonomous equation ($11'$). In that case one could
require that $y_\infty$ is a stationary solution, that is,
$g_0(\xi(y_\infty,t),y_\infty,t)\equiv0$.
However, since $Y$ is assumed to be a compact set, the limiting behaviour, as
$t\rightarrow+\infty$ is not relevant. It would be necessary to generalize
Theorems~1 and 2 to noncompact pieces of the slow manifold.

\subsection{Deformations: Tykhonov's theorem}\label{t}
As explained in the introduction, the classical Tykhonov's theorem concerns
the one-parameter deformation (2) of the unperturbed problem (3), under the
assumption of uniqueness of solutions of system (2), so that we can
consider the solution $(x(t,\varepsilon),y(t,\varepsilon))$ as depending on
the parameter $\varepsilon$
and discuss its limit as $\varepsilon\rightarrow0$. Actually, Tykhonov
formulated his result only
for systems for which the right-hand side does not depend on $\varepsilon$.
In \cite{T} Tykhonov requires that the equilibrium point $x=\xi(y)$ of system
(\ref{fe}) is
asymptotically stable for all $y\in Y$, but he does not require the uniformity
of the basin of attraction over $Y$. The number $\eta$ whose existence is
assumed in Definition~1 will, in general, depend on $\mu$ and also on $y$.
This brings the following definition.

\proclaim Definition 3. The equilibrium point $x=\xi(y)$ of the equation
(\ref{f}) is said to be uniformly asymptotically stable over $Y$ if for every
$\mu>0$ there exists a
$\eta$ with the property that for any $y\in Y$, any solution $x(\tau)$ of
(\ref{f}) for
which $\Vert x(0)-\xi(y)\Vert<\eta$ can be continued for all $\tau>0$ and
satisfies the
inequality $\Vert x(\tau)-\xi(y)\Vert<\mu$ and
$\lim_{\tau\rightarrow\infty}x(\tau)=\xi(y)$.

Tykhonov \cite{T} proves that $\eta$ may be chosen independently of $y$ as
long as $Y$ is compact, that is, the asymptotic stability is uniform over
$Y$. However, this is false, and simple examples show that $\eta$ is not
always bounded away from zero on the compact set $Y$. We are therefore forced
to introduce the hypothesis that the asymptotic stability of $x=\xi(y)$ is
uniform over $Y$ (see \cite{W}, p. 255).
These matters have been fully discussed by Hoppensteadt \cite{H2} (see also
\cite{VB}).

Assume that $x=\xi(y)$ is uniformly asymptotically stable over $Y$. It is
easy to show that the basin of attraction is uniform over $Y$. Conversely,
Hoppensteadt \cite{H2} proved that if $Y$ is compact, then the asymptotic
stability of $x=\xi(y)$ for all $y\in Y$,
together with the existence of a uniform basin of attraction over $Y$, imply
that the asymptotic stability is uniform over $Y$ (see also the remark
following Lemma~6 of the present paper). Thus, to
formulate Tykhonov's theorem under the hypothesis that $x=\xi(y)$ is uniformly
asymptotically stable over $Y$ as given by Wasow \cite{W}, is the same as
formulating it under the
hypothesis that $x=\xi(y)$ is asymptotically stable for all $y\in Y$ and has a
uniform basin of attraction over $Y$ as given by Hoppensteadt \cite{H1,H2} or
as given in the present paper (see also \cite{E}, p. 235). Note that the
proof of Hoppensteadt \cite{H1} is based on
construction of Lyapunov functions and is quite different from Tykhonov
original proof \cite{T}. The reader who is not acquainted with Russian language
should consult Wasow \cite{W} who follows the presentation given in \cite{T}.

We notice that Tykhonov paper deals also with systems of the form
\begin{eqnarray*}
& \varepsilon_j \dot x_j = f_j(x_1,\cdots
,x_k,y)\quad j=1,\cdots ,k &\\
& \dot y= g(x_1,\cdots ,x_k,y)\,, & 
\end{eqnarray*}
where $x_j\in{\Bbb R}^{n_j}$, $y\in{\Bbb R}^m$ and $\varepsilon_1,\cdots
,\varepsilon_k$ are small positive parameters.
Tykhonov gives the behaviour of solutions when $\varepsilon_1\rightarrow0$ and
$\varepsilon_{j+1}/\varepsilon_j\rightarrow0$,
$j=1,\cdots ,k-1$. Tykhonov defines a hierarchy of boundary
layer equations and reduced problems that approximate the solutions at
various time scales. Such systems have been also studied by Hoppensteadt
\cite{H4,H5}. They will be considered in a forthcoming paper with emphasis on
the underlying functional spaces and topologies.

Of course, Theorem~2 may also be formulated in terms of one-parameter
deformations. We obtain approximations, on the infinite time interval, of the
perturbed problem solutions, under the assumption that the
reduced problem has an asymptotically stable equilibrium point.
This result is neither presented in Tykhonov's paper nor in Wasow's
book.
However, Hoppensteadt in a series of papers \cite{H1,H3,H5,H6} studied
extensively the approximations on the infinite time
interval. His studies
concern also non autonomous systems.


\section{Nonstandard results}

\subsection{A short tutorial on Internal Set Theory}
{\em Internal Set Theory }(IST) is an axiomatic approach to Nonstandard
Analysis,
proposed by Nelson \cite{IST}. We adjoin to ordinary mathematics (say ZFC) a
new undefined unary predicate {\em standard }(st). The axioms of IST are
the usual
axioms of ZFC plus three others which govern the use of the new predicate.
Hence {\em all theorems of }ZFC {\em remain valid}. What is new in IST is
an addition,
not a change. We call a formula of IST {\em external }if it involves the
new predicate ``st''; otherwise, we call it {\em internal}. Thus internal
formulas are the formulas of ZFC. IST is a {\em conservative extension }of
ZFC,
that is, every internal theorem of IST is a theorem of ZFC. Some of the
theorems which are proved in IST are external and can be reformulated so that
they become internal. Indeed, there is a {\em reduction algorithm }which
reduces
any external formula $A$ of IST to an internal formula $A'$, with the same
free variables, which satisfies $A\equiv A'$, that is, $A\Leftrightarrow
A'$ for all standard
values of the free variables. We give the reduction of the frequently
occurring formula $\forall
x~(\forall^{{\rm{st}}}y~A\Rightarrow\forall^{{\rm{st}}}z~B)$, where $A$ and
$B$ are
internal formulas:
\begin{equation}\label{r1}
\forall x~(\forall^{{\rm{st}}}y~A\Rightarrow\forall^{{\rm{st}}}z~B)
~\equiv~\forall z~\exists^{{\rm{fin}}}y'~\forall x~(\forall y\in
y'~A\Rightarrow B).
\end{equation}
A real number $x$ is called {\em infinitesimal}, denoted by $x\simeq O$, if
$|x|<\varepsilon$
for all standard $\varepsilon>0$, {\em limited }if $|x|<r$ for some 
standard $r$,
{\em appreciable }if it is limited and not infinitesimal, and
{\em unlimited}, denoted by $x\simeq\mp\infty$, if it is not limited. Let
$(E,d)$ be a
standard metric space. Two points $x$ and $y$ in $E$ are called {\em
infinitely
close}, denoted by $x\simeq y$, if $d(x,y)\simeq0$.

We may not use external formulas in the axiom schemes of ZFC, in particular
we may not use external formulas to define subsets. The notations
$\{x\in{\Bbb R}:x\mbox{ is limited}\}$ or $\{x\in{\Bbb R}:x\simeq0\}$ are
not allowed. Moreover, we can
prove that
there does not exist subsets $L$ and $I$ of ${\Bbb R}$ such that, for all
$x$ in ${\Bbb R}$,
$x$ is in $L$ if and only if $x$ is limited, or $x$ is in $I$ if and only if
$x$ is infinitesimal. This result is frequently used in proofs. Suppose
that
we have shown that a certain internal property $A$ holds for every limited
$r$; then we know that $A$ holds for some unlimited $r$, for otherwise we
could let $L=\{x\in{\Bbb R}:A\}$. This is called the {\em Cauchy
principle}. It has the
following consequence

\proclaim Lemma 1. {\em (Robinson's Lemma)}. Let $r(t)$ be a real function
such that
$r(t)\simeq0$ for all limited $t\geq0$, then there exists an unlimited
positive number
$\nu$ such that $f(t)\simeq0$ for all $t\in[0,\nu]$.

\noindent
{\bf Proof.} The set of all $s$, such that $|r(t)|<1/s$ for all $t\in[0,s]$, 
contains
all limited $s\geq1$. By the Cauchy principle, it must contain some
unlimited $\nu$. {\hfill$\Box$ \medskip}

Let $X$ be a standard topological space. A point $x$ in $X$ is said to be
{\em infinitely close }to a standard point $x_0$, denoted by $x\simeq x_0$,
if $x$ is
in every standard neighbourhood of $x_0$. Let $A$ be a standard open subset of
$X$. A point $x\in X$ is said to be {\em nearstandard in }$A$ if there
exists a
standard $x_0\in A$ such that $x\simeq x_0$. We recall that $A$ is
compact if and only if any point $x\in A$ is nearstandard in $A$. We recall
that $A$ is open if and only if any point $x\in X$ which is nearstandard in
$A$, belongs to $A$. For more informations on the nonstandard
approach to topological spaces, the reader is referred to \cite{S1}.

\subsection{Perturbations}

Classically, the intuitive notion of perturbation can only
be described via deformations or neighbourhoods. The first benefit we gain
from NSA is a natural and useful notion of perturbation. A perturbation of a
standard object is a nonstandard object which is (infinitely) close to it in
some sense to be precised. Since a perturbation is a simple nonstandard
object, its properties can be investigated directly, and do not require to
use extra-properties with respect to the parameters of the deformation as in
the classical approach.
\proclaim Definition 4.
An element $(D,f,g,\alpha,\beta)\in{\cal{T}}$ is said to be a perturbation
of the
standard element $(D_0,f_0,g_0,\alpha_0,\beta_0)\in{\cal{T}}$ if $D$
contains all the
nearstandard point in $D_0$, $f(x,y)\simeq f_0(x,y)$ and $g(x,y)\simeq
g_0(x,y)$ for all
$(x,y)$ which is nearstandard in $D_0$ and $\alpha\simeq\alpha_0$,
$\beta\simeq\beta_0$.

We note that $f_0(x,y)$ and $g_0(x,y)$ are well defined for all nearstandard
points $(x,y)$ in $D_0$. Indeed, $D_0$ is a standard open set so it contains
all the nearstandard points $(x,y)$ in $D_0$. With this
notion we can reformulate Theorems~1 and 2 as follows.

\proclaim Theorem~3. Let $f_0:D_0\rightarrow{\Bbb R}^n$,
$g_0:D_0\rightarrow{\Bbb R}^m$ and $\xi:Y\rightarrow{\Bbb R}^n$ be standard
continuous functions. Let $(\alpha_0,\beta_0)$ be standard in $D_0$. Assume
that
hypothesis (H1) to (H5) hold. Let $x_0(\tau)$ be the solution of the boundary
layer equation (\ref{ble1}). Let $y_0(t)$ be the solution of the reduced
problem (\ref{rp1}). Let $T$
be standard in $I$, $I$ being the positive interval of definition of $y_0$. Let
$\varepsilon>0$ be
infinitesimal. Let $(D,f,g,\alpha,\beta)$ be a perturbation of
$(D_0,f_0,g_0,\alpha_0,\beta_0)$. Any
solution $(x(t),y(t))$ of Problem (\ref{sf}) is defined at least on
$[0,T]$ and there exists $L>0$ such that $\varepsilon L\simeq0$,
$x(\varepsilon\tau)\simeq x_0(\tau)$ for $0\leq\tau\leq L$,
$x(t)\simeq\xi(y_0(t))$ for $\varepsilon L\leq t\leq T$ and $y(t)\simeq
y_0(t)$ for $0\leq t\leq T$.

\proclaim Theorem~4. Let $f_0:D_0\rightarrow{\Bbb R}^n$,
$g_0:D_0\rightarrow{\Bbb R}^m$ and $\xi:Y\rightarrow{\Bbb R}^n$ be standard
continuous functions. Let $y_\infty\in Y_0$ and $(\alpha_0,\beta_0)\in D_0$
be standard. Assume that
hypothesis (H1) to (H6) hold. Let $x_0(\tau)$ be the solution of the boundary
layer equation (\ref{ble1}). Let $y_0(t)$ be the solution of the reduced
problem (\ref{rp1}). Let $\varepsilon>0$
be infinitesimal. Let $(D,f,g,\alpha,\beta)$ be a perturbation of
$(D_0,f_0,g_0,\alpha_0,\beta_0)$.
Any solution $(x(t),y(t))$ of Problem (\ref{sf}) is defined for all
$t\geq0$
and and there exists $L>0$ such that $\varepsilon L\simeq0$,
$x(\varepsilon\tau)\simeq x_0(\tau)$ for $0\leq\tau\leq L$,
$x(t)\simeq\xi(y_0(t))$ for $t\geq\varepsilon L$ and $y(t)\simeq y_0(t)$
for $t\geq0$.

The proofs of Theorems 3 and 4 are postponed to Section~4. Theorems 3 and 4
are external
statements. As we have recalled, Nelson \cite{IST} proposed a reduction
algorithm that reduces external theorems to equivalent internal forms.
Let us show that the reduction of Theorem~3 (resp. Theorem~4) is
Theorem~1 (resp. Theorem~2). We need the following result.

\proclaim Lemma 2. The element $(D,f,g,\alpha,\beta)\in{\cal{T}}$ is a
perturbation of the
standard element $(D_0,f_0,g_0,\alpha_0,\beta_0)\in{\cal{T}}$ if and only
if $(D,f,g,\alpha,\beta)$ is
infinitely
close to $(D_0,f_0,g_0,\alpha_0,\beta_0)$ for the topology of uniform
convergence on
compacta.

\noindent
{\bf Proof.} Let $(D,f,g,\alpha,\beta)$ be a perturbation of
$(D_0,f_0,g_0,\alpha_0,\beta_0)$. Let
$\Delta$ be a standard compact subset of $D_0$. Any $(x,y)\in\Delta$ is
nearstandard in
$\Delta$, and so in $D_0$. Then $\Delta\subset D$, $\alpha\simeq\alpha_0$,
$\beta\simeq\beta_0$ and
$f(x,y)\simeq f_0(x,y)$, $g(x,y)\simeq g_0(x,y)$ for all $(x,y)\in\Delta$.
Let $a>0$ be a standard
real number. Then $\Vert f-f_0\Vert_\Delta<a$, $\Vert g-g_0\Vert_\Delta<a$,
$\Vert\alpha-\alpha_0\Vert<a$ and $\Vert\beta-\beta_0\Vert<a$. Hence
$(D,f,g,\alpha,\beta)\in V(\Delta,a)$ for all standard compact
$\Delta\subset D_0$ and all standard $a>0$,
that is, $(D,f,g,\alpha,\beta)\simeq(D_0,f_0,g_0,\alpha_0,\beta_0)$ for the
topology of uniform
convergence on compacta. Conversely, let $(D,f,g,\alpha,\beta)$ be
infinitely close to
$(D_0,f_0,g_0,\alpha_0,\beta_0)$ for the topology of uniform convergence on
compacta. Let
$(x,y)$ be nearstandard in $D_0$. There exists a standard element
$(x_0,y_0)\in D_0$
such that $(x,y)\simeq(x_0,y_0)$. Let $\Delta$ be a standard compact
neighbourhood of
$(x_0,y_0)$ such that $\Delta\subset D_0$. Then $\Delta\subset D$,
$(x,y)\in\Delta$ and $\Vert f(x,y)-f_0(x,y)\Vert<a$,
$\Vert g(x,y)-g_0(x,y)\Vert<a$ for any standard $a>0$. Thus $(x,y)\in D$ and
$f(x,y)\simeq f_0(x,y)$, $g(x,y)\simeq g_0(x,y)$. Since
$\alpha\simeq\alpha_0$ and $\beta\simeq\beta_0$, we
obtain that $(D,f,g,\alpha,\beta)$ is a perturbation of
$(D_0,f_0,g_0,\alpha_0,\beta_0)$.
{\hfill$\Box$ \medskip}

\noindent
{\bf Proof of Theorem 1.}
We adopt the following abbreviations: $u$ is
the variable
$(D_0,f_0,g_0,\alpha_0,\beta_0)$, $v$ is the variable
$(D,f,g,\alpha,\beta)$, and $B$ is the formula
\begin{quote}
If $\eta>0$ then any solution $(x(t),y(t))$ of Problem (\ref{sf}) is
defined
at least on $[0,T]$ and there exists $L>0$ such that $\varepsilon L<\eta$,
$\Vert x(\varepsilon\tau)-
x_0(\tau)\Vert<\eta$ for $0\leq\tau\leq L$, $\Vert x(t)-
\xi(y_0(t))\Vert<\eta$ for $\varepsilon L\leq t\leq T$ and $\Vert
y(t)-y_0(t)\Vert<\eta$
for $0\leq t\leq T$.
\end{quote}
According to the Lemma 2, to
say that $v$ is a perturbation of $u$ is the same
as saying $v$ is in any standard neighbourhood of $u$. To say that ``any
solution
$(x(t),y(t))$ of Problem (\ref{sf}) is defined at least on $[0,T]$ and
there exists $L>0$ such that $\varepsilon L\simeq0$,
$x(\varepsilon\tau)\simeq x_0(\tau)$ for $0\leq\tau\leq L$,
$x(t)\simeq\xi(y_0(t))$ for $\varepsilon L\leq t\leq T$ and $y(t)\simeq
y_0(t)$ for $0\leq t\leq T$''
is the same as saying $\forall^{{\rm{st}}}\eta~B$. Then Theorem~3 asserts that
$$\forall\varepsilon~\forall
v(\forall^{{\rm{st}}}\delta~\forall^{{\rm{st}}}{\cal{V}}~\varepsilon<\delta~\&~v
\in{\cal{V}}\Rightarrow\forall^{{\rm{st}}}\eta~B).$$
In this formula, $u$, $\xi$ and $T$ are standard parameters, $v$ ranges over
${\cal{T}}$, $\varepsilon$, $\delta$, and $\eta$ range over the strictly
positive
real numbers and ${\cal{V}}$ ranges over the neighbourhoods of $u$. By
(\ref{r1}),
this is equivalent to
$$\forall\eta~\exists^{{\rm{fin}}}\delta'~\exists^{{\rm{fin}}}{\cal{V}}'~\forall
\varepsilon~\forall
v~(\forall\delta\in\delta'~\forall{\cal{V}}\in{\cal{V}}'~\varepsilon<\delta~\&~v
\in{\cal{V}}\Rightarrow B).$$
For $\delta'$ and ${\cal{V}}'$ finite sets,
$\forall\delta\in\delta'~\forall{\cal{V}}\in{\cal{V}}'~\varepsilon<\delta~\&~v\in
{\cal{V}}$ is the same as
$\varepsilon<\delta~\&~v\in{\cal{V}}$
for $\delta=\min{\delta'}$ and ${\cal{V}}=\cap_{V\in{\cal{V}}'}V$, and so
our formula is equivalent to
$$\forall\eta~\exists\delta~\exists{\cal{V}}~\forall\varepsilon~\forall
v~(\varepsilon<\delta~\&~v\in{\cal{V}}\Rightarrow B).$$
This shows that for any standard $u$, $\xi$ and any standard
$T\in I$, the statement of Theorem 1 holds, thus by transfer, it holds for any
$u$, $\xi$ and any $T\in I$.
{\hfill$\Box$ \medskip}

The reduction of Theorem 4 to Theorem 2 follows almost verbatim the
reduction of Theorem~3 to Theorem~1 and is left to
the reader.

\subsection{Uniform asymptotic stability}
The external characterizations
of the notion of stability and attractivity of the equilibrium point $\xi(y)$
of equation (\ref{f}), given in Definition~1, are as follows.

\proclaim Lemma 3. Assume $f$, $\xi$ and $y$ are standard. The equilibrium
point $x=\xi(y)$ of the equation (\ref{f}) is \\
1. Stable if and only if any solution $x(\tau)$ of (\ref{f}) for which
$x(0)\simeq\xi(y)$ can be continued for all $\tau>0$ and satisfies
$x(\tau)\simeq\xi(y)$.\\
2. Attractive if it admits a standard basin of attraction, that is, a
standard neighbourhood ${\cal{V}}$ with the property that any solution
$x(\tau)$ of
system (\ref{f}) for which $x(0)$ is standard in ${\cal{V}}$ can be
continued for all
$\tau>0$ and satisfies $x(\tau)\simeq\xi(y)$ for all $\tau\simeq+\infty$.

\noindent
{\bf Proof.}
1. Let $B$ be the formula ``Any solution $x(\tau)$ of equation (\ref{f}) for
which $x(0)=\alpha$ can be continued for all $\tau>0$ and satisfies the
inequality
$\Vert x(\tau)-\xi(y)\Vert<\mu$''. The characterization of stability in the
lemma is
$$\forall\alpha~(\forall^{{\rm{st}}}\eta~\Vert\alpha-\xi(y)\Vert<\eta\Rightarrow
\forall^{{\rm{st}}}\mu~B).$$
In this formula $f$, $\xi$ and $y$ are standard parameters and $\eta$, $\mu$
range over the strictly positive real numbers. By (\ref{r1}), this is
equivalent to
$$\forall\mu~\exists^{{\rm{fin}}}\eta'~\forall\alpha~(\forall\eta\in\eta'~\Vert
\alpha-\xi(y)\Vert<\eta\Rightarrow B).$$
For $\eta'$ a finite set,
$\forall\eta\in\eta'~\Vert\alpha-\xi(y)\Vert<\eta$ is the same as
$\Vert\alpha-\xi(y)\Vert<\eta$ for
$\eta=\min \eta'$, and so our formula is equivalent to
$$\forall\mu~\exists\eta~\forall\alpha~(\Vert\alpha-\xi(y)\Vert<\eta\Rightarrow
B).$$
This is the usual definition of stability.\\
2. By transfer, the attractivity of an equilibrium is equivalent to the
existence of a standard basin of attraction. The characterization of a
standard basin of attraction ${\cal{V}}$ in the lemma is that any solution
$x(\tau)$ of
system (\ref{f}) for which $x(0)$ is standard in ${\cal{V}}$ can be
continued for all
$\tau>0$ and satisfies
$$\forall\tau~(\forall^{{\rm{st}}}r~\tau>r\Rightarrow\forall^{{\rm{st}}}\mu~\Vert x(\tau)-\xi(y)\Vert<\mu).$$
In this formula $\xi$ and $x(\cdot)$ are standard parameters and $r$, $\mu$
range
over
the strictly positive real numbers. By (\ref{r1}), this is equivalent to
$$\forall\mu~\exists^{{\rm{fin}}}r'~\forall\tau~(\forall r\in
r'~\tau>r\Rightarrow\Vert x(\tau)-\xi(y)\Vert<\mu).$$
For $r'$ a finite set $\forall r\in r'~\tau>r$ is the same as $\tau>r$ for
$r=\max r'$, and
so our formula is equivalent to
$$\forall\mu~\exists r~\forall\tau~(\tau>r\Rightarrow\Vert
x(\tau)-\xi\Vert<\mu).$$
We have shown that for all standard $\alpha$ in ${\cal{V}}$ (and
consequently, by transfer,
for
all $\alpha$ in ${\cal{V}}$) any solution $x(\tau)$ of problem (\ref{f})
for which $x(0)=\alpha$,
can be continued for all $\tau>0$ and satisfies
$\lim_{\tau\rightarrow+\infty}x(\tau)=\xi(y)$. This is the
usual definition of a basin of attraction.
{\hfill$\Box$ \medskip}

Let hypothesis (H1) be satisfied. Let $\pi(\tau,\alpha,y)$ be the unique
noncontinuable solution of equation (\ref{f}) such that
$\pi(0,\alpha,y)=\alpha$. This
solution is
defined on the interval $I(\alpha,y)$. It follows from the basic theorems of
differential equations that the function $\pi$ is continuous with respect
to the
initial
condition $\alpha$ and the parameters $y$. The external formulation of this
result
is as follows.

\proclaim Lemma 4.
 Assume $f_0$ is
standard. Let $y_0$ and $\alpha_0$ be standard, then for all standard
$\tau\in I(\alpha_0,y_0)$
and all $\alpha\simeq\alpha_0$, $y\simeq y_0$, we have $\tau\in
I(\alpha,y)$ and $\pi(\tau,\alpha,y)\simeq\pi(\tau,\alpha_0,y_0)$.

\noindent
{\bf Proof.}
The reduction of the Lemma 4 is the usual continuity of solutions with
respect to initial conditions and parameters. This lemma is a particular
case of the {\em Short Shadow Lemma }(see Section~4).
{\hfill$\Box$ \medskip}

\proclaim Lemma 5. Assume that hypothesis (H1) is satisfied. Assume that $f$,
$\xi$ and $y$ are standard. Then the equilibrium point $x=\xi(y)$ is
asymptotically stable if
and only if there exists a standard $a>0$ with
the property that for any $\alpha$ in the ball ${\cal{B}}$ of center
$\xi(y)$ and radius $a$,
the solution $x(\tau)$ of system (\ref{f}) for which $x(0)=\alpha$, can be be
continued for all $\tau>0$ and satisfies $x(\tau)\simeq\xi(y)$ for all
$\tau\simeq+\infty$.

\noindent
{\bf Proof.}
Assume that $\xi$ is asymptotically stable.
Then it is attractive, and so it admits a standard basin of attraction
${\cal{V}}$.
Let $a>0$ be standard such that the closure of the ball ${\cal{B}}$ of
center $\xi(y)$
and radius $a$ is included in ${\cal{V}}$. Let $\alpha\in{\cal{B}}$ and let
$\alpha_0$ be standard in ${\cal{V}}$
such that $\alpha\simeq\alpha_0$. Let $x(\tau)=\pi(\tau,\alpha,y)$ and
$x_0(\tau)=\pi(\tau,\alpha_0,y)$. By
the attractivity of $\xi(y)$,
the solution $x_0(\tau)$ is defined for all $\tau>0$ and satisfies
$x_0(\tau)\simeq\xi(y)$ for
all $\tau\simeq+\infty$. By Lemma~4, $x(\tau)\simeq x_0(\tau)$
for all limited $\tau>0$. By
Robinson's Lemma, there exists $\nu\simeq+\infty$ such that $x(\tau)\simeq
x_0(\tau)$ for all
$\tau\in[0,\nu]$.
Thus $x(\tau)\simeq\xi(y)$ for all unlimited $\tau\leq\nu$. By stability of
$\xi(y)$ we have
$x(\tau)\simeq\xi(y)$ for all $\tau>\nu$. Hence $x(\tau)\simeq\xi(y)$ for
all $\tau\simeq+\infty$. Conversely,
assume $\xi(y)$ satisfies the property in the lemma. By Lemma~3, the ball
${\cal{B}}$ is a standard basin of attraction of $\xi(y)$. Hence $\xi(y)$
is attractive.
Let
$\alpha\simeq\xi(y)$. Then, by hypothesis, $x(\tau)\simeq\xi(y)$ for all
$\tau\simeq+\infty$, and, by
Lemma~4, $x(\tau)\simeq\pi(\tau,\xi(y),y)=\xi(y)$ for all limited $\tau$. By
Lemma~3, $\xi(y)$ is stable. Thus $\xi(y)$ is asymptotically stable.
{\hfill$\Box$ \medskip}

Let us return now to the discussion of uniform asymptotic stability over $Y$
of the equilibrium $\xi(y)$ of equation (\ref{f}). Assume $f_0$ and $\xi$
are standard. According to Lemma~5, hypothesis (H3) is equivalent to

\proclaim (H$3'$). There exists a
standard $a>0$ with the property that for all standard $y\in Y$, any solution
$x(\tau)$
of (\ref{f}) for which $\Vert x(0)-\xi(y)\Vert<a$ can be continued for all
$\tau>0$ and
satisfies $x(\tau)\simeq\xi(y)$ for all $\tau\simeq+\infty$.

The following result is not used in the present paper. We give it as a
complement of the previous discussions on uniform asymptotic stability.
Moreover this result is connected also to the discussion following
Definition~3, on the various hypothesis under which Tykhonov's theorem
can be formulated.

\proclaim Lemma 6. Let hypothesis (H1) be satisfied. Assume $f_0$
and $\xi$ are standard and $Y$ compact. If the equilibrium
point $x=\xi(y)$ of the equation (\ref{f}) is asymptotically stable and the
basin of attraction is uniform over $Y$ then there exists a
standard $a>0$ with the property that for all $y\in Y$ any solution
$x(\tau)$ of
(\ref{f}) for which $\Vert x(0)-\xi(y)\Vert<a$ can be continued for all
$\tau>0$ and
satisfies $x(\tau)\simeq\xi(y)$ for all $\tau\simeq+\infty$.

\noindent
{\bf Proof.}
Assume $\xi(y)$ is asymptotically stable and has a basin of attraction
which is
uniform over $Y$. Let $a>0$ such that for all $y\in Y$, any solution
$x(\tau)$ of
system (\ref{f})
for which $\Vert x(0)-\xi(y)\Vert<a$ can be continued for all $\tau>0$ and
satisfies
$\lim_{\tau\rightarrow+\infty}x(\tau)=\xi(y)$. By transfer, there exists a
standard $a>0$ with this
property. Let $y\in Y$ and let $\alpha$ be such that
$\Vert\alpha-\xi(y)\Vert<a$. Let
$x(\tau)=\pi(\tau,\alpha,y)$.
Since $Y$ is a standard compact set, there exists $y_0$ standard in
$Y$ and $\alpha_0$ standard in the ball of center $\xi(y_0)$ and radius $a$
such that
$y\simeq y_0$ and $x(0)\simeq\alpha_0$.
Let
$x_0(\tau)=\pi(\tau,\alpha_0,y_0)$.
By the Lemma~3, the solution $x_0(\tau)$ can be continued for all $\tau>0$ and
satisfies
$x_0(\tau)\simeq\xi(y_0)$ for all $\tau\simeq+\infty$.
According to the Lemma~4, $x(\tau)\simeq x_0(\tau)$ for all limited
$\tau>0$.
According to the Robinson's Lemma, there exists $\nu\simeq+\infty$ such
that $x(\tau)\simeq x_0(\tau)$
for all $\tau\in[0,\nu]$. By attractivity $x_0(\nu)\simeq\xi(y_0)$. Then
$x(\nu)\simeq\xi(y_0)\simeq\xi(y)$.
According to the Lemma~4
\begin{equation}\label{18}
x(\nu+s)\simeq\xi(y_0)\simeq\xi(y),\quad \mbox{for all limited }s.
\end{equation}
Assume
there exists $\nu_1>\nu$ such that
$x(\nu_1)\not\simeq\xi(y)$, that is $\gamma=\Vert x(\nu_1)-\xi(y)\Vert$ is
appreciable.
Let $m$ be the smallest value of $\tau\in[\nu,\nu_1]$ such that
$\Vert x(m)-\xi(y)\Vert=\gamma$. Thus
$x(m)\not\simeq\xi(y)$. If $s=m-\nu$
was limited then, by property (\ref{18}), one would have
$x(m)=x(\nu+s)\simeq\xi(y)$, which contradicts $x(m)\not\simeq\xi(y)$.
Thus $s$ is unlimited and $x(m+\tau)$ lies in the ball
${\cal{B}}=\{x:\Vert x-\xi(y)\Vert\leq\gamma\}$
for all
$\tau\in[-s,0]$.
By the Lemma ~4 we have $x(m+\tau)\simeq x_0(m+\tau)$
for all limited $\tau$.
Hence $ x_0(m+\tau)$ lies in the (standard) ball ${\cal{B}}=\{x:\Vert
x-\xi(y_0)\Vert\leq\gamma_0\}$, where
$\gamma_0>0$ is the standard part of the appreciable number $\gamma$, for
all standard
negative $\tau$ and
then (by transfert) for all $\tau\leq0$. Let $\tau_0\simeq-\infty$.
According to Lemma~5,
$x_0(m+\tau_0+\tau)\simeq\xi(y_0)$ for all $\tau\simeq+\infty$. For
$\tau=-\tau_0$, we obtain $x_0(m)\simeq\xi(y_0)$, a
contradiction with $x_0(m)\not\simeq\xi(y_0)$.
{\hfill$\Box$ \medskip}

The external characterizations of the notion of uniform asymptotic stability
over $Y$ of the equilibrium point $\xi(y)$ of equation (\ref{f}), given in
Definition~3, is as follows: there exists a standard $a>0$ with the property
that for all $y\in Y$ any solution $x(\tau)$ of (\ref{f}) for which $\Vert
x(0)-\xi(y)\Vert<a$
can be continued for all $\tau>0$ and satisfies $x(\tau)\simeq\xi(y)$ for
all $\tau\simeq+\infty$.
Thus, the previous lemma asserts that if $Y$ is compact, then the asymptotic
stability of $x=\xi(y)$ for all $y\in Y$, together with the existence of a
uniform
basin of attraction over $Y$, imply that the asymptotic stability is uniform
over $Y$. The proof by Hoppensteadt
\cite{H2} of this result is based
on a theorem of Massera \cite{M} on construction of Liapunov functions for
asymptotically stable equilibrium points.

\section{Proofs of Theorems 3 and 4}

\subsection{Preliminary lemmas}
Let us discuss now the {\em Short Shadow Lemma}, which is the fundamental
tool in
regular perturbation theory. Let us consider the initial value
problems:
\begin{eqnarray} 
&\dot{x}=F_0(x),\quad x(0)=\alpha_0\,,&\label{4}\\
&\dot{x}=F(x),\quad x(0)=\alpha\,, \label{5}
\end{eqnarray}
where
$F_0:D_0\rightarrow{\Bbb R}^d$ and $F:D\rightarrow{\Bbb R}^d$ are
continuous functions, $D_0$ and $D$
open subsets of ${\Bbb R}^d$, $F_0$ standard, $\alpha\in D$ and $\alpha_0$
standard in $D_0$.
Problem (\ref{5}) is said to be a regular perturbation of problem (\ref{4})
when $D$ contains all the nearstandard points in $D_0$,
$\alpha\simeq\alpha_0$, and
$F(x)\simeq F_0(x)$ for all $x$ nearstandard in $D_0$.
We assume that problem (\ref{4}) has a unique solution. Let
$\phi_0:I\rightarrow{\Bbb R}^d$ be its noncontinuable solution. Will the
solutions of
problem (\ref{5}) also exist on $I$ and be close to $\phi_0$~? This
question is
answered by the Short Shadow Lemma, which is one of the first results of
nonstandard asymptotic analysis of differential equations. This
result appeared in the nonstandard literature under various formulations (see
\cite{vdB1,D,DR,DL,LG}). For our purpose it is more convenient to
adopt the formulation given in \cite{S2,S}.

\proclaim Lemma~7. {\em(Short Shadow Lemma)
}Let problem (\ref{5}) be a regular perturbation of problem
(\ref{4}). Then for any nearstandard $t\in I$, any solution $\phi$ of problem
(\ref{5}) is defined and satisfies $\phi(t)\simeq\phi_0(t)$.

The perturbed equation may depend also on the time
$t\in]a,b[$, where $]a,b[$ is a possibly nonstandard interval. In that case
problem (\ref{5}) is replaced by problem
\begin{equation}\label{6}
\dot{x}=F(x,t)\,\quad x(t_0)=\alpha,
\end{equation}
where $F:D\times]a,b[\rightarrow{\Bbb R}^d$ is continuous and $t_0\in]a,b[$.
Problem (\ref{6}) is said to be a regular perturbation of problem (\ref{4})
when $D$ contains all the nearstandard points in $D_0$,
$\alpha\simeq\alpha_0$, and
$F(x,t)\simeq F_0(x)$ for all $t\in]a,b[$ and $x$ nearstandard in $D_0$.

\proclaim Lemma~8.
Let problem (\ref{6}) be a regular perturbation of problem
(\ref{4}). Then for any nearstandard $s\in I$, any solution $\phi$ of problem
(\ref{6}) is defined and satisfies $\phi(t_0+s)\simeq\phi_0(s)$ as long as
$t_0+s\in]a,b[$.

\noindent
{\bf Proof.}
This result is a corollary of the Short Shadow Lemma. To see this, we consider
$t$ as a dependant parameter and append the trivial
equation $\dot t=1$. The unperturbed equation would be considered as defined
on the standard set $D_0\times\Lambda_0$, where $\Lambda_0$ is the standard
open interval whose
standard elements are those elements $s_0$ of ${\Bbb R}$ such that
$s+t_0\in\Lambda$ for all
$s\simeq s_0$.
{\hfill$\Box$ \medskip}


Let $\mu>0$. The set $\Vert x-\xi(y)\Vert\leq\mu,~y\in Y$
will be called a $\mu$-{\em tube }around ${\cal{L}}$. The set
$\Vert x-\xi(y)\Vert=\mu,~y\in Y$ constitutes the {\em lateral boundary }of
the $\mu$-tube.
Let $f_0:D_0\rightarrow{\Bbb R}^n$, $g_0:D_0\rightarrow{\Bbb R}^m$ be
standard continuous functions on the standard
open subset $D_0$ of ${\Bbb R}^{n+m}$. Let
$f:D\rightarrow{\Bbb R}^n$, $g:D\rightarrow{\Bbb R}^m$ be continuous
perturbations of $f_0$ and $g_0$, that is $D$
contains all the nearstandard points in $D_0$ and
$f(x,y)\simeq f_0(x,y)$, $g(x,y)\simeq g_0(x,y)$ for all nearstandard
$(x,y)$ in $D_0$.
For the proof of Theorem~3, we need the
following results. The first result (Lemma~9) states that any solution
$(x(t),y(t))$ of system
\begin{eqnarray} 
&\varepsilon\dot x =  f(x,y)\,,&\label{a}\\
&\dot y =  g(x,y)\,,\nonumber
\end{eqnarray}
that comes infinitely close to the slow manifold ${\cal{L}}$ remains
infinitely close to it as long as $y(t)$ is nearstandard in $Y_0$, that is,
$y$ is not infinitely close to the boundary of $Y$. The
second result (Lemma~10) states
that the $y$-component of a solution $(x(t),y(t))$ of (\ref{a}) that is
infinitely
close to the slow manifold ${\cal{L}}$, is infinitely close to a solution
of the
slow equation (\ref{s}).

\proclaim Lemma~9. Let hypothesis (H1) to (H3) be satisfied. Let
$(x(t),y(t))$ be a solution of (\ref{a}) with the properties that $y(t)$
is nearstandard in $Y_0$
for $t_0\leq t\leq t_1$, and
$x\left(t_0\right)\simeq\xi\left(y\left(t_0\right)\right)$ then
$x\left(t\right)\simeq\xi\left(y\left(t\right)\right)$ for $t_0\leq t\leq t_1$.

\noindent
{\bf Proof.}
Let $y_0$ be standard in $Y_0$ such that $y(t_0)\simeq y_0$. Thus
$x(t_0)\simeq\xi(y_0)$. As
a function of $\tau$, $(x(t_0+\varepsilon\tau),y(t_0+\varepsilon\tau))$ is
a solution of system
\begin{eqnarray}
&x' =  f(x,y)\,,&\label{b}\\
&y' =  \varepsilon g(x,y)\,,&\nonumber 
\end{eqnarray}
where $'=d/d\tau$, with initial condition $(x(t_0),y(t_0))$. This
problem is a regular perturbation of system
\begin{eqnarray} 
&x' =  f(x,y)\,,&\label{c}\\
&y' = 0\,, &\nonumber 
\end{eqnarray}
with initial condition $(\xi(y_0),y_0)$. By the Short Shadow Lemma
\begin{equation}\label{d}
x(t_0+\varepsilon\tau)\simeq\xi(y_0),~y(t_0+\varepsilon\tau)\simeq
y_0\quad \mbox{for all limited }\tau.
\end{equation}
Since $y(t)$ is nearstandard in $Y_0$, for all $t\in[t_0,t_1]$,
$Y$ is a standard compact neighbourhood of
$y[t_0,t_1]$.
Let $a$ be a standard positive number satisfying the properties in
hypothesis (H$3'$). Assume the statement of the
Lemma is false. Then there must exist $s$, $t_0<s<t_1$ such that
$x(s)\not\simeq\xi(y(s))$, that is,
$\gamma=\Vert x(s)-\xi(y(s))\Vert$ is appreciable. We may choose $s$ such that 
$\gamma$ is standard and $0<\gamma\leq a$. Let ${\cal{T}}$ be the 
$\gamma$-tube around ${\cal{L}}$. Since
$(x(t_0),y(t_0))$ belongs to ${\cal{T}}$, $(x(s),y(s))$ belongs to the lateral
boundary of ${\cal{T}}$, and
no point $y(t)$ with $t\in[t_0,t_1]$ belongs to the boundary of $Y$, there
must
exist the smallest value $m$ of $t$, in $[t_0,t_1]$, such that $(x(m),y(m))$
belongs to the lateral boundary of ${\cal{T}}$, that is
$\Vert x(m)-\xi(y(m))\Vert=\gamma$. By compactness of this boundary, there
exists a standard $\left(x_1,y_1\right)$ belonging to the lateral boundary of
${\cal{T}}$, that is, $\Vert x_1-\xi(y_1)\Vert=\gamma$, such that
$\left(x\left(m\right),y\left(m\right)\right)\simeq\left(x_1,y_1\right)$. If
$\tau_0=(m-t_0)/\varepsilon$
was limited then, by property (\ref{d}), one would has
$x(m)=x(t_0+\varepsilon\tau_0)\simeq\xi(y(t_0))$,
$y(m)=y(t_0+\varepsilon\tau_0)\simeq y(t_0)$. Since $\xi$ is standard
continuous, $\xi(y(m))\simeq\xi(y(t_0))\simeq x(m)$, which contradicts
$x(m)\not\simeq\xi(y(m))$.
Thus $\tau_0$ is unlimited and
$(x(m+\varepsilon\tau),y(m+\varepsilon\tau))$ belongs to ${\cal{T}}$ for
all $\tau\in[-\tau_0,0]$.
As a function of $\tau$, $(x(m+\varepsilon\tau),y(m+\varepsilon\tau))$ is a
solution of system
(\ref{b})
with initial condition $(x(m),y(m))$. This
problem is a regular perturbation of system (\ref{c}) with initial
condition $(x_1,y_1)$. The solution of this
problem is $(x_1(\tau),y_1)$, where $x_1(\tau)=\pi(\tau,x_1,y_1)$.
By the Short Shadow Lemma we have
$$x(m+\varepsilon\tau)\simeq x_1(\tau)~~y(m+\varepsilon\tau)\simeq
y_1\mbox{~for all limited~}\tau\leq0\,.$$
By Robinson's Lemma, there exists $\tau_1\simeq-\infty$, which can be chosen
such that $-\tau_0\leq\tau_1$, satisfying $x(m+\varepsilon\tau_1)\simeq
x_1(\tau_1)$. Thus
$x_1(\tau_1)$ belongs to the ball of center $\xi(y_1)$ and radius $a$.
According to hypothesis (H$3'$),
$x_1(\tau_1+\tau)\simeq\xi(y_1)$ for all $\tau\simeq+\infty$. Take
$\tau=-\tau_1$, then
$x(m)\simeq x_1(0)=x_1(\tau_1-\tau_1)\simeq\xi(y_1)\simeq\xi(y(m))$. This
is a contradiction with
$x(m)\not\simeq\xi(y(m))$.
{\hfill$\Box$ \medskip}

\proclaim Lemma~10\,. Let hypothesis (H4) be satisfied. Let $y^0$ be standard in
$Y_0$. Let $(x(t),y(t))$ be a solution of (\ref{fs}) with the
property that $x\left(t\right)\simeq\xi\left(y\left(t\right)\right)$ for
$t_0\leq t\leq t_1$
and $y(t_0)\simeq y^0$. Let $y_0(t)$ be the solution of the slow equation
(\ref{s})
with initial condition $y^0$ which is assumed to be defined on the standard
interval $0\leq t\leq T$. Then $y(t_0+s)\simeq y_0(s)$ for all $s\leq T$
such that $t_0+s\leq t_1$.

\noindent
{\bf Proof.}
We can write $x(t)=\xi(y(t))+\alpha(t)$, where $\alpha(t)\simeq0$ for
$t_0\leq t\leq t_1$. By the second
equation in (\ref{a}) this implies that
$$\frac{dy}{dt}(t)=g(\xi(y(t))+\alpha(t),y(t)),$$
that is, $y(t)$ is a solution of the non-autonomous equation
\begin{equation}\label{l}
\frac{dy}{dt}=g(\xi(y)+\alpha(t),y).
\end{equation}
The solution under consideration can be written $y(t_0+s)$, its initial
condition is then $y(t_0)$. The problem consisting of equation (\ref{l})
together with the initial condition $y(t_0)$ is a regular perturbation (for
$t_0\leq t\leq t_1$) of the problem consisting of the slow equation
(\ref{s}) together
with the initial condition $y^0$, whose solution $y_0(s)$ is assumed to be
defined and limited on the interval $[0,T]$. By Lemma~8,
$y(t_0+s)\simeq y_0(s)$ as long as $s\leq T$ and $t_0+s\leq t_1$.
{\hfill$\Box$ \medskip}


\subsection{Proof of Theorem~3}
Let $(x(t),y(t))$ be a solution of problem (\ref{sf}), then
$(x(\varepsilon\tau),y(\varepsilon\tau))$ is a solution of the problem
consisting of system (\ref{b})
together with initial conditions $x(0)=\alpha$, $y(0)=\beta$.
This problem is a regular perturbation of the problem
consisting of system
(\ref{c}) together with initial conditions
$x(0)=\alpha_0$, $y(0)=\beta_0$, which is nothing more than the boundary
layer equation
(\ref{ble1}). According to the hypothesis (H5), the
solution $x_0(\tau)$ of the boundary layer equation is defined for all
$\tau\geq0$,
and
\begin{equation}\label{app}
x_0(\tau)\simeq\xi(\beta_0)~~{\rm for}~\tau\simeq+\infty.
\end{equation}
By the Short Shadow Lemma $x(\varepsilon\tau)$ and $y(\varepsilon\tau)$ are
defined and satisfy
$x(\varepsilon\tau)\simeq x_0(\tau)$, $y(\varepsilon\tau)\simeq\beta_0$ for
all limited $\tau$. By Robinson's Lemma,
there exists $L\simeq+\infty$ which can be chosen such that $\varepsilon
L\simeq0$, with the
properties that
\begin{equation}\label{at}
x(\varepsilon\tau)\simeq x_0(\tau),~y(\varepsilon\tau)\simeq\beta_0,~~{\rm
for}~0\leq\tau\leq L.
\end{equation}
According to (\ref{app}), when $t_0=\varepsilon L$, the solution has come
infinitely close
to the slow manifold ${\cal{L}}$. Let $t_1$ be the largest value (maybe
$t_1=+\infty$) such
that $y(t)$
lies in $Y$ for $t_0\leq t\leq t_1$. By Lemma~9, the solution remains
infinitely close
to ${\cal{L}}$ for $t_0\leq t\leq t_1$ as long as $y(t)$ is nearstandard in
$Y_0$.
By Lemma~10,
$y(t)$ is infinitely close to the solution $y_0(t)$ of the reduced problem,
as long as $t\leq T$ and $t\leq t_1$. If $t_1<T$, then
$y\left(t_1\right)\simeq y_0\left(t_1\right)$ and $y_0\left(t_1\right)$ is
nearstandard
in $Y_0$, thus $y(t)\in Y_0$ for some $t>t_1$, which contradicts the
definition
of $t_1$. Thus $t_1\geq T$, and $y(t)$ is defined for $0\leq t\leq T$ and
satisfies
\begin{equation}\label{ay}
y(t)\simeq y_0(t)~~{\rm for}~0\leq t\leq T.
\end{equation}
Since $x(t)\simeq\xi(y(t))$ for $t_0\leq t\leq T$, the approximation
(\ref{ay}) of
$y(t)$ implies that
\begin{equation}\label{ax}
x(t)\simeq x_0(t)=\xi(y_0(t))~~{\rm for}~t_0\leq t\leq T.
\end{equation}
Thus (\ref{at}), (\ref{ay}) and (\ref{ax}) complete the proof of the theorem.

\subsection{Proof of Theorem~4}
According to hypothesis (H6), the solution $y_0(t)$ of the reduced
problem is defined for all $t\geq0$ and satisfies $y_0(t)\simeq y_\infty$
for $t\simeq+\infty$.
By Theorem~3, the approximations
\begin{eqnarray*}
& y(t)\simeq y_0(t)~~{\rm for}~0\leq t\leq T\,,&\\
& x(t)\simeq x_0(t)=\xi(y_0(t))~~{\rm for}~\varepsilon L\leq t\leq T\,,& 
\end{eqnarray*}
hold for each limited $T>0$. By Robinson's Lemma, they hold also for
some $T\simeq+\infty$. Thus $x(T)\simeq x_\infty$ and $y(T)\simeq y_\infty$.
Starting from $(x(T),y(T))$ and applying again Theorem~3, one obtains
\begin{equation}\label{20}
x(T+s)\simeq x_\infty,~~y(T+s)\simeq y_\infty~~\mbox{for ~all~limited~}s\geq0\,.
\end{equation}
It remains to prove that $x(t)$ and $y(t)$ are defined for all $t\geq T$
and satisfy
$$x(t)\simeq x_\infty,~~y(t)\simeq y_\infty~~{\rm for}~t\geq T.$$
Assume that this is false. Then there must exist $s\geq T$ such that
$y(s)\not\simeq y_\infty$,
that is
$\mu=\Vert y(s)-y_\infty\Vert$ is appreciable. We can chose $s$ so that the
ball ${\cal{B}}$ of center
$y_\infty$ and radius $\mu$ is included in the basin of attraction of
$y_\infty$.
Let $m$ be the smallest value $t\geq T$
such that $\Vert y(m)-y_\infty\Vert=\mu$. Then $y(t)$ is limited for all
$T\leq t\leq m$. By Lemma~9, $x(t)\simeq\xi(y(t))$ for $T\leq t\leq m$. If
$s_0=m-T$ was limited
then, by property (\ref{20}), one would have
$x(m)=x(T+s_0)\simeq x_\infty$, $y(m)=y(T+s_0)\simeq y_\infty$, which
contradicts $y(m)\not\simeq y_\infty$.
Thus $s_0$ is unlimited and $(x(m+s),y(m+s))$ belongs to the ball
${\cal{B}}$ for all
$s\in[-s_0,0]$.
Let $y_1(s)$ be the solution of the slow equation (\ref{s}) with
initial condition $y_1(0)=y(m)$. By Lemma~10 one has
$$y(m+s)\simeq y_1(s)\quad\mbox{for all limited }s\leq0\,.$$
By Robinson's Lemma, there exists $s_1\simeq-\infty$, which one can
choose such that $-s_0\leq s_1$, satisfying $y(m+s_1)\simeq y_1(s_1)$. Thus
$y_1(s_1)$ is limited. By the asymptotic stability of $y_\infty$,
$y_1(s_1+s)\simeq y_\infty$ for all $s\simeq+\infty$. Take $s=-s_1$, then
$y(m)=y_1(0)=y_1(s_1-s_1)\simeq y_\infty$. This is a contradiction with
$y(m)\not\simeq y_\infty$.


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\bigskip

{\sc Claude Lobry}\\
Centre International de Math\'ematiques Pures et Appliqu\'ees\\
``Le Dubellay'', Bat. B, 4 avenue Joachim\\
06100 Nice, Fance.\\
E-mail address: clobry@sophia.inria.fr
\medskip

{\sc Tewfik Sari}\\
Laboratoire de Math\'ematiques, Universit\'e de Haute Alsace\\
4, rue des Fr\`eres Lumi\`ere\\
68093 Mulhouse, France. \\
E-mail address:  T.Sari@univ-mulhouse.fr
\medskip 

{\sc Sefiane Touhami}\\
Laboratoire de Math\'ematiques, Universit\'e de Haute Alsace\\
4, rue des Fr\`eres Lumi\`ere\\
68093 Mulhouse, France. \\
E-mail address: S.Touhami@univ-mulhouse.fr

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