\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 139, pp. 1--17.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2004/139\hfil On Pontryagin-Rodygin's theorem]
{On Pontryagin-Rodygin's theorem for convergence of solutions of slow
and fast systems}
\author[T. Sari, K. Yadi\hfil EJDE-2004/139\hfilneg]
{Tewfik Sari, Karim Yadi} % in alphabetical order
\address{Tewfik Sari \hfill\break
Laboratoire de Math\'ematiques et Applications\\
Universit\'e de Haute Alsace \\
4, rue des fr\`{e}res Lumi\`{e}re\\
68093, Mulhouse, France}
\email{Tewfik.Sari@uha.fr}
\address{Karim Yadi \hfill\break
Laboratoire de Math\'ematiques et Applications\\
Universit\'e de Haute Alsace \\
4, rue des fr\`{e}res Lumi\`{e}re\\
68093, Mulhouse, France}
\email{K.Yadi@uha.fr}
\date{}
\thanks{Submitted May 12, 2004. Published November 26, 2004.}
\subjclass[2000]{34D15, 34E15, 34E18}
\keywords{Singular perturbations; asymptotic stability; nonstandard analysis}
\begin{abstract}
In this paper we study fast and slow systems for which the fast
dynamics has limit cycles, for all fixed values of the slow variables.
The fundamental tool is the Pontryagin and Rodygin theorem which
describes the limiting behavior of the solutions in the continuously
differentiable case, when the cycles are exponentially stable.
We extend this result to the continuous case, and exponential
stability is replaced by asymptotic stability. We give two examples
with numerical simulations to illustrate the problem. Our results are
formulated in classical mathematics. They are proved using Nonstandard
Analysis.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\section{Introduction}
This paper will focus on slow and fast systems of the form
\begin{equation}
\varepsilon\,\frac{dx}{dt}=f(x,y,\varepsilon),\quad\frac{dy}{dt}
=g(x,y,\varepsilon), \label{a}
\end{equation}
where $x\in\mathbb{R}^{n}$, $y\in\mathbb{R}^{m}$ and $\varepsilon$ is a small
positive parameter. The variable $x$ is called a \textit{fast variable}, $y$
is called a \textit{slow variable}. The change of time $\tau=t/\varepsilon$
transforms system (\ref{a}) into
\begin{equation}
\frac{dx}{d\tau}=f(x,y,\varepsilon),\quad\frac{dy}{d\tau}=\varepsilon
g(x,y,\varepsilon). \label{b}
\end{equation}
This system is a one parameter deformation of the unperturbed system
\begin{equation}
\frac{dx}{d\tau}=f(x,y,0),\quad\frac{dy}{d\tau}=0, \label{d}
\end{equation}
which is called the \textit{fast equation}.
In the case where solutions of (\ref{d}) tend toward an equilibrium point
$\xi(y)$, where $x=\xi(y)$ is a root of equation
\begin{equation}
f(x,y,0)=0, \label{e}
\end{equation}
Tykhonov's theorem \cite{tykhonov,wasow} gives the limiting behavior of system
(\ref{a}). A fast transition brings the solution near the slow manifold (\ref{e}). Then, a
slow motion takes place near the slow manifold and is
approximated by the solution of the reduced equation
\begin{equation}
\frac{dy}{dt}=g(\xi(y),y,0).\label{f0}
\end{equation}
This result was obtained in \cite{tykhonov} for continuous vector fields $f$
and $g$, under the assumption that $\xi(y)$ is an asymptotically stable
equilibrium of the fast equation (\ref{d}), uniformly in $y$. It was extended
in \cite{lobry} to all systems that belong to a small neighborhood of the
unperturbed system. For differentiable systems, if the variational equation
has eigenvalues with negative real part for all $y$ in the domain of interest,
then the uniform asymptotic stability of the equilibrium $\xi(y) $ holds.
In the case where solutions of (\ref{d}) tend toward a cycle $\Gamma_{y}$,
Pontryagin and Rodygin's theorem \cite{pontryagin} gives the limiting behavior
of system (\ref{a})~ : after a fast transition that brings the solutions
near the cycles, the solutions
of system (\ref{a}) are approximated by the solutions of the average system
\begin{equation}
\frac{dy}{dt}=\frac{1}{T(y)}\int_{0}^{T(y)}g(x^{\ast}(\tau,y),y,0)\,d\tau,
\label{f}
\end{equation}
where $x^{\ast}(\tau,y)$ is a periodic solution of the fast equation (\ref{d})
corresponding to the cycle $\Gamma_{y}$ and $T(y)$ is its period. This result
was obtained for at least continuously differentiable vector fields $f$ and
$g$, under the assumption that the cycles $\Gamma_{y}$ are asymptotically
stable in the linear approximation, that is, the variational equation
corresponding to the cycle has multipliers with moduli less than 1 with a
single exception. To our knowledge the continuous case with asymptotic
stability instead of exponential stability was not considered in the literature.
Assume that the equilibrium $\xi(y)$ (resp. the cycle $\Gamma_{y}$) loses its
stability, but remains nondegenerate. Neishtadt \cite{neishtadt} proved, in
analytic systems, that there is a delayed loss of stability of the solutions
of (\ref{a})~ : the solutions remain for a long time near the unstable
equilibrium (resp. the unstable cycle) and the slow variable $y$ remains
approximated by the solution of the reduced equation (\ref{f0}) (resp. the
averaged system (\ref{f})).
The aim of this work is to extend the result of \cite{pontryagin} to
continuous vector fields and to define a topology such that the description of
the solutions holds for systems that belong to a small neighborhood of the
unperturbed system. Following \cite{lobry}, we define in Section \ref{main} a
suitable function space of \emph{Initial Value Problems} (IVPs) in order to
study small neighborhoods of the unperturbed problem. The main results
concerning approximations of solutions on finite and infinite time interval
(Theorem \ref{standard1}, Theorem \ref{standard2}) are stated. In the present
work, the results are formulated in classical mathematics and proved within
\emph{Internal Set Theory} (IST) \cite{nelson} which is an axiomatic approach
of \emph{Nonstandard Analysis} (NSA) \cite{robinson}. The idea to use NSA in
perturbation theory of differential equations goes back to the seventies with
the Reebian school \cite{lutz}. It has become today a well-established tool in
asymptotic theory (see the five-digits classification 34E18 of the 2000
Mathematical Subject Classification). We give in Section \ref{nonstandard} a
short tutorial on IST in order to characterize the notion of stability and to
translate our main results in nonstandard words (Theorem \ref{nonstandard1}
and Theorem \ref{nonstandard2}). Section \ref{preliminaries} consists of
presenting some lemmas in view of the proofs of Theorem \ref{nonstandard1} and
Theorem \ref{nonstandard2}. In Section \ref{applications}, we apply our result
to two examples and we give numerical simulations.
\section{Results\label{main}}
Let us consider the differential system
\begin{equation}
\begin{gathered}
\varepsilon\,\dot{x} =f(x,y), \quad x(0)=\alpha,\\
\dot{y}=g(x,y), \quad y(0)=\beta,
\end{gathered}
\label{perturbed}
\end{equation}
where $\varepsilon$ is a positive real number in $]0,\varepsilon_{0}]$,
$f:\Omega\rightarrow\mathbb{R}^{n}$, $g:\Omega\rightarrow\mathbb{R}^{m}$ are
continuous on an open subset $\Omega$ of $\mathbb{R}^{n+m}$ and $(\alpha
,\beta)\,\in\Omega$. The dot ($^{\cdot}$) means $d/dt$. The set
\begin{align*}
\mathcal{T} =\{&(\Omega,f,g,\alpha,\beta):\Omega
\mbox{ open subset of }\mathbb{R}^{n+m},(\alpha,\beta)\,\in\Omega,\\
& f:\Omega\rightarrow\mathbb{R}^{n},~g:\Omega\rightarrow\mathbb{R}
^{m}\mbox{ continuous}\}
\end{align*}
is provided with the \emph{topology of uniform convergence on compacta}
\cite{lobry}. This topology is the topology for which the neighborhood system
of an element $(\Omega_{0},f_{0},g_{0},\alpha_{0},\beta_{0})$ is generated by
the sets
\begin{align*}
V(D,a) =\{&(\Omega,f,g,\alpha,\beta)\in\mathcal{T}:\; D\subset
\Omega\text{, }\| f-f_{0}\| _{D}0$,
the mapping $y\rightarrow T(y)$ is continuous, and
the cycle $\Gamma_{y}$ corresponding to the
periodic solution $x^{\ast}(\tau,y)$ is asymptotically stable and
its basin of attraction is uniform over $G$.
\end{itemize}
From Assumption (B) it follows that the cycle $\Gamma_{y}$ depends
continuously on $y$ and is locally unique, that is, there exists an
neighborhood $W$ of $\Gamma_{y}$ such that the equation (\ref{fast}) has no
other cycle in $W$.
\begin{definition} \rm
The periodic solution $x^{\ast}(\tau,y)$ of \textit{(\ref{fast}) }is said to
be orbitally asymptotically stable (in the sense of Lyapunov) if its orbit
$\Gamma_{y}$ is :
\noindent1. Stable, i.e. for every $\mu>0$, there exists $\eta>0$ such that
any solution $\tilde{x}(\tau)$ of \textit{(\ref{fast})} for which
$\mathop{\rm dis}(\tilde{x}(0)$,$\Gamma_{y})<\eta$ can be continued for all
$\tau\geq0$ and satisfies the inequality $\mathop{\rm dis}(\tilde{x}(\tau
)$,$\Gamma_{y})<\mu$.
\noindent2. and Attractive, i.e. $\Gamma_{y}$ admits a neighborhood
$\mathcal{V}$ (basin of attraction) such that any solution $\tilde{x}(\tau)$
of \textit{(\ref{fast})} for which $\tilde{x}(0)\in\mathcal{V}$ can be
continued for all $\tau\geq0$ and satisfies $\underset{\tau\rightarrow\infty
}{\lim}\mathop{\rm dis}(\tilde{x}(\tau)$,$\Gamma_{y})=0$.
\noindent Moreover, the basin of attraction of the orbit $\Gamma_{y}$ is
uniform over $G$ if there exists a real number $a>0$ such that, for all $y$ in
$G$, the set $\{x\in\mathbb{R}^{n}$ : $\mathop{\rm dis}(x,\Gamma_{y})\leq a\}$ is
in the basin of attraction of $\Gamma_{y}$.
\end{definition}
We define the \textit{slow equation} on the interior $G_{0}$ of $G$ by the
averaged system
\begin{equation}
\overset{\cdot}{y}=\bar{g}_{0}(y):=\frac{1}{T(y)}\int_{0}^{T(y)}g_{0}
(x^{\ast}(\tau,y),y)\,d\tau, \label{average}
\end{equation}
and we add the following two assumptions:
\begin{itemize}
\item[(C)] The slow equation (\ref{average}) has the uniqueness of the
solutions with prescribed initial conditions.
\item[(D)] $\beta_{0}$ is in $G_{0}$ and $\alpha_{0}$ is in the basin
of attraction of $\Gamma_{\beta_{0}}$.
\end{itemize}
We refer to the \textit{boundary layer equation} as
\begin{equation}
x'=f_{0}(x,\beta_{0}),\text{ \ }x(0)=\alpha_{0},
\label{boundary layer}
\end{equation}
and to the \textit{reduced problem} as
\begin{equation}
\overset{\cdot}{y}=\bar{g}_{0}(y),\; y(0)=\beta_{0}. \label{reduced}
\end{equation}
We can state the first result.
\begin{theorem} \label{standard1}
Let $(\Omega_{0},f_{0},g_{0},\alpha_{0},\beta_{0})$ be in
$\mathcal{T}$. Assume that $(A)-(D)$ are satisfied. Let $\tilde{x}(\tau)$ and
$\bar{y}(t)$ be the respective solutions of (\ref{boundary layer}) and
(\ref{reduced}) and $L\in I$, where $I$ is the positive interval of definition
of $\bar{y}(t)$. Then for all $\eta>0$, there exist $\varepsilon^{\ast}>0$ and
a neighborhood $\mathcal{V}$ of $(\Omega_{0},f_{0},g_{0},\alpha_{0},\beta
_{0})$ in $\mathcal{T}$ such that for all $\varepsilon<\varepsilon^{\ast}$ and
all $(\Omega,f,g,\alpha,\beta)$ in $\mathcal{V}$, any solution $(x(t),y(t))$
of (\ref{perturbed}) is defined at least on $[0$,$L]$ and there exists
$\omega>0$ such that $\varepsilon\omega<\eta$, $\| x(\varepsilon\tau)-\tilde
{x}(\tau)\| <\eta$ for $0\leq\tau\leq\omega$, $\| y(t)-\bar{y}(t)\| <\eta$ for
$0\leq t\leq L$ and $\mathop{\rm dis}(x(t),\Gamma_{\bar{y}(t)})<\eta$ for
$\varepsilon\omega\leq t\leq L$.
\end{theorem}
Suppose in addition that there exists a point $\bar{y}_{\infty}$ such that
$\bar{g}_{0}(\bar{y}_{\infty})=0$. Under the following assumption, the
previous theorem holds for all $t\geq0$.
\begin{itemize}
\item[(E)] The point $\bar{y}_{\infty}$ is an
asymptotically stable equilibrium of (\ref{average}) and $\beta_{0}$
is in its basin of attraction.
\end{itemize}
\begin{theorem}\label{standard2}
Let $(\Omega_{0},f_{0},g_{0},\alpha_{0},\beta_{0})$ be in
$\mathcal{T}$. Let $\bar{y}_{\infty}$ be in $G_{0}$. Assume that $(A)-(E) $
are satisfied. Let $\tilde{x}(\tau)$ and $\bar{y}(t)$ be the respective
solutions of (\ref{boundary layer}) and (\ref{reduced}). Then for all $\eta
>0$, there exist $\varepsilon^{\ast}>0$ and a neighborhood $\mathcal{V}$ of
$(\Omega_{0},f_{0},g_{0},\alpha_{0},\beta_{0})$ in $\mathcal{T}$ such that for
all $\varepsilon<\varepsilon^{\ast}$ and all $(\Omega,f,g,\alpha,\beta)$ in
$\mathcal{V}$, any solution $(x(t),y(t))$ of (\ref{perturbed}) is defined for
all $t\geq0$ and there exists $\omega>0 $ such that $\varepsilon\omega<\eta$,
$\| x(\varepsilon\tau)-\tilde{x}(\tau)\| <\eta$ for $0\leq\tau\leq\omega$,
$\| y(t)-\bar{y}(t)\| <\eta$ for $t\geq0$ and $\mathop{\rm dis}(x(t),\Gamma_{\bar
{y}(t)})<\eta$ for $t\geq\varepsilon\omega$.
\end{theorem}
The proofs of the two theorems are postponed to Subsections \ref{ns1} and
\ref{ns2}.
\section{Nonstandard formulations of the results\label{nonstandard}}
\subsection{A short tutorial on IST}
As it was outlined in the introduction, \textit{Internal Set Theory} (IST )
\cite{nelson} is an extension of ordinary mathematics, that is, Zermelo-Fraenkel set theory plus
axiom of choice (ZFC). The theory IST gives an axiomatic approach of Robinson's
\textit{Nonstandard Analysis} \cite{robinson}. We adjoin to ZFC a new
undefined unary predicate \textit{standard }(st) and add to the usual axioms
of ZFC three others for governing the use of the new predicate. \textit{All
theorems of }ZFC \textit{remain valid in }IST. What is new in IST is an
addition, not a change. We call a formula of IST \textit{internal} in the case
where it does not involve the new predicate \textquotedblleft
st\textquotedblright\ ; otherwise, we call it \textit{external}. The theory
IST is a \textit{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 \textit{reduction algorithm} due to Nelson which reduces any
external formula $F(x_{1}$,...,$x_{n})$ of IST without other free variables
than $x_{1}$,...,$x_{n}$ to an internal formula $F'(x_{1}$
,...,$x_{n})$ with the same free variables, such that $F\equiv F'$,
that is, $F\Leftrightarrow F'$ for all standard values of the free
variables. We will need the following reduction formula which occurs
frequently:
\begin{equation}
\forall x\; (\forall^{\mathrm{st}}y\; A\Rightarrow\forall
^{\mathrm{st}}z\; B)\equiv\forall z\; \exists^{\mathop{\rm 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 $\forall
^{\mathrm{st}}$ means \textquotedblleft for all standard\textquotedblright
\ and $\exists^{\mathop{\rm fin}}$ means \textquotedblleft there is a
finite\textquotedblright.
\noindent A real number $x$ is \textit{infinitesimal, }denoted by $x\simeq0$,
if $|x|r)\Rightarrow
\forall^{\mathrm{st}}\mu\; \mathop{\rm dis}(\tilde{x}(\tau),\Gamma_{y}
)<\mu\,.
\]
In this formula, $\tilde{x}(.)$ and $\Gamma_{y}$ are standard parameters while
$r$, $\mu$ range over the positive real numbers. By (\ref{reduction}), this is
equivalent to
\[
\forall\mu\; \exists^{\mathop{\rm fin}}r'\; \forall\tau\; (\forall r\in r'\; \tau>r\Rightarrow\mathop{\rm dis}(\tilde{x}
(\tau),\Gamma_{y})<\mu).
\]
But to say, for $r'$ a finite set, $\forall r\in r'$ $\tau>r$
is the same as to say $\tau>r$ for $r=\max r'$ and the formula is
equivalent to
\[
\forall\mu\; \exists r\; \forall\tau\; (\tau>r\Rightarrow
\mathop{\rm dis}(\tilde{x}(\tau),\Gamma_{y})<\mu).
\]
Hence, for all standard $\alpha$ in $\mathcal{V}$, any solution $\tilde
{x}(\tau)$ of the equation (\ref{fast}) for which $\tilde{x}(0)=\alpha$, can
be continued for all $\tau\geq0$\textit{\ }and satisfies $\underset
{\tau\rightarrow\infty}{\lim}\mathop{\rm dis}(\tilde{x}(\tau)$,$\Gamma_{y})=0$. By
transfer, this property remains true for all $\alpha$ in $\mathcal{V}$. This
is the usual definition of the orbital attractivity.
\end{proof}
The following lemma is needed to reformulate the Assumption (B) and its
proof is postponed to subsection \ref{fundamental}.
\begin{lemma} \label{oas}
Assume that hypothesis $(A)$ is satisfied and that $f_{0}$ and
$x^{\ast}(\tau,y)$ are standard. Then the periodic solution $x^{\ast}(\tau,y)$
of \textit{(\ref{fast}) is }orbitally asymptotically stable if and only if
there exists a standard $a>0$\textit{\ such that any solution }$\tilde{x}
(\tau)$\textit{\ of (\ref{fast}) for which }$\mathop{\rm dis}(\tilde{x}
(0)$\textit{, }$\Gamma_{y})0$,
the mapping $y\rightarrow T(y)$ is continuous, and
there exists a standard $a>0$ such that, for all
standard $y$, any solution $\tilde{x}(\tau)$ of
(\ref{fast}) for which $\mathop{\rm dis}(\tilde{x}(0)$, $\Gamma_{y}
)0$ be infinitesimal and $(\Omega,f,g,\alpha,\beta)\in\mathcal{T}$ be a
perturbation of $(\Omega_{0},f_{0},g_{0},\alpha_{0},\beta_{0})\in\mathcal{T}$.
Then any solution $(x(t),y(t))$ of (\ref{perturbed}) is defined at least on
$[0$,$L]$ and there exists $\omega>0$ such that $\varepsilon\omega\simeq0$,
$x(\varepsilon\tau)\simeq\tilde{x}(\tau)$ for $0\leq\tau\leq\omega$,
$y(t)\simeq\bar{y}(t)$ for $0\leq t\leq L$ and $x(t)\simeq\Gamma_{\bar{y}(t)}$
for $\varepsilon\omega\leq t\leq L$.
\end{theorem}
\begin{theorem} \label{nonstandard2}
Let $(\Omega_{0},f_{0},g_{0},\alpha_{0},\beta_{0})$ be a
standard element of $\mathcal{T}$. Let $\bar{y}_{\infty}$ be standard in
$G_{0}$. Assume that $(A)-(E)$ are satisfied. Let $\tilde{x}(\tau)$ and
$\bar{y}(t)$ be the respective solutions of (\ref{boundary layer}) and
(\ref{reduced}). Let $\varepsilon>0$ be infinitesimal and $(\Omega
,f,g,\alpha,\beta)\in\mathcal{T}$ be a perturbation of $(\Omega_{0}
,f_{0},g_{0},\alpha_{0},\beta_{0})\in\mathcal{T}$. Then any solution
$(x(t),y(t))$ of (\ref{perturbed}) is defined for all $t\geq0$ and there
exists $\omega>0$ such that $\varepsilon\omega\simeq0$, $x(\varepsilon
\tau)\simeq\tilde{x}(\tau)$ for $0\leq\tau\leq\omega$, $y(t)\simeq\bar{y}(t)$
for $t\geq0$ and $x(t)\simeq\Gamma_{\bar{y}(t)}$ for $t\geq\varepsilon\omega$.
\end{theorem}
We propose to show that Theorem \ref{nonstandard1}, which is external, reduces
by Nelson's algorithm to its internal equivalent Theorem \ref{standard1} while
we let to the reader the reduction of Theorem \ref{nonstandard2} to Theorem
\ref{standard2}. We need the following lemma~:
\begin{lemma} \label{lem3.7}
The element $(\Omega,f,g,\alpha,\beta)$ of $\mathcal{T}$ is a perturbation of
the standard element $(\Omega_{0},f_{0},g_{0},\alpha_{0},\beta_{0})$ of
$\mathcal{T}$ if and only if $(\Omega,f,g,\alpha,\beta)$ is infinitely close
to $(\Omega_{0},f_{0},g_{0},\alpha_{0},\beta_{0})$ for the topology of uniform
convergence on compacta, that is, $(\Omega,f,g,\alpha,\beta)$ is in any
standard neighborhood of $(\Omega_{0},f_{0},g_{0},\alpha_{0},\beta_{0})$.
\end{lemma}
The proof of this lemma can be found in \cite[Lemma 2, page 11.]{lobry}.
\begin{proof}[Reduction of Theorem \ref{nonstandard1}]
We design by $F$ the formula:
\lq\lq Any solution $(x(t),y(t))$ of
(\ref{perturbed}) is defined at least on $[0$,$L]$ and
there exists $\omega>0$ such that $\varepsilon\omega<\eta$,
$\| x(\varepsilon\tau)-\tilde{x}(\tau)\| <\eta$ for $0\leq\tau
\leq\omega$, $\| y(t)-\bar{y}(t)\| <\eta$ for $0\leq t\leq
L$ and $\mathop{\rm dis}(x(t),\Gamma_{\bar{y}(t)})<\eta$ for
$\varepsilon\omega\leq t\leq L$''
and respectively by $u_{0}$ and $u$ the variables $(\Omega_{0},f_{0},g_{0}
,\alpha_{0},\beta_{0})$ and $(\Omega,f,g,\alpha,\beta)$ of $\mathcal{T}$. We
also design by $F'$ the formula
\lq\lq any solution $(x(t),y(t))$ of (\ref{perturbed}) is defined at least on
$[0$, $L]$ and there exists $\omega>0$ such that
$\varepsilon\omega\simeq0$, $x(\varepsilon\tau)\simeq\tilde{x}
(\tau)$ for $0\leq\tau\leq\omega$, $y(t)\simeq\bar{y}(t)$ for
$0\leq t\leq L$ and $x(t)\simeq\Gamma_{\bar{y}(t)}$ for
$\varepsilon\omega\leq t\leq L$''
On the other hand, to say that \lq\lq $\varepsilon$ is
infinitesimal'' is the same as to say that \lq\lq
$\forall^{\mathrm{st}}$ $\varepsilon^{\ast}$, $\varepsilon<$ $\varepsilon
^{\ast}$'', to say that \lq\lq $u$ is a perturbation
of $u_{0}$'' is the same as to say that \lq\lq $u$
is in any standard neighborhood $\mathcal{V}$ of $u_{0}$''.
Finally, the formula $F'$\ is equivalent to the formula $\forall
^{\mathrm{st}}\eta$ $F$. Then, Theorem \ref{nonstandard1} can be formalized by
\begin{equation}
\forall\varepsilon\; \forall u\; (\forall^{\mathrm{st}}
\varepsilon^{\ast}\; \forall^{\mathrm{st}}\mathcal{V}\;
K\Rightarrow\forall^{\mathrm{st}}\eta\; F), \label{formula}
\end{equation}
where $K$ designates the formula $\varepsilon<\varepsilon^{\ast}$ $\&$
$u\in\mathcal{V}$. Here, $u_{0}$ and $L$ are standard parameters, $u$ ranges
over $\mathcal{T}$, while $\varepsilon$ and $\varepsilon^{\ast}$ range over
the positive real numbers and $\mathcal{V}$ ranges over the neighborhoods of
$u_{0}$. Using the reduction formula (\ref{reduction}), (\ref{formula}) is
equivalent to
\[
\forall\eta\; \exists^{\mathop{\rm fin}}\varepsilon^{\ast\prime}\;
\exists^{\mathop{\rm fin}}\mathcal{V}'\; \forall\varepsilon\; \forall u\;
(\forall\varepsilon^{\ast}\in\varepsilon^{\ast\prime}\;
\forall\mathcal{V\in V}'\; K\Rightarrow F).
\]
But, $\varepsilon^{\ast\prime}$ and $\mathcal{V}'$ being finite sets,
there exists $\varepsilon^{\ast}$ and $\mathcal{V}$ such that $\varepsilon
^{\ast}=\min\varepsilon^{\ast\prime}$ and $\mathcal{V}=\underset
{V\in\mathcal{V}'}{\cap}V$ and the last formula becomes equivalent to
\[
\forall\eta\; \exists\varepsilon^{\ast}\; \exists\mathcal{V}\;
\forall\varepsilon\; \forall u\; (K\Rightarrow F).
\]
Hence, the statement of Theorem \ref{standard1} holds for any standard $u_{0}$
and $L\in I$. By transfer, it holds for any $u_{0}$ and $L\in I$.
\end{proof}
\section{Proofs of Theorems \ref{nonstandard1} and \ref{nonstandard2}
\label{preliminaries}}
\subsection{Fundamental lemmas\label{fundamental}}
We present in this subsection two fundamental lemmas of the
\textit{nonstandard perturbation theory of differential equations}. The
\textit{stroboscopic method} was proposed by J. L. Callot and G. Reeb and
improved by R. Lutz and T. Sari (see \cite{callot-sari}, \cite{lutz},
\cite{sari}, \cite{sari2}).
Let $\mathcal{O}$ be a standard open subset of $\mathbb{R}^{n}$,
$F:\mathcal{O}\rightarrow\mathbb{R}^{n}$ a standard continuous function. Let
$J$ be an interval of $\mathbb{R}$ containing $0$ and $\phi:J\rightarrow
\mathbb{R}^{n}$ a function such that $\phi(0)$ is nearstandard in
$\mathcal{O}$, that is, there exists a standard $x_{0}\in\mathcal{O}$ such
that $\phi(0)\simeq x_{0}$. Let $I$ be a connected subset of $J$, eventually
external, such that $0\in I$.
\begin{definition}[Stroboscopic property] \label{def4.1}
Let $t$ and $t'$ be in $I$. The function $\phi$
is said to satisfy the stroboscopic property $\mathcal{S}(t$, $t')$ if
$t'\simeq t$, and $\phi(s)\simeq\phi(t)$ 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)=x_{0}. \label{F}
\end{equation}
\begin{theorem}[Stroboscopy Lemma]\label{stroboscopy}
Suppose that
\begin{itemize}
\item[(i)] There exists $\mu>0$ such that, whenever $t\in I$ is limited and $\phi(t)$
is nearstandard in $\mathcal{O}$, there is $t'\in I$ 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 $[0,L]\subset I$ and $\phi(t)\simeq x(t)$ for all $t\in
[0,L]$.
\end{theorem}
An other tool which is related to the theory of regular perturbations is
needed. Let us define the two initial value problems
\begin{gather}
\frac{dx}{dt}=F_{0}(x),\; x(0)=a_{0}\in\mathcal{O}, \label{unperturbed}\\
\frac{dx}{dt}=F(x),\; x(0)=a\in\mathcal{O}. \label{per}
\end{gather}
The so-called \textit{Short Shadow Lemma }answers to the problem of comparing
the solutions of (\ref{unperturbed}) and (\ref{per}) when $F$ is close to
$F_{0}$ and the initial condition $a$ is close to $a_{0}$ (see \cite{sari2}).
\begin{theorem}[Short Shadow Lemma] \label{thm4.3}
Let $\mathcal{O}$ be a standard open subset of
$\mathbb{R}^{n}$ and let $F_{0}:\mathcal{O\rightarrow}\mathbb{R}^{n}$ be
standard and continuous. Let $a_{0}\in$ $\mathcal{O}$ be standard. Assume that
the initial value problem (\ref{unperturbed}) has a unique solution $x_{0}(t)$
and let $J=[0$, $\omega)$, $0<\omega\leq+\infty$, be its maximal positive
interval of definition. Let $F:\mathcal{O\rightarrow}\mathbb{R}^{n}$ be
continuous such that $F(x)\simeq F_{0}(x)$ for all $x$ nearstandard in
$\mathcal{O}$. Then, every solution $x(t)$ of the initial value problem
(\ref{per}) with $a\simeq a_{0}$, is defined for all $t$ nearstandard in $J$
and satisfies$\ x(t)\simeq x_{0}(t)$.
\end{theorem}
With the help of the last theorem, we give now the proof of Lemma~\ref{oas}.
\begin{proof}[Proof of Lemma \ref{oas}]
Assume that the periodic solution $x^{\ast}(\tau,y)$
is orbitally asymptotically stable. By attractivity, its orbit $\Gamma_{y}$
has a standard basin of attraction $\mathcal{V}$. Let $a>0$ be standard such
that the closure of the set $\mathcal{A}=\{x\in\mathbb{R}^{n}$ :
$\mathop{\rm dis}(x$, $\Gamma_{y})0$. By
Robinson's Lemma, there exists $\upsilon\simeq+\infty$ such that $\tilde
{x}(\tau)\simeq\tilde{x}_{0}(\tau)$ for all $\tau$ in $[0$, $\upsilon]$. Thus,
$\mathop{\rm dis}(\tilde{x}(\tau)$, $\Gamma_{y})\simeq0$ for all unlimited
$\tau\leq\upsilon$. By stability of the closed orbit, this approximation still
holds for all $\tau>\upsilon$. Hence, $\mathop{\rm dis}(\tilde{x}(\tau)$,
$\Gamma_{y})\simeq0$ for all $\tau\geq0$. Conversely, if the orbit $\Gamma
_{y}$ is assumed to satisfy the property in the lemma, then by definition, the
standard set $\mathcal{A}$ is in the basin of attraction of $\Gamma_{y}$.
Hence, the considered closed orbit is attractive. Let $\tilde{x}(\tau)$ be a
solution of (\ref{fast}) such that $\tilde{x}(0)=\alpha$, where $\alpha$ is
infinitely close to a standard $\alpha_{0}\in\Gamma_{y}$. Since $\alpha
\in\mathcal{A}$, by hypothesis, $\tilde{x}(\tau)$ can be continued for all
$\tau\geq0$ and satisfies $\mathop{\rm dis}(\tilde{x}(\tau)$, $\Gamma_{y})\simeq0$
for all $\tau\simeq+\infty$. On the other hand, if $\tilde{x}_{0}(\tau)$ is
the maximal solution of (\ref{fast}) such that $\tilde{x}_{0}(0)=\alpha_{0}$,
its trajectory is the closed orbit $\Gamma_{y}$. Hence, by the Short Shadow
Lemma, $\mathop{\rm dis}(\tilde{x}(\tau)$, $\Gamma_{y})\simeq0$ for all limited
$\tau\geq0$. Finally$,\Gamma_{y}$ is stable.
\end{proof}
\subsection{Preparatory lemmas}
Let $\mathcal{C}=\underset{y\in G}{\cup}(\Gamma_{y}\times\{y\})$ and consider
the system
\begin{equation}
\begin{gathered}
\varepsilon\,\dot{x} =f(x,y),\\
\dot{y} =g(x,y).
\end{gathered} \label{singular}
\end{equation}
The following lemma asserts that a trajectory of (\ref{singular}) which comes
infinitely close to $\mathcal{C}$ remains close to it as long as $y$ is not
infinitely close to the boundary of $G$.
\begin{lemma}\label{lemma1}
Let Assumptions $(A)$ and $(B')$ be satisfied. Let
$(x(t),y(t))$ be a solution of (\ref{singular}) such that $y(t)$ is
nearstandard in $G_{0}$ for $t\in[ t_{0\text{,}}t_{1}]$ and
$x(t_{0})\simeq\Gamma_{y(t_{0})}$, then $x(t)\simeq\Gamma_{y(t)}$ for all $t$
in $[t_{0\text{,}}t_{1}]$.
\end{lemma}
\begin{proof}
Let $y_{0}$ be standard in $G_{0}$ and let $x_{0}$ be standard in
$\Gamma_{y_{0}}$ such that $x(t_{0})\simeq x_{0}$ and $y(t_{0})\simeq y_{0}$.
As a function of $\tau$, $(x(t_{0}+\varepsilon\tau)$,$y(t_{0}+\varepsilon
\tau))$ is the solution of system
\begin{equation}
\begin{gathered}
x' =f(x,y),\\
y' =\varepsilon g(x,y),
\end{gathered}
\label{regular}
\end{equation}
with initial condition $(x(t_{0}),y(t_{0}))$. This IVP is a regular
perturbation of system
\begin{equation}
\begin{gathered}
x' =f_{0}(x,y),\\
y' =0,
\end{gathered} \label{regular0}
\end{equation}
with initial condition $(x_{0}$,$y_{0})$. According to the Short Shadow Lemma,
we obtain
\begin{equation}
x(t_{0}+\varepsilon\tau)\simeq\Gamma_{y_{0}},\; y(t_{0}+\varepsilon
\tau)\simeq y_{0}\text{ for limited }\tau\geq0. \label{neuf}
\end{equation}
Assume that there exists $s\in]t_{0},t_{1}]$ such that $\mathop{\rm dis}
(x(s)$,$\Gamma_{y(s)})=\gamma_{0}$ is not infinitesimal. Since the asymptotic
stability of the cycles $\Gamma_{y}$ is uniform over $G$, there exists $a>0$
satisfying the property stated in Assumption $(B')$. Let
$\gamma<\gamma_{0}$ be such that $0<\gamma\leq a$ and $\gamma\not \simeq 0$
and let chose $s\in]t_{0},t_{1}]$ such that $\mathop{\rm dis}(x(s)$,$\Gamma
_{y(s)})=\gamma$. Since $\mathop{\rm dis}(x(t_{0})$,$\Gamma_{y(t_{0})})\simeq0$
and $y(t)$ is nearstandard in $G_{0}$ for all $t\in[ t_{0},s]$, there
exists a smallest $m\in]t_{0},t_{1}]$ such that $\mathop{\rm dis}(x(m)$,
$\Gamma_{y(m)})=\gamma$ and a standard $(x_{1}$,$y_{1})$ such that $y_{1}\in
G_{0}$ and $(x_{1}$,$y_{1})\simeq(x(m),y(m))$. If $\tau_{0}=(m-t_{0})/\varepsilon$
was limited, by (\ref{neuf}) one will have $x(m)\simeq
\Gamma_{y_{0}}$ and $y(m)\simeq y_{0}$, thus $x(m)\simeq\Gamma_{y(m)}$. This
contradicts $\mathop{\rm dis}(x(m)$, $\Gamma_{y(m)})=\gamma$. The value $\tau_{0}$
is then unlimited. Let us consider the solution $(x(m+\varepsilon\tau)$,
$y(m+\varepsilon\tau))$ of (\ref{regular}) with initial condition $(x(m)$,
$y(m))$. This problem is a regular perturbation of (\ref{regular0}) with
initial condition $(x_{1}$,$y_{1})$, of maximal solution $(\tilde{x}(\tau)$,
$y_{1})$. According to the Short Shadow Lemma, $x(m+\varepsilon\tau
)\simeq\tilde{x}(\tau)$ and $y(m+\varepsilon\tau)\simeq y_{1}$ for all limited
$\tau\leq0$. By Robinson's Lemma, there exists $\tau_{1}<0$ unlimited, which
can be chosen such that $-\tau_{0}<\tau_{1}$, still satisfying
$x(m+\varepsilon\tau_{1})\simeq\tilde{x}(\tau_{1})$. By noting that
$\mathop{\rm dis}(x(m+\varepsilon\tau),\Gamma_{y(m+\varepsilon\tau)})<\gamma$ for
all $\tau\in[-\tau_{0}$, $0[$, we will have in particular $\mathop{\rm dis}
(\tilde{x}(\tau_{1})$, $\Gamma_{y_{1}})<\gamma\leq a$. According to Assumption
$(B')$, $\tilde{x}(\tau_{1}+\tau)$ is defined for all $\tau\geq0$ and
satisfies $\tilde{x}(\tau_{1}+\tau)\simeq\Gamma_{y_{1}}$ for all $\tau$
positive and unlimited. In particular, for $\tau=-\tau_{1}$, $\tilde
{x}(0)\simeq\Gamma_{y_{1}}$ i.e. $x(m)\simeq\Gamma_{y_{1}}\simeq\Gamma_{y(m)}
$. This contradicts the fact that $\mathop{\rm dis}(x(m)$, $\Gamma_{y(m)})=\gamma$.
\end{proof}
The following lemma asserts that the $y$-component of a trajectory of
(\ref{singular}) which is infinitely close to $\mathcal{C}$ is approximated by
a solution of the slow equation (\ref{average}).
\begin{lemma}\label{lemma2}
Let Assumptions (A), (B') and (C) be satisfied.
Let $(x(t),y(t))$ be a solution of (\ref{singular}) such that $y(t_{0})$ is
nearstandard in $G_{0}$. Let $y_{0}$ be standard in $G_{0}$ such that
$y(t_{0})\simeq y_{0}$. Let $\bar{y}(t)$ be the solution of (\ref{average})
with initial condition $y_{0}$ and defined on the standard interval $[0,L]$.
Let $t_{1}\geq t_{0}$ such that $t_{1}\leq t_{0}+L$ and $x(t)\simeq
\Gamma_{y(t)}$ for $t\in[ t_{0\text{,}}t_{1}]$. Then $y(t_{0}
+s)\simeq\bar{y}(s)$ for all $0\leq s\leq L$ such that $t_{0}+s\leq t_{1}$.
\end{lemma}
\begin{proof}
Let $(x(t),y(t))$ be a solution of (\ref{singular}) such that $y(t_{0})$ is
nearstandard in $G_{0}$.
Let us consider the external set
$$
I=\{t\geq t_{0}: (x(s),y(s)) \mbox{ is defined
and }x(s)\simeq\Gamma_{y(s)}\mbox{ for all }s\in[t_{0},t]\}$$
which contains, by hypothesis, the interval $[t_{0\text{,}}t_{1}]$.
Let us show that $y(t)$
satisfies the hypothesis $(i)$ of the Stroboscopy Lemma (Theorem
\ref{stroboscopy}). Let $\mu=\varepsilon{\min_{y\in G}}T(y)$. Since
$T$ is continuous and $G$ is compact, $\mu>0$. Let $t_{\lambda}$ limited be in
$I$ such that $y(t_{\lambda})$ is nearstandard in $G_{0}$. The change of
variables
\begin{equation}
\tau=\frac{t-t_{\lambda}}{\varepsilon},\quad
X(\tau)=x(t_\lambda+\varepsilon\tau),\quad
Y(\tau)=\frac{y(t_{\lambda}+\varepsilon\tau)-y(t_{\lambda})}{\varepsilon},
\label{neuf'}
\end{equation}
transforms the problem (\ref{singular}) with initial condition
$(x(t_{\lambda }),y(t_{\lambda}))$ into
\begin{gather}
X'=f(X,y(t_{\lambda})+\varepsilon Y), \quad X(0)=x(t_\lambda),\\
Y'=g(X,y(t_{\lambda})+\varepsilon Y), \quad Y(0)=0.
\end{gather}
For $\tau$ and $Y$ limited, this
problem is a regular perturbation of
\begin{equation}
\begin{gathered}
X' =f_{0}(X,y_{\lambda}),\quad X(0)=x_\lambda,\\
Y' =g_{0}(X,y_{\lambda}),\quad Y(0)=0,
\end{gathered} \label{dix}
\end{equation}
where $x_{\lambda}$ and $y_{\lambda}$
are standard and such that
$x_{\lambda}\simeq x(t_{\lambda})$, $y_{\lambda}\simeq y(t_{\lambda})$.
The Short Shadow Lemma gives
\begin{equation}\label{app1}
X(\tau)\simeq X_{0}(\tau),\quad
Y(\tau)\simeq Y_{0}(\tau),
\end{equation}
for all limited $\tau$, where $(X_0(\tau),Y_0(\tau))$ is the
solution of (\ref{dix}).
Knowing that
$x(t_{\lambda})\simeq\Gamma_{y(t_{\lambda})}\simeq\Gamma_{y_{\lambda}}$
and that $x_{\lambda}$ and
$\Gamma_{y_{\lambda}}$ are standard, we obtain that $x_{\lambda}\in
\Gamma_{y_{\lambda}}$. The first equation of (\ref{dix}) is nothing else than
the fast equation (\ref{fast}) with initial condition $x_{\lambda}$ and
parameter $y=y_{\lambda}$. There exists $\tau_{\lambda}\in[0,T(y_{\lambda})]$
such that $x^{\ast}(\tau_{\lambda}$, $y_{\lambda
})=x_{\lambda}$. Hence,
\begin{equation}\label{app2}
X_{0}(\tau)=x^{\ast}(\tau+\tau_{\lambda},y_{\lambda}),\quad
Y_{0}(\tau)=\int_{0}^{\tau}g_{0}(x^{\ast}(s+\tau_{\lambda},y_{\lambda}),y_{\lambda})ds.
\end{equation}
Using (\ref{app1}) and (\ref{app2}) and the periodicity of $x^{\ast}$,
we obtain
\begin{equation}
Y(T(y_{\lambda}))\simeq\int_{0}^{T(y_{\lambda})}g_{0}(x^{\ast}(s,y_{\lambda
}),y_{\lambda})ds. \label{onze}
\end{equation}
Let us consider now the instant $t_{\nu}:=t_{\lambda}+\varepsilon
T(y_{\lambda})$. We claim that $t_{\nu}\in I$, that is, $x(s)\simeq
\Gamma_{y(s)}$ for all $s$ in $[t_0,t_{\nu}]$. Since $t_\lambda$ is in $I$, this property
holds for all $s$ in $[t_0,t_{\lambda}]$. Its remains to show that its holds also for
all $s$ in $[t_\lambda,t_{\nu}]$.
Indeed, let
$s=t_{\lambda}+\varepsilon\tau$. We have
$y(s)=y(t_{\lambda})+\varepsilon Y(\tau)\simeq y(t_{\lambda})$,
for all $\tau$ in
$[0,T(y_{\lambda})]$. Since $y(t_{\lambda})$ is nearstandard in $G_{0}$ and
$x(t_{\lambda})\simeq\Gamma_{y(t_{\lambda})}$, by Lemma \ref{lemma1} we have
$x(s)\simeq\Gamma_{y(s)}$ for all $s$ in $[t_{\lambda},t_{\nu}]$.
We have proved
that, for $t_{\lambda}$ limited in $I$ and $y(t_{\lambda})$ nearstandard in
$G_{0}$, there exists $t_{\nu}$ such that $0\simeq t_{\nu}-t_{\lambda}\geq\mu
$, $[t_{\lambda},t_{\nu}]\subset I$, $y(s)\simeq y(t_{\lambda})$ for all
$s$ in $[t_{\lambda},t_{\nu}]$. By (\ref{onze}), we have
\[
\frac{y(t_{\nu})-y(t_{\lambda})}{t_{\nu}-t_{\lambda}}=\frac{Y(T(y_{\lambda
}))}{T(y_{\lambda})}\simeq\bar{g}_{0}(y_{\lambda})\newline\simeq\bar{g}
_{0}(y(t_{\lambda})).
\]
By the Stroboscopy Lemma, $[t_{0},t_{0}+L]\subset I$ and $y(t_{0}+s)\simeq
\bar{y}(s)$ for all $0\leq s\leq L$. Therefore, this approximation holds for
all $s$ such that $t_{0}+s\leq t_{1}$.
\end{proof}
\subsection{Proof of Theorem \ref{nonstandard1}\label{ns1}}
Let $(x(t),y(t))$ be a solution of the system (\ref{perturbed})$.$ Then
$(x(\varepsilon\tau)$, $y(\varepsilon\tau))\,$\ is a solution of
(\ref{regular}) with initial condition $(\alpha$, $\beta)$. This problem is a
regular perturbation of (\ref{regular0}) with initial
conditions $(\alpha_{0}$, $\beta_{0})$. By the Short Shadow Lemma,
$x(\varepsilon\tau)\simeq\tilde{x}(\tau)$ and $y(\varepsilon\tau)\simeq
\beta_{0}$ for all limited $\tau\geq0$. By Robinson's Lemma, there exists
$\omega$ positive unlimited such that those approximations still hold for all
$\tau\in[0$, $\omega]$. One can chose $\omega$ such that $\varepsilon
\omega\simeq0$. On the other hand, by Assumptions $(B')$ and $(D)$,
$\tilde{x}(\tau)$ is defined for all $\tau\geq0$ and satisfies $\tilde{x}
(\tau)\simeq\Gamma_{\beta_{0}}$ for all $\tau$ positive and unlimited. This
last property is true in particular for $\tau=\omega$, which means that after
a time $t_{0}:=\varepsilon\omega$ the solution of (\ref{perturbed}) is
infinitely close to $\Gamma_{\beta_{0}}\subset\mathcal{C}$.
Assume
that there exists $s\in]t_{0},L]$ such that
$y(s)\not\simeq\bar{y}(s)$. Let $r>0$ be standard such that
$\| y(s)-\bar{y}(s)\| \geq r$.
Since $\bar{y}(t)$
is nearstandard in $G_{0}$, we can chose $r$ small enough such that the tubular neighborhood
$$
\mathcal{B}=\{(t,y): t\in[0, L], y\in G_{0}\mbox{ and }
\| \bar{y}(t)-y\| 0$. By Robinson's Lemma, those approximations still hold
for a certain $L\simeq+\infty$. Thus, using Assumption $(E)$, $y(L)\simeq
\bar{y}(L)\simeq\bar{y}_{\infty}$ and $x(L)\simeq\Gamma_{\bar{y}_{\infty}}$.
Applying again Theorem \ref{nonstandard1} to the solution starting from
$(x(L)$, $y(L))$ gives
\begin{equation}\label{quatorze}
y(L+k)\simeq\bar{y}_{\infty},\quad
x(L+k)\simeq\Gamma_{\bar{y}_{\infty}}\text{ for all limited }k\geq0.
\end{equation}
Suppose that there exists $s\geq L$ such that $y(s)\ $is not infinitely close
to $\bar{y}_{\infty}$ and let us find a contradiction. Then we can suppose
that $\| y(s)-\bar{y}_{\infty}\| =\mu$ standard. The value $s$ is chosen such
that the ball $\mathcal{B}$ of center $\bar{y}_{\infty}$ and radius $\mu$ is
contained in the basin of attraction of $\bar{y}_{\infty}$. Let $m$ be the
smallest value of such numbers $s$ (this $m$ exists by compactness of
$\partial\mathcal{B}$ and $\| y(m)-\bar{y}_{\infty}\| =\mu$). It can be seen
from (\ref{quatorze}) that $k_{0}:=m-L$ is positive unlimited. The solution
starting by $(x(m)$, $y(m))$ satisfies $y(m+k)\in\mathcal{B}$ for all $k$ in
$[-k_{0}$, $0]$. Let $\bar{y}(k)$ be the solution of the slow equation
(\ref{average}) with initial condition $\bar{y}(0)=y_{0}(m)$, where $y_{0}(m)$
is a standard verifying $y_{0}(m)\simeq y(m)$. Lemma \ref{lemma2} asserts that
$y(m+k)\simeq\bar{y}(k)$ for all limited $k\leq0$ provided $x(m+k)\simeq
\Gamma_{y(m+k)}$, which can be established by contradiction as in the proof of
Lemma \ref{lemma1}. By Robinson's Lemma, there exists $k_{1}<0$ unlimited such
that $y(m+k_{1})\simeq\bar{y}(k_{1})$ which may be chosen such that
$-k_{0}\leq k_{1}$. This means that $\bar{y}(k_{1})$ is in $\mathcal{B}$, thus
in the basin of attraction of $\bar{y}_{\infty}$. Assumption $(E)$ then gives
$\bar{y}(k_{1}+k)\simeq\bar{y}_{\infty}$ for all unlimited $k>0$. In
particular, for $k=-k_{1}$, one has $\bar{y}(0)\simeq\bar{y}_{\infty}$. But
$\bar{y}(0)=y_{0}(m)$ and $y_{0}(m)\simeq y(m)$, then $y(m)\simeq\bar
{y}_{\infty}$, which is absurd.
\section{Applications\label{applications}}
\subsection{A system with delayed loss of stability}
The aim of this example is to illustrate both the result of Theorem
\ref{standard1} and the delayed loss of stability phenomenon pointed out in
the introduction. Let us consider the three dimensional slow-fast system
\begin{equation}\label{systemS}
\begin{gathered}
\varepsilon\dot{x}_{1}=x_{2}-yx_{1}(1-x_{1}^{2}-x_{2}^{2})^{3},\\
\varepsilon\dot{x}_{2}=-x_{1}-yx_{2}(1-x_{1}^{2}-x_{2}^{2})^{3},\\
\dot{y}=x_{1}^{2}.
\end{gathered}
\end{equation}
The fast equation is
\begin{equation}
\begin{gathered}
x_{1}'=x_{2}-yx_{1}(1-x_{1}^{2}-x_{2}^{2})^{3},\\
x_{2}'=-x_{1}-yx_{2}(1-x_{1}^{2}-x_{2}^{2})^{3},
\end{gathered}
\label{B}
\end{equation}
where $y$ is a parameter. In terms of the polar coordinates $(x_{1}
=r\cos\theta,x_{2}=r\sin\theta)$, the equation (\ref{B}) is written as
\begin{equation}
\begin{gathered}
r'=-ry(1-r^{2})^{3},\\
\theta'=-1.
\end{gathered}
\label{C}
\end{equation}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=6cm]{sarifig101}
\includegraphics[width=6cm]{sarifig102}
\vspace*{-0.1cm}
\caption{A solution of (\ref{systemS})
with $\varepsilon=0.1$, $x_1^0=2$, $x_2^0=2, y^0=-1$
in the phase space $(x_1,x_2,y)$.
The functions $r(t,\varepsilon)$ and $y(t,\varepsilon)$ are approximated respectively by
the functions $\bar{r}(t)$, and $\bar{y}(t)$ even after $t=2$ where
the cycles become unstable.}
\setlength{\unitlength}{1cm}
\vspace{-7.5cm}
\begin{picture}(12,7.3){\small
\put(11,5.6){$\bar{r}(t)$}
\put(10.3,3.9){$\bar{y}(t)$}
\put(9.8,5.8){$r(t,\varepsilon)$}
\put(8.4,4.1){$y(t,\varepsilon)$}
\put(4.5,2.4){$x_1$}
\put(0,5.0){$x_2$}
\put(1.2,2.4){$y$}}
\end{picture}
\end{center}
\end{figure}
From (\ref{C}) it is seen that the fast equation (\ref{B}) admits a unique
cycle $\Gamma_{y}$ for all $y\neq0$, namely the circle of center the origin and
radius $1$. This cycle corresponds for instance to the $2\pi$-periodic solution
$x^{\ast}(\tau,y)=(\cos\tau,-\sin\tau)$. The cycles are asymptotically stable
for all $y<0$, while they are unstable for $y>0$. If $y=0$, the origin of
(\ref{B}) is a center.\ Notice that the cycles $\Gamma_{y}$ are not
exponentially stable, so that the result of Pontryagin and Rodygin does not
apply. Note that the basin of attraction of $\Gamma_{y}$ is the whole plan
$(x_{1},x_{2})$ for all $y$ $<0$, except the origin. The asymptotic stability
is therefore uniform over any interval $G$ of $]-\infty,0[$. We consider the
IVP consisting of the system (\ref{systemS}) together with the initial condition
$(x_1^0,x_2^0,y^0)$, such that $y^0<0$. The reduced problem is defined by
\[
\dot{y}=\frac{1}{2\pi}\int_{0}^{2\pi}\cos^{2}\tau d\tau=\frac{1}{2},\quad y(0)=y^0.
\]
Its solution is $\bar{y}(t)=y^0+t/2$. According to
Theorem~\ref{standard1}, the solution of (\ref{systemS}) satisfies
${\lim_{\varepsilon\rightarrow0}}y(t,\varepsilon)=\bar{y}(t)$ as long
as $0\leq t\leq L<-2y^0$. By Remark \ref{remark}, $(x_{1}(t,\varepsilon
),x_{2}(t,\varepsilon))$ stays near the cycle $\Gamma_{\bar{y}(t)}$ while
performing rapid oscillations along it of period approximately
$2\pi\varepsilon$, that is, $r(t)$ is approximated by the solution of the averaged
equation
\begin{equation}\label{averr}
\varepsilon\dot{r}=-r\bar{y}(t)(1-r^{2})^{3},\quad r(0)=\sqrt{(x_1^0)^2+(x_2^0)^2}.
\end{equation}
\begin{figure}[ht]
%\vspace{5cm}
\begin{center}
\includegraphics[width=6cm]{sarifig201}
\includegraphics[width=6cm]{sarifig202}
\vspace*{-0.1cm}
\caption{A solution of (\ref{systemS}) with $\varepsilon=0.01$, $x_1^0=2$, $x_2^0=2, y^0=-1$
in the phase space $(x_1,x_2,y)$.
The functions $r(t,\varepsilon)$ and $y(t,\varepsilon)$ are approximated respectively by
the functions $\bar{r}(t)$, and $\bar{y}(t)$ asymptotically until the exit-time $t=4$ of
the averaged system.}
\setlength{\unitlength}{1cm}
\vspace{-7.8cm}
%\hspace{-3cm}
\begin{picture}(12,7.3){\small
\put(11.2,5.3){$\bar{r}(t)$}
\put(10.2,3.9){$\bar{y}(t)$}
\put(10.2,5.4){$r(t,\varepsilon)$}
\put(8.9,4.2){$y(t,\varepsilon)$}
\put(4.5,2.5){$x_1$}
\put(0,5.1){$x_2$}
\put(1.2,2.5){$y$}}
\end{picture}
\end{center}
\end{figure}
The solution of (\ref{averr}) is denoted by $\bar{r}(t)$. Its satisfies the
property
$\bar{r}(-4y^0-t)=\bar{r}(t)$, hence $\bar{r}(-4y^0)=\bar{r}(0)$. Since
$\bar{y}(-4y^0)=-y^0$, if a trajectory of the averaged system approaches the cycles
of radius 1 for some value $y^0<0$, then it remains near the cycles as long as
$y^0<\bar{y}(t)<-y^0$. Notice that for
$0<\bar{y}(t)<-y^0$ the cycles are unstable~: there is a delayed loss of
stability for the averaged system and the {\em entrance-exit}
function near the cycles is defined by $y^0\mapsto -y^0$.
According to
Theorem~\ref{standard1}, the solution of (\ref{systemS}) is approximated
by the averaged solution as long $0\leq t<-2y^0$, that is, as long as $y^0\leq \bar{y}(t)<0$.
The numerical simulations in Figures 1 and 2 show that the actual solution
$(r(t,\varepsilon),y(t,\varepsilon))$
is approximated by the averaged solution $(\bar{r}(t),\bar{y}(t))$ even after time $t=-2y^0$
where the cycles become unstable. This approximation holds asymptotically until the {\em exit-time}
$t=-4y_0$ of the averaged system.
The rolling up of the trajectory
$(x_{1}(t,\varepsilon)$, $x_{2}(t,\varepsilon)$, $y(t,\varepsilon))$ around
the cycles $\Gamma_{y}$ still holds for positive values of $y$, although the
cycles became unstable. If we consider $y$ as a dynamical bifurcation
parameter, the delayed loss of stability phenomenon established by
Neishtadt \cite{neishtadt} turns to be still available. Recall that in this example
the stability of the cycles is just asymptotic and not exponential, so that the result of Neishtadt
does not apply. This problem deserves a special investigation.
\subsection{A model from population ecology}
Let us consider the following three trophic levels food chain model
\begin{equation}
\begin{gathered}
\varepsilon\dot{x}_{1}=U(x_{1})-x_{2}V_{1}(x_{1}),\\
\varepsilon\dot{x}_{2}=\alpha_{1}x_{2}V_{1}(x_{1})-D_{2}x_{2}-yV_{2}(x_{2}),\\
\dot{y}=\alpha_{2}yV_{2}(x_{2})-Dy,
\end{gathered} \label{model}
\end{equation}
where $\varepsilon$ is a small positive parameter. The nonnegative variables
$x_{1}$, $x_{2}$ and $y$ are the respective densities of the prey, the
predator and the superpredator. The function $U(x_{1})$ is the \textit{growth function} of the
prey. The functions $V_{1}(x_{1})$ and $V_{2}(x_{2})$ are
the \textit{functional responses} of
the predator and the superpredator respectively. The parameters $D_{2}$ and $D$ are
the respective death rates
of the predator and the superpredator and $\alpha_{1}$ and $\alpha_{2}$
are conversion coefficients of the biomass respectively from the prey to the
predator and from the predator to the superpredator. For more details on this
kind of models and these biological constants, all positive, see for example
\cite{bs,muratori}. Note that the presence of the small parameter $\varepsilon$
emphasizes the fact that the multiplications of the prey and the predator are
of same order and much faster than the growth of the superpredator.
\begin{figure}[ht]
%\vspace{7cm}
\begin{center}
\includegraphics[width=7cm]{sarifig300}
\vspace*{-0.1cm}
\caption{The growth function $U$ and the functional responses $V_1$ and $V_2$ of
the three trophic levels food chain model (\ref{model}).}
\end{center}
\end{figure}
We assume that the functions $U$, $V_{1}$ and $V_{2}$ are continuous.
Nonsmooth righthand side of differential equations (and even discontinuous
righthand sides) are of interest in the
biological literature (see for example \cite{gs}).
We assume also that these functions satisfy the
following properties (see Figure~3):
\begin{itemize}
\item
$U(0)=V_{1}(0)=V_{2}(0)=0$,
\item
there exists $K>0$ such
that $U(K)=0$ and $U$ is positive on $]0,K[$ and negative on $]K,+\infty[$,
\item
The functions $V_1$ and $V_2$ are strictly increasing and
$\lim_{x_{1}\rightarrow+\infty}V_{1}(x_{1})$ and
$\lim_{x_{2}\rightarrow+\infty}V_{2}(x_{2})$ are finite.
\end{itemize}
Such properties are satisfied by the model
with \textit{logistic growth} of the prey
and \textit{Holling type II}
functional responses of the predator and the superpredator~:
\begin{equation}\label{modelP}
U(x_{1})=rx_{1}(1-x_{1}/K),\quad
V_{1}(x_{1})=\frac{a_{1}x_{1}}{b_{1}+x_{1}},\quad
V_{2}(x_{2})=\frac{a_{2}x_{2}}{b_{2}+x_{2}},
\end{equation}
where $r$, $K$, $a_{1}$, $a_{2}$, $b_{1}$ and $b_{2}$
are biological positive parameters.
\begin{figure}[ht]
%\vspace{5cm}
\begin{center}
\includegraphics[width=6cm]{sarifig401}
\includegraphics[width=6cm]{sarifig402}
\vspace*{-0.1cm}
\caption{A numerical simulation of model
(\ref{model}-\ref{modelP}) with the following values of the parameters : $r=K=10$,
$\alpha_{1}=0.4$, $a_{1}=5$, $b_{1}=2.5$, $D_{2}=1$, $\alpha_{2}=0.5$,
$a_{2}=10$, $b_{2}=5$ and $D=2$. The figure at left corresponds to
$\varepsilon=0.05$ and the figure at right corresponds to $\varepsilon=0.01$.
The initial condition is $x_1^0=10$, $x_2^0=6$, $y^0=0.1$.}
\setlength{\unitlength}{1cm}
\vspace{-7cm}
%\hspace{-3cm}
\begin{picture}(12,7.3)\small
\put(4.9,3.3){$x_1$}\put(11,3.3){$x_1$}
\put(0,5.8){$x_2$}\put(6.1,5.8){$x_2$}
\put(1.5,3.0){$y$}\put(7.6,3.0){$y$}
\end{picture}
\end{center}
\end{figure}
When $z=0$, the fast equation associated
to (\ref{model}-\ref{modelP}) is the classical prey-predator system. For this model,
under suitable
conditions on the parameters, the uniqueness and the exponential stability of a limit
cycle have been proved in \cite{liou}. For $z>0$, numerical simulations
\cite{kuznetsov} give strong evidence that the model still have a limit cycle for certain values of
the parameters, but there is no theoretical results on the existence of a cycle, nor on its stability.
More details are given in \cite{yadi}.
We return to the general model (\ref{model})
and we assume that the fast equation
\begin{equation}
\begin{gathered}
x_{1}'=U(x_{1})-x_{2}V_{1}(x_{1}),\\
x_{2}'=\alpha_{1}x_{2}V_{1}(x_{1})-D_{2}x_{2}-yV_{2}(x_{2}),
\end{gathered} \label{submodel}
\end{equation}
satisfies Assumption $(B)$. More precisely we assume that
there exist $\alpha_{1}$, $\alpha_{2}$,
$D$, $D_{2}$ and a compact interval
$G$ of $[0,+\infty[$ with a non empty interior such
that, for all $y\in G$, the fast system (\ref{submodel})
has a unique cycle $\Gamma_{y}$ which is asymptotically stable
with a uniform basin of attraction over $G$.
Let $(x_{1}^{\ast}(\tau,y),x_{2}^{\ast}(\tau,y))$ be a $T(y)$-periodic
solution of orbit $\Gamma_{y}$ and define on $G_{0}$ the function
$$M(y)=\frac{1}{T(y)}\int_{0}^{T(y)}(\alpha_{2}yV_{2}(x_{2}^{\ast}(\tau,y))-Dy)d\tau.$$
According to Theorem \ref{standard1}, it follows that for every
initial condition $(x_{1}^{0},x_{2}^{0},y^{0})$ such that $y^{0}\in G_{0}$ and
$(x_{1}^{0},x_{2}^{0})$ is in the basin of attraction of $\Gamma_{y^{0}}$, the
evolution of the superpredator in the model (\ref{model}) is approximated by
the solution $\bar{y}(t)$ of the reduced problem
\[
\dot{y}=M(y),\quad y(0)=y^{0}.
\]
More exactly, if $((x_{1}(t,\varepsilon),x_{2}(t,\varepsilon),y(t,\varepsilon
))$ is the solution of the IVP, we have
$$\lim_{\varepsilon\rightarrow 0}y(t,\varepsilon)=\bar{y}(t)\mbox{ for all }0\leq t\leq T$$
$$\lim_{\varepsilon\rightarrow 0}\mathop{\rm dis}((x_{1}(t,\varepsilon
),x_{2}(t,\varepsilon)),\Gamma_{\bar{y}(t)})=0\mbox{ for all }0