\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 106, pp. 1--20.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2010/106\hfil Slow exit and entrance points]
{Connecting fast-slow systems and Conley Index theory via transversality}
\author[C. Kuehn\hfil EJDE-2010/106\hfilneg]
{Christian Kuehn}
\address{Christian Kuehn \newline
Center for Applied Mathematics,
657 Frank H.T. Rhodes Hall,
Cornell University,
Ithaca, NY 14853, USA}
\email{ck274@cornell.edu}
\thanks{Submitted May 19, 2010. Published August 4, 2010.}
\subjclass[2000]{34C26, 34E15, 37B30, 34A26, 34C45}
\keywords{Fast-slow system; Conley index; Fenichel normal form;
transversality}
\begin{abstract}
Geometric Singular Perturbation Theory (GSPT) and
Conley Index Theory are two powerful techniques to analyze
dynamical systems. Conley already realized that using his index
is easier for singular perturbation problems. In this paper,
we will revisit Conley's results and prove that the GSPT technique
of Fenichel Normal Form can be used to simplify the application
of Conley index techniques even further. We also hope that our
results provide a better bridge between the different fields.
Furthermore we show how to interpret Conley's conditions in
terms of averaging. The result are illustrated by the two-dimensional
van der Pol equation and by a three-dimensional Morris-Lecar model.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\section{Introduction}
We start by outlining the context and results of the paper.
Singular perturbation theory often involves a distinguished
parameter, usually denoted $\epsilon$, which is assumed to be
small and positive. For $\epsilon=0$, the dynamical system is
``degenerate'' but it might also be easier to analyze. Ordinary
differential equations (ODEs) with two different time scales,
so-called fast-slow systems, are a very important class of
singular perturbation problems. In this context, the parameter
$\epsilon$ describes the separation of time scales. One successful
strategy to analyze fast-slow systems is to understand the case
$\epsilon=0$ and using this knowledge to try to prove perturbation
results for $\epsilon>0$ sufficiently small; many geometric and
asymptotic methods follow this pattern.
Conley Index Theory can be applied to wide classes of dynamical
systems which do not have to be singularly perturbed. The goal is
to convert the problem into an algebraic/topological question.
Conley already proved that this approach can be substanially
simplified for singular perturbation problems. Mischaikow and
co-workers developed Conley's index theory for fast-slow systems
recently even further. The problem with the broad applicability of
the theory is that it is still very technical. It has been
demonstrated in low-dimensional examples that easily applicable
geometric versions of the theory should be possible. One possible
generalization to higher dimensions has been proposed but requires
a rather complicated and lengthy topological construction.
The goal of the current paper is to start building a simpler
bridge between GSPT and Conley Index Theory; in particular, we are
going to show that a fundamental result due to Conley can be
simplified using Fenichel Normal Form. The result we prove shows
that if we isolate an invariant set by a suitable neighbourhood
$N$, then $N$ is of the form required by Conley Index Theory if
the slow motion for $\epsilon=0$ is transverse to the boundary of
$N$. We also investigate the case when the fast motion of the
system has periodic orbits and discuss further generalizations. We
hope that the new results will be of interest from the viewpoint
of GSPT as well as Conley Index Theory.
The paper is structured as follows. In Section 2 we describe the
background from fast-slow systems theory and in Section 3 a
similar exposition is given for Conley Index Theory with a focus
on the application to fast-slow systems. Both introductions are a
little more detailed than strictly necessary to accomodate the two
different perspectives with respect to background knowledge. In
Section 4 we prove the main result for equilibrium points of the
fast motion. In Section 5 a result for the case of periodic orbits
is given and a generalization to any bounded invariant set is
outlined. Furthermore the role of averaging is explained. In
Section 6 we demonstrate the applicability of the result by two
examples.
\section{Fast-Slow Systems}
Several viewpoints have influenced the development of multiple
time scale or fast-slow systems starting with asymptotic analysis
\cite{MisRoz,Eckhaus} using techniques like matched asymptotic
expansions \cite{KevorkianCole,Lagerstrom}. A geometric theory
focusing on invariant manifolds was developed
\cite{Fenichel4,Jones,TikhonovVasilevaSveshnikov} which is now
commonly known as Fenichel theory due to Fenichel's seminal work
\cite{Fenichel4}. There was also significant influence by a group
using nonstandard analysis
\cite{DienerDiener,BenoitCallotDienerDiener}.
We shall focus on the geometric viewpoint in this paper. The term
``Geometric Singular Perturbation Theory'' (GSPT) is used to
encompass Fenichel theory and further geometric methods developed
over the last three decades in the context of multiple time scale
problems. The general formulation of a \emph{fast-slow system}
of ordinary differential equations (ODEs) is
\begin{equation}
\label{eq:basic1}
\begin{array}{rcrcl}
\epsilon \dot{x}&=&\epsilon\frac{dx}{d\tau}&=&f(x,y,\epsilon),\\
\dot{y}&=&\frac{dy}{d\tau}&=&g(x,y,\epsilon),\\
\end{array}
\end{equation}
where $(x,y)\in\mathbb{R}^m\times \mathbb{R}^n$ and $\epsilon$
is a small parameter $0<\epsilon\ll 1$ representing the ratio
of time scales. The functions
$f:\mathbb{R}^m\times\mathbb{R}^n \times \mathbb{R}\to \mathbb{R}^m$ and
$g:\mathbb{R}^m\times\mathbb{R}^n\times \mathbb{R}\to \mathbb{R}^n$ will
be assumed to be sufficiently smooth. The variables $x$ are fast
and the variables $y$ are slow and we can change
in \eqref{eq:basic1} from the slow time scale $\tau$ to the fast
time scale $t=\tau/\epsilon$ which yields:
\begin{equation} \label{eq:basic2}
\begin{array}{lclcl}
x'&=&\frac{dx}{dt}&=&f(x,y,\epsilon),\\
y'&=&\frac{dy}{dt}&=&\epsilon g(x,y,\epsilon).\\
\end{array}
\end{equation}
We will also denote the vector field \eqref{eq:basic2} by $z'=F(z)$
where $F=(f,\epsilon g)$ and $z=(x,y)$. The first major idea to
analyze \eqref{eq:basic1}-\eqref{eq:basic2} is to consider
the \emph{singular limit} as $\epsilon\to 0$.
\begin{definition} \label{def2.1} \rm
Setting $\epsilon=0$ in \eqref{eq:basic2} gives
\begin{equation} \label{eq:basic_fss}
\begin{array}{lcl}
x'&=&f(x,y,0),\\
y'&=&0,\\
\end{array}
\end{equation}
which is system of ODEs parametrized by the slow variables $y$.
We call \eqref{eq:basic_fss} the \emph{fast subsystem} or
\emph{layer equations}. The associated flow is called the
\emph{fast flow}.
\end{definition}
\begin{definition} \label{def2.2} \rm
Considering the singular limit $\epsilon =0$ for \eqref{eq:basic1}
yields:
\begin{equation} \label{eq:basic_sf}
\begin{array}{lcl}
0&=& f(x,y,0),\\
\dot{y}&=& g(x,y,0).\\
\end{array}
\end{equation}
System \eqref{eq:basic_sf} is a differential-algebraic equation (DAE)
called \emph{slow subsystem} or \emph{reduced system}.
The associated flow is called the \emph{slow flow}.
\end{definition}
One goal of GSPT is to use the fast and slow subsystems to
understand the dynamics of the full system
\eqref{eq:basic1}-\eqref{eq:basic2} for $\epsilon>0$.
\begin{definition} \label{def2.3} \rm
The algebraic constraint of \eqref{eq:basic_sf} defines
the \emph{critical manifold}
\[
C:=\{(x,y)\in\mathbb{R}^m\times \mathbb{R}^n:f(x,y,0)=0\}.
\]
\end{definition}
Note that it is possible that $C$ is not an actual
manifold \cite{KruSzm4} but we shall not consider this case here.
The points in $C$ are equilibrium points for the fast subsystem
\eqref{eq:basic_fss}.
\begin{example} \label{ex:fss} \rm
Consider the following very simple planar fast-slow system
\begin{equation} \label{eq:ex_fss}
\begin{array}{rcl}
\epsilon\dot{x}&=& y-x^2,\\
\dot{y}&=& -1.\\
\end{array}
\end{equation}
The critical manifold $C=\{(x,y)\in\mathbb{R}^2:y=x^2\}$ is a parabola.
Observe that the slow flow on $C$ is $\dot{y}=-1$ so that under
this flow any initial condition on $C$ will ``flow down'' to
the origin $(x,y)=(0,0)$; see single arrows in Figure \ref{fig:fig1}(a). The fast subsystem is $x'=y-x^2$ which has one stable equilibrium and one unstable one for $y>0$, a saddle-node (or fold) bifurcation for $y=0$ and no equilibria for $y<0$; the flow is indicated by double arrows in Figure \ref{fig:fig1}(a).
\end{example}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=1\textwidth]{fig1}
\end{center}
\caption{\label{fig:fig1}(a) Critical manifold $C$ (dark grey)
of \eqref{eq:ex_fss}; the fast and slow flows are indicated
by double and single arrows respectively. (b) $C$ (dark grey)
and a slow manifold $M_\epsilon$ (light grey) obtained from Fenichel's
Theorem \ref{thm:fenichel1} are shown together with a trajectory
$\gamma$ of \eqref{eq:ex_fss} starting at $\gamma(0)=(1,0.4)$
at parameter value $\epsilon=0.05$. Observe that $\gamma$ quickly
approaches $M_\epsilon$, then tracks it (actually
$O(e^{-K/\epsilon})$-close) until the fast-slow structure breaks
down near the fold point $(x,y)=(0,0)$.}
\end{figure}
\begin{definition} \label{def2.5} \rm
A subset $M\subset C$ is called \emph{normally hyperbolic}
if the $m\times m$ matrix $(D_xf)(p)$ of first partial derivatives
with respect to the fast variables has no eigenvalues with zero
real part for all $p\in M$; this condition is equivalent to
requiring that points $p\in S$ are hyperbolic equilibria of the
fast subsystem \eqref{eq:basic_fss}.
\end{definition}
We call a normally hyberbolic subset $M$ attracting if all
eigenvalues of $(D_xf)(p)$ have negative real parts for $p\in M$;
similarly $M$ is called repelling if all eigenvalues have positive
real parts. If $M$ is normally hyperbolic and neither
attracting nor repelling we say it is of saddle-type. A typical
class of points where normal hyperbolicity fails are
\emph{fold points}. They are defined as the points where the
critical manifold $C$ is locally parabolic with respect to the
fast directions. In other words, at a fold point $p_*$ one
requires that $f(p_*, 0) = 0$ and that $(D_xf)(p_*, 0)$ is of rank
$m - 1$ with left and right null vectors $w$ and $v$, such that
$w\cdot [(D_{xx}f)(p)(v, v)] \ne 0$ and $w \cdot [(D_yf)(p)] \ne 0$.
\begin{example}[Example \ref{ex:fss} continued] \label{exa2.6} \rm
The critical manifold $C=\{y=x^2\}$ splits into one repelling part,
one attracting part and a fold point at the origin:
\[
C=C_l\cup\{(0,0)\} \cup C_r
\]
where $C_l=C\cap \{x<0\}$ is repelling, $C_r=C\cap \{x>0\}$ is
attracting and $(x,y)=(0,0)$ is a fold point which is easily
verified since
\[
D_xf=\frac{\partial f}{\partial x}=-2x\quad \text{and} \quad
D_{xx}f=\frac{\partial^2 f}{\partial x^2}=-2.
\]
\end{example}
We continue with the general case. If $(D_xf)$ has maximal rank
the implicit function theorem applied to $f(x,y,0)=0$ locally
provides a function $h(y)=x$ so that $C$ can be expressed as a graph.
Hence the slow subsystem \eqref{eq:basic_sf} can be more succinctly
expressed as:
\begin{equation}
\label{eq:basic_sf1}
\dot{y}=g(h(y),y,0)
\end{equation}
We shall also refer to the flow induced by \eqref{eq:basic_sf1} as
slow flow. To relate the dynamics of the slow flow to the dynamics
of the full system for $\epsilon>0$ the next theorem is of fundamental
importance.
\begin{theorem}[Fenichel's Theorem, \cite{Fenichel4,Jones}]
\label{thm:fenichel1}
Suppose $M=M_0$ is a compact normally hyperbolic submanifold of the critical manifold $C$. Then for $\epsilon>0$ sufficiently small the following holds:
\begin{itemize}
\item[(F1)] There exists a locally invariant manifold
$M_\epsilon$ diffeomorphic to $M_0$. \emph{Local invariance}
means that $M_\epsilon$ can have boundaries through which
trajectories enter or leave.
\item[(F2)] $M_\epsilon$ has a distance $O(\epsilon)$ from $M_0$.
\item[(F3)] The flow on $M_\epsilon$ converges to the slow
flow as $\epsilon \to 0$.
\item[(F4)] $M_\epsilon$ is $C^r$-smooth for any $r<\infty$
(as long as $f,g\in C^\infty$).
\item[(F5)] $M_\epsilon$ is normally hyperbolic and has the same
stability properties with respect to the fast variables as
$M_0$ (attracting, repelling or saddle-type).
\item[(F6)] For fixed $\epsilon>0$, $M_\epsilon$ is usually not
unique but all manifolds satisfying (F1)-(F5) lie at a Hausdorff
distance $O(e^{K/\epsilon})$ from each other for some $K>0$, $K=O(1)$.
\end{itemize}
We call a manifold $M_\epsilon$ a \emph{slow manifold}. Note that
all asymptotic notation refers to $\epsilon\to0$. The same
conclusions as for $M_0$ hold (locally) for its stable and unstable
manifolds:
\[
W^s(M_0)=\cup_{p\in M_0} W^s(p),\quad W^u(M_0)=\cup_{p\in M_0} W^u(p)
\]
where we view points $p\in M_0$ as equilibria of the fast subsystem.
\end{theorem}
Figure \ref{fig:fig1}(b) shows an typical scenario where Fenichel's
Theorem applies; there we picked a compact submanifold $M_0\subset C_r$
and obtained an associated slow manifold. In addition to Fenichel's
Theorem we can also find coordinate changes the simplify a fast-slow
system considerably near a critical manifold.
\begin{theorem}[Fenichel Normal Form, \cite{Fenichel4,JonesKaperKopell}]
\label{thm:FNform}
Suppose the origin $0\in C$ is a normally hyberbolic point with
$m_u$ unstable and $m_s$ stable fast directions; choose a sufficiently
small compact normally hperbolic subset $M_0\subset C$ containing
the origin. Then there exists a smooth invertible coordinate change
$(x,y)\mapsto (a,b,v)\in\mathbb{R}^{m_u}\times \mathbb{R}^{m_s}\times
\mathbb{R}^n$ in a neighbourhood of $0$ so that a fast-slow
system \eqref{eq:basic2} can be written as
\begin{equation} \label{eq:FNform1}
\begin{array}{lcl}
a'&=& \Lambda(a,b,v,\epsilon)a ,\\
b'&=& \Gamma(a,b,v,\epsilon)b,\\
v'&=& \epsilon(m(v,\epsilon)+H(a,b,v,\epsilon)ab),\\
\end{array}
\end{equation}
where $\Lambda$, $\Gamma$ are matrix-valued functions. $\Lambda$
has $m_u$ positive and $\Gamma$ has $m_u$ negative eigenvalues.
$H$ is bilinear and given in coordinates by
\begin{equation}
\label{eq:Hbilinear}
H_i(a,b,v,\epsilon)ab=\sum_{r=1}^{m_s}\sum_{s=1}^{m_u}H_{irs}a_r b_s.
\end{equation}
\end{theorem}
The situation is illustrated in Figure \ref{fig:fig2}.
The manifold $M_0$ perturbs to a slow manifold $M_\epsilon$
by Fenichel's Theorem. Then this slow manifold is ``straightened''
together with its stable and unstable manifolds that become
coordinate planes.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=1\textwidth]{fig2}
\end{center}
\caption{Illustration of Theorem \ref{thm:FNform}.}
\label{fig:fig2}
\end{figure}
As a major overall conclusion of Fenichel Theory we get that
the flow near a normally hyperbolic critical manifold is
``completely determined'' by the singular limit systems for
$\epsilon=0$. We will show in Section \ref{sec:eepts} how this
result reappears for the Conley index theory of fast-slow systems.
\section{Conley Index Theory}
In this section we describe the basic constructions from Conley
Index Theory and how these have been adapted to fast-slow systems.
When Conley \cite{ConleyEaston} studied these techniques that
now bear his name it seems possible \cite{ConleyFastSlow1}
that he also had applications to singular perturbation problems
in mind. The idea of the theory is to convert a dynamical
problem (e.g. ``Does this dynamical system have a heteroclinic orbit'')
into an algebraic problem (e.g. ``What is the structure of a matrix?'').
Several successful applications exist; see \cite{MischaikowMrozek}
for a recent survey. We are going to outline only the basic
techniques of the theory focusing on isolating neighbourhoods for
fast-slow ODEs. In this context, the current theory can be found in
\cite{Conley,MischaikowMrozekReineck,ConleyFastSlow1,ConleyFastSlow2,KokubuMischaikowOka}.
For a more detailed introduction in the case of general
finite-dimensional dynamical systems we refer to
\cite{MischaikowMrozek}; the infinite-dimensional case is
considered in \cite{Rybakowski}.
Let $\phi:\mathbb{R}\times \mathbb{R}^k \to \mathbb{R}^k$ be a flow with
$\phi=\phi(t,z)$. A compact set $N\subset \mathbb{R}^k$ is called
an \emph{isolating neighbourhood} if
\[
\operatorname{Inv}(N,\phi)
:=\{z\in\mathbb{R}^k|\phi(\mathbb{R},z)\subset N\}\subset \operatorname{int}(N)
\]
where $\operatorname{int}(N)$ denotes the interior of $N$.
If we set $S:=\operatorname{Inv}(N,\phi)$ then $S$ is called an
\emph{isolated invariant set}. The situation is illustrated in
Figure \ref{fig:fig3} where $\operatorname{Inv}(N,\phi)$
is an unstable node.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.4\textwidth]{fig3}
\end{center}
\caption{\label{fig:fig3}An isolating neighbourhood $N$
(shaded disk) of an unstable node $q$ is shown. The node $q$ is
the invariant set of of $N$ i.e. $\operatorname{Inv}(N,\phi)=\{q\}$.
The boundary of $N$ (dashed circle) is denoted by $L$ and $(N,L)$
is easily seen to form an index pair.}
\end{figure}
\begin{definition} \label{def3.1} \rm
\label{defn:indexpair}
Let $S$ be an isolated invariant set. A pair of compact sets
$(N,L)$ with $L\subset N$ is called an \emph{index pair} for
$S$ if the following conditions hold:
\begin{itemize}
\item[(a)] $S=\operatorname{Inv}(\operatorname{cl}(N-L))$ and
$N-L$ is a neighbourhood of $S$.
\item[(b)] $L$ is positively invariant in $N$ i.e. for any
$z\in L$ and $\phi([0,t],z)\subset N$ then $\phi([0,t],z)\subset L$.
\item[(c)] $L$ is an \emph{exit set} for $N$ i.e. for any $z\in N$
and $t_1>0$ such that $\phi(t_1,z)\not\in N$ then there exists
$t_0\in[0,t_1]$ for which $\phi([0,t_0],z)\subset N$ and
$\phi(t_0,z)\in L$.
\end{itemize}
\end{definition}
We define the \emph{Conley index} of $S$ as
\[
CH_*(S):=H_*(N,L)
\]
where $H_*$ is relative homology \cite{Hatcher,Spanier}. Note that
an alternative way to define the Conley index would be to
consider cohomology and set $CH^*(S):=H^*(N,L)$. The strategy to
use the Conley index for dynamical systems usually proceeds along
the following lines:
\begin{itemize}
\item[(S1)] Find an isolating neighbourhood $N$.
\item[(S2)] Determine an index pair $(N,L)$.
\item[(S3)] Calculate the Conley index.
\item[(S4)] Use the calculation to prove a result about
$\operatorname{Inv}(N,\phi)$.
\end{itemize}
We will focus on (S1)-(S2) in the context of fast-slow systems.
The main question is whether we can use the fast-slow structure
to find an index pair. Re-writing a general fast-slow system
\eqref{eq:basic2} on the fast time scale with $z=(x,y)$ will
be convient
\begin{equation} \label{eq:sipsumform}
z'=F_0(z)+\sum_{i=1}^j \epsilon^i F_i(z)+o(\epsilon^j).
\end{equation}
We denote the flow of \eqref{eq:sipsumform} by
$\phi_\epsilon:\mathbb{R}\times \mathbb{R}^{m+n}\to \mathbb{R}^{m+n}$.
Observe that $\phi_0$ is the flow of the fast subsystem
\eqref{eq:basic_fss}. It is easy to see that if $N$ is an
isolating neighbourhood for $\phi_0$ then it is also an
isolating neighbourhood for $\phi_\epsilon$. The problem is that
usually $N$ will not be an isolating neighbourhood for $\epsilon=0$
but it still can be an isolating neighbourhood $\epsilon>0$.
\begin{definition} \label{def3.2} \rm
A compact set $N\subset \mathbb{R}^{m+n}$ is called a
\emph{singular isolating neighbourhood} if $N$ is not an
isolating neighbourhood for $\phi_0$ but there exists $\bar{\epsilon}$
such that $N$ is an isolating neighbourhood for $\phi_\epsilon$
with $\epsilon\in (0,\bar{\epsilon}]$.
\end{definition}
The next example illustrates, without proof, a singular
isolating neighbourhood in a fast-slow system.
\begin{example} \label{ex:VdPb} \rm
A time reversed version of Van der Pol's \cite{vanderPol,vanderPol_RO}
equation is
\begin{equation}
\label{eq:VdPsip_b}
\begin{array}{lcl}
x'&=& x^3/3-x-y,\\
y'&=& \epsilon x.\\
\end{array}
\end{equation}
We shall consider the Van der Pol equation in more detail
in Section \ref{sec:VdP}. For now it is useful to look ahead
to Figure \ref{fig:VdP} that shows the critical manifold $C_0$
of \eqref{eq:VdPsip_b} and an orbit for $\epsilon=0$ composed
of fast and slow subsystem trajectories. To prove that this orbit
perturbs we want to construct a singular isolating neighbourhood;
the dashed lines in Figure \ref{fig:VdP} indicate a possible guess
for a such a neighbourhood $N$. In Section \ref{sec:VdP} we are
going to prove that $N$ is a singular isolating neighbourhood.
For now it is important to observe that $N$ is not an isolating
neighbourhood for $\epsilon=0$ on the fast time scale (check it!).
\end{example}
To check whether a compact set is a singular isolating neighbourhood
we temporalily decide to define the complications away.
\begin{definition} \label{defn:sept} \rm
Let $N$ be a compact set and let
$z\in \operatorname{Inv}(N,\phi_0)=:S$. We say that $z$ is a
\emph{slow exit (entrance) point} if there is a neighbourhood $U$
of $z$ and an $\bar{\epsilon}>0$ such that for all
$\epsilon\in (0,\bar{\epsilon}]$ there is a time $T(\epsilon,U)>0$
($T(\epsilon,U)<0$) such that
\[
\phi_\epsilon(T(\epsilon,U),U)\cap U=\emptyset.
\]
Let $S^-$ $(S^+)$ denote the set of slow exit (entrance) points.
Furthermore define the following sets
\[
S_\partial :=S\cap \partial N \quad \text{and}
\quad S_\partial^\pm:=S_\partial \cap S^\pm.
\]
\end{definition}
To understand what slow exit and entrance points are and what
characterizes them is the main goal of this paper and should be
clear after Section \ref{sec:eepts}. Obviously Definition
\ref{defn:sept} `cheats' by prescribing the dynamics of the slow
motion under perturbation. Therefore the next result is very easy
to prove.
\begin{theorem} \label{thm:sinbhd}
If $\operatorname{Inv}(N,\phi_0)\cap \partial N$ consists of slow
exit and entrance points then $N$ is a singular isolating
neighbourhood i.e. it is an isolating neighbourhood for the
full fast-slow system for sufficiently small $\epsilon>0$.
\end{theorem}
Hence we have reduced the problem to characterizing slow
exit/entrance points in a more computable way. The next definition
provides a technical notion which will be necessary for this task.
\begin{definition} \label{def3.6} \rm
The \emph{average} of a function $h$ on $S\subset \mathbb{R}^{m+n}$,
denoted $\operatorname{Avg}(h,S)$, is the limit as $T\to \infty$ of the set
of numbers
\[
\Big\{\frac1T \int_0^T h(\phi_0(s,z))ds:z\in S\Big\}.
\]
We say that $h$ has \emph{strictly positive averages} on $S$
if $\operatorname{Avg}(h,S)\subset (0,\infty)$.
\end{definition}
In a seminal paper \cite{Conley} Conley was able to give computable
conditions for slow exit and entry points.
\begin{theorem}[\cite{Conley}]\label{thm:csept}
A point $z\in S$ is a slow exit point if there exists a compact
set $K_z\subset S$ invariant under $\phi_0$, a neighbourhood $U_z$
of the chain recurrent set $\mathcal{R}(K_z)$ of $K_z$, an
$\bar{\epsilon}>0$ and a function
$l:\operatorname{cl}(U_z)\times [0,\bar{\epsilon}]\to \mathbb{R}$
such that the following conditions are satisfied:
\begin{itemize}
\item[(a)] $\omega(z,\phi_0)=\bigcap_{t\in \mathbb{R}}
\operatorname{cl}\left(\{\phi_0(s,z):s>t\}\right)\subset K_z$;
here $\operatorname{cl}(.)$ denotes closure.
\item[(b)] $l$ is of the form
$l(w,\epsilon)=l_0(w)+\epsilon l_1(w)+\ldots+\epsilon^j l_j(w)$.
\item[(c)] If $L_0=\{w|l_0(w)=0\}$ then
$K_z\cap \operatorname{cl}(U_z)=S\cap L_0\cap \operatorname{cl}(U_z)$
and furthermore $l_0|_{S\cap \operatorname{cl}(U_z)}\leq 0$.
\item[(d)] Let $G_j(w)=\nabla_z l_0(w)\cdot F_j(w)
+\nabla_z l_1(w)\cdot F_{j-1}(w)+\ldots+\nabla_z l_j(w)\cdot F_0(w)$.
Then for some $k$, $G_k=0$ if $k0$ do enter
crucially in (d). But as we shall see in the next section,
the idea was intuitively correct. In fact, one should state that
only the fast and slow flows of the singular subsystems play a role
in the calculations.
Using Theorem \ref{thm:csept} we can often identify an isolating
neighbourhood for a fast-slow system. Mischaikow, Mrozek
and Reineck \cite{MischaikowMrozekReineck} give an analogous
construction for index pairs.
\begin{definition} \label{def} \rm
A pair of compact sets $(N,L)$ with $L\subset N$ is called a
\emph{singular index pair} if $\operatorname{cl}(N-L)$ is a singular
isolating neighbourhood and there exists an $\bar{\epsilon}>0$
such that for all $\epsilon\in (0,\bar{\epsilon}]$
\[
H_*(N,L)=CH_*(\operatorname{Inv}(\operatorname{cl}(N-L),\phi_\epsilon)).
\]
\end{definition}
The singular index pair should be characterized by similar conditions
as the usual index pair described in Definition \ref{defn:indexpair}.
From the exit set requirement we know that $L$ has to contain
the \emph{immediate exit set} of $N$
\[
N^-:=\{z\in \partial N : \phi_0((0,t),z)\not\subset N
\text{ for all }t>0\}.
\]
Regarding positive invariance, it turns out that give $Y\subset N$
one has to consider the \emph{pushforward set} in $N$ under the flow $Y$
defined by
\[
\rho(Y,N,\phi_0):=\{z\in N : \exists w\in Y,t\geq 0 \text{ s.t. }
\phi_0([0,t],w)\subset N,\phi_0(t,w)=z\}.
\]
Basically $\rho(Y,N,\phi_0)$ consists of points in $N$ that can
be reached from $Y$ by a positive trajectory in $N$; observe that
by construction we must have $Y\subset \rho(Y,N,\phi_0)$. In addition,
we also must consider a special version of the unstable manifold of
a point lying in $N$
\[
W^u_N(Y):=\{z\in N : \phi_0((-\infty,0),z)\subset N \text{ and }
\alpha(z,\phi_0)\subset Y\}.
\]
Again we observe that $Y\subset W^u_N(Y)$. Before we can state the
theorem about characterizing singular index pairs, one last
definition is needed.
\begin{definition} \label{def3.9} \rm
A slow entrance point $z$ is called a \emph{strict slow entrance point}
if there exists a neighbourhood $V$ of $z$ and an $\bar{\epsilon}>0$
such that if $v\in V\cap N$ and $\epsilon\in(0,\bar{\epsilon}]$ then
there exists a time $t_v(\epsilon)$ such that
\[
\phi_\epsilon([0,t_v(\epsilon)],v)\subset N.
\]
The set of strict slow entrance points will be denoted by
$S^{++}_\partial$.
\end{definition}
\begin{theorem}[\cite{MischaikowMrozekReineck}] \label{thm:sipmis}
Let $N$ be a singular isolating neighbourhood. Assume
\begin{itemize}
\item[(A)] $S_\partial^-$ consists of C-slow exit points.
\item[(B)] $S_\partial \subset S_\partial^{++}\cup S_\partial^-$.
\item[(C)] $(S_\partial^{++}-S_\partial^-)\cap \operatorname{cl}(N^-)
=\emptyset$.
\end{itemize}
For each $z\in S_\partial^-$, let $K_z$ be a slow exit guide for $z$.
Define
\[
L:=\rho(\operatorname{cl}(N^-),N,\phi_0)\cup
W^u_N\left(\cup_{z\in S^-_\partial}\mathcal{R}(K_z)\right).
\]
If $L$ is closed then (N,L) is a singular index pair.
\end{theorem}
Observe that for Theorem \ref{thm:csept} and
Theorem \ref{thm:sipmis} it is crucial to determine which points
are slow exit/entrance points. The conditions (a)-(d) given in
Theorem \ref{thm:csept} are complicated. The goal of this paper
is to simplify these conditions.
\section{Equilibrium Exit Points}
\label{sec:eepts}
Let $N$ be a compact set and let
$z_0=(x_0,y_0)\in \operatorname{Inv}(N,\phi_0)=:S$ where $\phi_0$
denotes the flow of the fast subsystem. Let $C$ denote the critical
manifold. We make the following assumptions:
\begin{itemize}
\item [(A1)] $z_0\in C\cap \partial N$.
\item [(A2)] $C$ is a normally hyperbolic manifold at $z_0$ and locally
given as a graph $x=h(y)$.
\item [(A3)] $\partial N$ is smooth and parallel to the fast fibers
near $z_0$.
\item [(A4)] The slow flow $\dot{y}=g(h(y),y)$ is transverse to $\partial N$
near $z_0$. Let $\vec{n}$ denote the outward unit normal to $N$ at $z_0$;
there are two cases:
\begin{itemize}
\item [(A4.1)] $\vec{n}\cdot (0,g(h(y_0),y_0))>0$, slow flow directed
outward near $z_0$.
\item [(A4.2)] $\vec{n}\cdot (0,g(h(y_0),y_0))<0$, slow flow directed
inward near $z_0$.
\end{itemize}
\end{itemize}
\begin{theorem}\label{thm:eepts}
Under conditions {\rm (A1)-(A4)} the point $z_0$ is a slow
exit/entrance point. If {\rm (A4.1)} holds we have a slow exit
point and for {\rm (A4.2)} we get a slow entrance point.
\end{theorem}
We remark that Theorem \ref{thm:eepts} only requires knowledge
about the fast and slow subsystems for $\epsilon=0$.
This justifies more clearly than Theorem \ref{thm:csept} that
a special Conley index theory for fast-slow systems is possible.
\begin{proof}[Proof of Theorem \ref{thm:eepts}]
Suppose without loss of generality that (A4.1) holds so that we
are trying to show that $z_0\in C$ is a slow exit point.
We also work in a sufficiently small neighbourhood of $z_0$ for
the rest of the proof. The goal is to verify the conditions (a)-(d)
of Theorem \ref{thm:csept}. Using (A2) we apply Fenichel's Normal
Form Theorem \ref{thm:FNform} to re-write the fast-slow system near
$0$ as
\begin{equation} \label{eq:FNformPf1}
\begin{array}{lcl}
x'&=& \Omega(x,y,\epsilon)x,\\
y'&=& \epsilon(m(y,\epsilon)+H(x,y,\epsilon)x),\\
\end{array}
\end{equation}
where $H$ is bilinear as described in \eqref{eq:Hbilinear} and
we have re-written the Fenichel coordinates as $x=(a,b)\in\mathbb{R}^m$
and $v=y\in\mathbb{R}^n$ with $\Omega=(\Lambda,\Gamma)$. Due to (A3),
we obtain that $\partial N$ is locally given by $\{y=0\}$ and also
locally we have $N=\{y_i\leq 0\text{ for all $i=1,\ldots,n$}\}$;
see Figure \ref{fig:fig4}.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.8\textwidth]{fig4}
\end{center}
\caption{\label{fig:fig4}Illustration of the situation near a slow
exit point at the origin in $\mathbb{R}^3$. The compact set $N$ is
locally given by $\{y\leq 0\}$. The outer normal vector $e_1$ to
$\partial N$ is also shown. The slow flow will point along this
normal vector. The parabolic surface $L_0$ is the zero set
$l(x,y)=0=l_0(x,y)$.}
\end{figure}
Note that (A4.1) implies that near $0$ we can rectify the slow flow
so that \eqref{eq:FNformPf1} becomes:
\begin{equation} \label{eq:FNformPf2}
\begin{array}{lcl}
x'&=& \Omega(x,y,\epsilon)x,\\
y'&=& \epsilon(e_1+H(x,y,\epsilon)x),\\
\end{array}
\end{equation}
where $e_1=(1,0,\ldots,0)\in\mathbb{R}^n$. As a slow exit guide
set $K=K_0:=\{0\}$ and observe that since $0\in C$ it is an
equilibrium point for the fast subsystem. Therefore
$\omega(0,\phi_0)=\{0\}$ and so $\omega(0,\phi)=K$ which verifies (a).
Define the function $l$ by
\[
l(x,y)=y_1-\sum_{j=1}^m (x_j)^2=l_0(x,y).
\]
Notice that $l=l_0$ and so we find that (see also Figure \ref{fig:fig4})
\[
L_0=\{(x,y) : l_0(x,y)=0\}
=\{y_1=\sum_{j=1}^m (x_j)^2\}.
\]
Let $U$ be a sufficiently small neighbourhood around $0$ then
$U\cap K=\{0\}$. We also have locally
$S\cap \operatorname{cl}(U)=N\cap C$ and therefore
\[
S\cap \operatorname{cl}(U)\cap L_0=\{0\}
\]
since $0= \sum_{j=1}^m (x_j)^2$ holds if and only if $x_j=0$
for all $j=1,2,\ldots, m$. Obviously
$l_0|_{S\cap \operatorname{cl}(U)}\leq 0$ and so (b)-(c) hold.
For the last step, observe that $\mathcal{R}(0)=\{0\}$ and hence
we have to verify that condition (d) holds at the origin; i.e.,
\[
G_j(z)=\nabla_z l_0(z)\cdot F_j(z)+\nabla_z l_1(z)\cdot
F_{j-1}(z)+\ldots+\nabla_z l_j(z)\cdot F_0(z)
\]
satisfies that for some $k$, $G_k(0)=0$ if $k0$.
We compute the gradient of $l_0$,
\[
\nabla_z l_0(x,y)=(-2x_1,-2x_2,\ldots,-2x_m,1,0,\ldots,0).
\]
Since $\phi_0$ describes the fast flow, the first term $F_0$
in \eqref{eq:sipsumform} for the normal form \eqref{eq:FNformPf2}
is given by
\[
F_0(x,y)=(\Omega(x,y,0)x,0)^T.
\]
This gives that $\nabla_z l_0(0)\cdot F_0(0)=0\cdot \Omega(x,y,0)x
+ e_1\cdot 0=0$. Hence $G_0(0)$ is identically zero.
Next, we show that $G_1(0)$ is positive. We have
\[
G_1(z)=\nabla_z l_0(z) \cdot F_1(z)=(-2x_1,-2x_2,\ldots,-2x_m,1,0,
\ldots,0)\cdot(0, e_1+H(x,y,\epsilon)x)^T.
\]
Since $H(0,0,\epsilon)0=0$ we immediately get
$G_1(0)=e_1\cdot (e_1)^T=1>0$ verifying (d). Therefore the original
point $z_0$ is a slow exit point.
\end{proof}
The only condition that does not seem not quite natural for
Theorem \ref{thm:eepts} is (A3). To illustrate that it is necessary
consider the following example.
\begin{example} \label{ex:repelling} \rm
Consider a fast-slow system with $(x,y)\in\mathbb{R}^2$ given by
\begin{gather*}
x'= x,\\
y'= \epsilon.
\end{gather*}
The solution is given by $(x(t),y(t))=(x(0)e^t,y(0)+\epsilon t)$.
Fix some $m>0$ and let $N=\{(x,y)\in\mathbb{R}^2|y\leq mx\}$
locally near $0$ i.e. we truncate $N$ outside a suitable
neighbourhood to make it compact. Now the origin is not a slow exit
point although (A1)-(A2) and (A4.1) hold. Indeed, pick a
neighbourhood $U$ of $0$ then there is $(x(0),y(0))\in U$ such
that $x(0),y(0)>0$. For $\epsilon>0$ sufficiently small we can
easily assure that $y(0)+\epsilon t0$
such that the trajectory starting at $(x(0),y(0))$ stays in $N$.
\end{example}
Example \ref{ex:repelling} also indicates that the problem should
not occur for attracting critical/slow manifolds.
\begin{proposition}\label{prop:simple}
Suppose {\rm (A1), (A2), (A4)} hold. Furthermore assume that
$\partial N$ is locally linear and has an angle of order
$O(1)$ to $C$ at $z_0$ and that $C$ is attracting at $z_0$.
Then $z_0$ is a slow exit/entrance point. If {\rm (A4.1)} holds we
have a slow exit point and for {\rm (A4.2)} we get a slow
entrance point.
\end{proposition}
\begin{proof}
In this case the proof is much simpler and we do not need Conley's
Theorem \ref{thm:csept}. Again we can restrict without loss of
generality to the case (A4.1). We will work under the assumption
for the rest of this proof that $\epsilon>0$ has been chosen so
that Fenichel Theory applies. Applying Fenichel's Normal Form
Theorem as in Theorem \ref{thm:eepts} gives
\begin{equation}
\label{eq:FNformPf1a}
\begin{array}{lcl}
x'&=& \Omega(x,y,\epsilon)x,\\
y'&=& \epsilon(m(y,\epsilon)+H(x,y,\epsilon)x),\\
\end{array}
\end{equation}
where now $\Omega(0,0,0)$ has $m$ negative eigenvalues.
Let $U$ be a small neighbourhood around the origin.
By Fenichel's Theorem \ref{thm:fenichel1} (F1) there exists
a slow manifold $C_\epsilon$. Let $\gamma$ be a trajectory
with an initial condition in $U$. Since $\Omega(0,0,0)$ has $m$
negative eigenvalues Fenichel's Theorem (F5) shows that $C_\epsilon$
is attracting. Therefore $\gamma$ gets attracted exponentially
to $C_\epsilon$ or lies in $C_\epsilon$. Observe that $\partial N$
is not tangent to $C_\epsilon$ as it has an $O(1)$ angle to $C_0$.
By (A4.1) and Fenichel's Theorem (F3) we find that $\gamma$ must
leave $N$ after a time $T_\gamma(\epsilon)$. Since $U$ is bounded
we can take the maximum of all times over $\operatorname{cl}(U)$
\[
T(\epsilon,U):=\max_{\gamma(0)\in \operatorname{cl}(U)}T_\gamma(\epsilon).
\]
This verifies Definition \ref{defn:sept}.
\end{proof}
Unfortunately Proposition \ref{prop:simple} is rarely helpful.
One reason is that often it is convenient to make the critical
manifold repelling near slow exit points; see examples in
Section \ref{sec:appl}. The main reason is that in many important
cases critical manifolds of saddle-type appear
\cite{GuckenheimerKuehn2}. In fact, one of the most well-known
examples, the 3D FitzHugh-Nagumo equation, has two fast variables
and one slow variable with a critical/slow manifold of saddle
type \cite{GuckenheimerKuehn1,GuckenheimerKuehn3}.
\section{Periodic Orbit Exit Points}
In this section we shall not aim for the most general results but
show some characterizations of slow exit points in the case of
periodic orbits. We restrict to the case of fast-slow systems
in $\mathbb{R}^3$ with two fast variables; i.e.,
$(x,y)\in\mathbb{R}^2\times \mathbb{R}$ and
\begin{equation} \label{eq:basic3}
\begin{array}{lcl}
x_1'&=& f_1(x,y) ,\\
x_2'&=& f_2(x,y),\\
y'&=& \epsilon g(x,y).\\
\end{array}
\end{equation}
Let $\gamma_y(t)\in \mathbb{R}^2$ denote a periodic orbit for
the fast subsystem with period $T_y$ so that
\[
\gamma_y(0)=\gamma_y(T_y),\quad \gamma'_y(t)=f(\gamma_y(t),y).
\]
Let $N:=[-K,K]^2\times [-K,0]\subset \mathbb{R}^3$ for $K>0$
so that the following assumptions hold (see Figure \ref{fig:fig5}):
\begin{itemize}
\item [(B1)] There exists family of hyperbolic periodic orbits
$\{\gamma_y\}$ for $y\in[-\delta_0,\delta_0]$ for some
$\delta_0>0$ in the fast subsystem.
\item [(B2)] $\{\gamma_y\}_{y\in[-\delta_0,0]}\subset N$ and
$\gamma_0\subset\operatorname{int}([-K,K]^2\times\{0\})$.
\end{itemize}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.8\textwidth]{fig5}
\end{center}
\caption{\label{fig:fig5}Sketch of the situation near periodic
orbit $\gamma_0\subset \partial N$ in the fast subsystem.
The parabolic surface $L_0$ given by the zero set $l_0(x,y)=0$
is defined by rotating the given parabola along $\gamma_0$.}
\end{figure}
\begin{proposition} \label{prop:porbit}
Suppose {\rm (B1)-(B2)} hold and assume that
\begin{equation} \label{eq:avg_cond}
\frac{1}{T_0} \int_0^{T_0} g(\gamma_0(s),0)ds>0,
\end{equation}
where $g$ is as given in \eqref{eq:basic3}.
Then all points in $\gamma_0$ are slow exit points for $N$.
\end{proposition}
\begin{proof}
The proof is very similar to the argument for Theorem \ref{thm:eepts}.
Let $z_e=(x_e,0)\in \gamma_0$ be any point in the periodic
orbit contained in $\partial N$. Observe that
$\omega(x_e,\phi_0)=\gamma_0$ and let $K=\gamma_0=\mathcal{R}(K)$.
Let $U$ be an annular neighbourhood of $K$ contained in $N$,
for example we can set
\[
U=\big\{(x,y)\in\mathbb{R}^3:\min_{z_\gamma}\|z_\gamma-(x,y)^T\|_2 <\delta_1
\text{ for $z_\gamma\in\gamma_y$ with $y\in[-\delta_2,\delta_2]$}\big\}
\]
for $\delta_1,\delta_2>0$ sufficiently small. let
$\pi_0:\mathbb{R}^3\to \mathbb{R}^3$ denote the orthogonal projection
onto $\gamma_0$. Now define
\[
l(z)=l(x,y)=l_0(x,y):=y-\sum_{j=1}^2(x_j-\pi_0(z)_j)^2
\]
where the subscript $j$ indicates the $x_j$-coordinate of a point.
In the notation of Theorem \ref{thm:csept} we easily check that
(c) holds and we also observe that for (d) we have $G_0(z)\equiv 0$
since $\nabla_z l_0=(0,0,1)^T$ and $F_0=(f_1,f_2,0)$. We also find
that $G_1=(0,0,g(z))\cdot (0,0,1)^T$ and on $K$ we indeed have
\[
\frac{1}{T_0} \int_0^{T_0} g(\gamma_0(s),0)ds>0
\]
which verifies (d) and shows that $z_e$ is a slow exit point.
Noting that $z_e$ was arbitrary on $\gamma_0$ finishes the proof.
\end{proof}
Note that Proposition \ref{prop:porbit} has the rather obvious
interpretation that a point is a slow exit point for a periodic
orbit of the fast subsystem if it lies on the boundary of the
compact set $N$ and the average slow drift moves it outside of $N$.
It is more interesting to re-interpret the condition
\eqref{eq:avg_cond}.
We want to deal with families of periodic orbits in the fast subsystem.
For normally hyperbolic parts of the critical manifold we know that
there is a slow flow on a slow manifold that is $O(\epsilon)$-close.
Next, we recall an analog of the result for periodic orbits of the
fast subsystem. The idea is to find a flow that approximates
the flow on the family of periodic orbits. Consider the fast system
\begin{equation}
\label{eq:avg_fss}
\frac{dx}{dt}=x'=f(x,y)
\end{equation}
such that \eqref{eq:avg_fss} has a continuous family of periodic
orbits $\gamma_y(t)$ for each value of $y$ in some neighbourhood
$D_0$ of $y=y_0$ with period $T_y$ that is uniformly bounded so
that there are constants $T^a,T^b>0$ such that $T^a\leq T_y\leq T^b$.
For simplicity we shall also assume that each orbit $\gamma_y(t)$
is asymptotically stable with respect to the fast variables.
It seems plausible that the full fast-slow system should have
solutions $(x(\tau),y(\tau))$ such that the fast motion is
approximated by the family of rapid oscillating periodic orbits:
\[
x(\tau)\approx \gamma_y\left(\frac{\tau}{\epsilon}\right).
\]
Formally plugging this result into the slow equation yields
\[
\dot{y}=g(\gamma_y(\tau/\epsilon),y).
\]
The idea is that the slow motion on the family of periodic
orbits can be obtained by averaging out the fast oscillations.
Hence we might consider
\begin{equation} \label{eq:avg_int1}
\dot{Y}=\bar{g}(Y):=\frac{1}{T_Y}\int_0^{T_Y}g(\gamma_Y(t),Y)dt.
\end{equation}
It will be convenient to make a change of variable $t=T_Y\theta$
and to set $\Gamma_Y(\theta)=\gamma_Y(T(Y)\theta)$.
This transforms \eqref{eq:avg_int1} to
\begin{equation} \label{eq:avg_int2}
\dot{Y}=\int_0^1g(\Gamma_Y(\theta,Y))d\theta.
\end{equation}
Assume that the solution $Y(\tau)$ with initial condition
$Y(0)=Y_0$ stays inside $D_0$ for $0\leq \tau\leq \tau_1$.
Then a classical theorem shows that our averaging procedure
really produces the correct result with an error of order
$O(\epsilon)$.
\begin{theorem}[\cite{PontryaginRodygin,BerglundGentz}]
\label{thm:avg_fs}
Let $x_0$ be sufficiently close to $\Gamma_{Y_0}(\theta_0)$
for some $\theta_0$. Then there exists a function $\theta(\tau)$
that satisfies the differential equation
\[
\epsilon \dot{\theta}=\frac{1}{T_Y}+O(\epsilon).
\]
Furthermore, the following estimates hold
\begin{equation}
\begin{array}{lcl}
x(\tau)&=& \Gamma_Y(\theta(\tau))+O(\epsilon),\\
y(\tau)&=& Y(\tau)+O(\epsilon),\\
\end{array}
\end{equation}
for $O(\epsilon|\log \epsilon|)\leq \tau\leq \tau_1$.
\end{theorem}
Therefore, we observe that the averaged systems
\eqref{eq:avg_int1}-\eqref{eq:avg_int2} appear in the
condition \eqref{eq:avg_cond}. This means that points on the
periodic orbit contained in $\partial N$ are slow exit or
entry points if the averaged flow is transverse to $\partial N$;
we should view this averaged flow as a ``slow flow'' on the
family of fast periodic orbits. Hence we have provided an
analog for the condition of slow exit and entry points on
the critical manifold.
The next generalization step is now obvious. Consider a general
family of invariant sets (e.g. tori) in the fast susbsytem.
We can again average over the invariant measure of this family
in the case of periodic orbits; the transversality conditions
of this averaged flow will be exactly analogous to the previous cases.
In practical applications this scenario does not seem to be
needed very often as it does require three or more fast dimensions
or a family of fast subsystems with one additional free parameter
beyond the $y$-variables to be generic.
\section{Examples} \label{sec:appl}
\subsection{The Van der Pol Equation}
\label{sec:VdP}
We re-consider Example \ref{ex:VdPb}. The time reversed version
of Van der Pol's \cite{vanderPol,vanderPol_RO} equation is
\begin{equation} \label{eq:VdPsip}
\begin{array}{lcl}
x'= x^3/3-x-y,\\
y'= \epsilon x.\\
\end{array}
\end{equation}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=1\textwidth]{fig6}
\end{center}
\caption{\label{fig:VdP}Critical manifold $C_0$ (grey), singular
periodic orbit (black) and singular isolating neighbourhood
(dashed blue) for Van der Pol's equation \eqref{eq:VdPsip}.}
\end{figure}
The critical manifold $C_0$ of \eqref{eq:VdPsip} is given by
\[
C_0=\{(x,y)\in\mathbb{R}^2:y=x^3/3-x\}.
\]
Two fold points are located at $p_\pm=(\pm 1,\mp 2/3)$.
They naturally split the critical manifold into three parts
\[
C_l=C_0\cap \{x<-1\}, \quad C_m=C_0\cap \{-1\leq x \leq 1\},
\quad C_r=C_0\cap \{x>1\}.
\]
A singular periodic orbit for \eqref{eq:VdPsip} exists for
$\epsilon=0$ consisting of concatenations of solutions of the
fast and slow subsystems. We choose $N$ as a compact annulus
containing the singular periodic orbit as indicated in
Figure \ref{fig:VdP}. In this case, we have
\[
S=\operatorname{Inv}(N,\phi_0)=C_0\cap N.
\]
\begin{proposition} \label{prop:VdP}
$N$ is a singular isolating neighbourhood and $(N,\partial N)$
is a singular index pair.
\end{proposition}
\begin{proof}
Note that $\partial N \cap S$ consists of four points, two
on $C_m$ and one point each on $C_l$ and $C_r$.
Since (A1)-(A4) hold, Theorem \eqref{thm:eepts} implies that all
four points are slow exit points as the slow flow is transverse
at each point and pointing outwards with respect to $N$.
Therefore, Theorem \ref{thm:eepts} implies that $N$ is a
singular isolating neighbourhood. Next, we can apply
Theorem \ref{thm:sipmis} to see that $L=\partial N$ and
so $(N,L)$ is a singular index pair.
\end{proof}
A direct Conley index calculation for $(N,L)$, in combination
with the existence of a Poincar\'{e} section, can now be
used to show that for $\epsilon>0$ sufficiently the singular
periodic orbit perturbs to a periodic orbit of the full
system \eqref{eq:VdPsip}; see \cite{MischaikowMrozek} for
the detailed calculation.
\emph{Remark:} Proposition \ref{prop:VdP} is well-known
in Conley index and serves as one of the basic examples how
to apply the theory to show the existence of a non-trivial
invariant set in a dynamical system. We emphasize here that
with our characterization of slow exit points in Theorem
\ref{thm:eepts}, the problem has been reduced to the minimum
amount of work regarding the checking of theorems; our
transversality condition of the slow flow is much easier to
understand and check than the conditions of Conley's
Theorem \ref{thm:csept}.
\subsection{A Bursting Model}
We consider a modified Morris-Lecar model first proposed by Rinzel
and Ermentrout \cite{RinzelErmentrout}:
\begin{equation}
\label{eq:terman}
\begin{array}{lcl}
x_1'&=& y-0.5(x_1+0.5)-2x_2(x_1+0.7)-\\
& &0.5\left(1+\tanh\left(\frac{x_1+0.01}{0.15}\right)\right)(x_1-1),\\
x_2'&=& 1.15\left(0.5\left(1+\tanh\left(\frac{x_1-0.1}{0.145}\right)\right)-x_2\right)\cosh\left(\frac{x_1-0.1}{0.29}\right),\\
y'&=& \epsilon(k-x_1),\\
\end{array}
\end{equation}
where $k$ is a parameter and $0\leq \epsilon\ll 1$. We note
that \eqref{eq:terman} exhibits special periodic orbits that
are examples of bursting oscillations; see \cite{Izhikevich}
for more details. Terman \cite{Terman} and Guckenheimer and
Kuehn \cite{GuckenheimerKuehn2} investigated
\eqref{eq:terman} further focusing on the deformation of the
periodic orbits under parameter variation related to a phenomenon
called ``spike adding''. We shall not discuss these results
further but refer to the original references. The important
point in the current context is that the periodic orbits play
a key role in the dynamics. Our goal is to construct a singular
isolating neighbourhood for \eqref{eq:terman}.
In Figure \ref{fig:bif_diag} we show a bifurcation diagram for
the fast subsystem
\begin{equation} \label{eq:terman_fss}
\begin{array}{lcl}
x_1'&=& y-0.5(x_1+0.5)-2x_2(x_1+0.7)\\
&&-0.5\Big(1+\tanh\big(\frac{x_1+0.01}{0.15}\big)\Big)(x_1-1),\\
x_2'&=& 1.15\Big(0.5\Big(1
+\tanh\big(\frac{x_1-0.1}{0.145}\big)\Big)-x_2\Big)
\cosh\big(\frac{x_1-0.1}{0.29}\big),
\end{array}
\end{equation}
where we regard $y$ as a parameter. The diagram shows the
continuation of an equilibrium point which traces out a projection
of the critical manifold $C_0=\{x_1'=0=x_2'\}$. All continuation
calculations have been carried out using MatCont \cite{MatCont}.
The two fold points $p_{l,r}$ are fold (or saddle-node) bifurcations
of the fast subsystem and at these points normal hyperbolicity is lost.
They are located at
\begin{gather*}
p_l \approx (-0.0337,-0.0207,0.1365)=:(x_{1,l},x_{2,l},y_{l}),\\
p_r \approx (-0.2449,0.0832,0.0085)=:(x_{1,r},x_{2,r},y_{r}).
\end{gather*}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.9\textwidth]{fig7}
\end{center}
\caption{ \label{fig:bif_diag}Bifurcation diagram for the equilibria
of \eqref{eq:terman_fss}. Note that the S-shaped curve of equilibria
also represents the projection of the critical manifold into
$(x_2,y)$-space.}
\end{figure}
The normally hyperbolic parts of $C_0$ are separated by the fold
points into three branches
\[
C_b=C_0\cap\{yy_l\}
\]
representing the lower, middle and upper parts of the S-shaped curve
in Figure \ref{fig:bif_diag}. It is easy to check that $C_b$
is attracting and $C_m$ is of saddle-type. At $y_H\approx 0.075658$
we find a subcritical Hopf bifurcation of \eqref{eq:terman_fss}.
The upper part of the critical manifold $C_u$ is repelling for
$yy_H$. The unstable periodic orbits
generated in the Hopf bifurcation of \eqref{eq:terman_fss} undergo
further bifurcations as indicated in Figure \ref{fig:porbits}.
The bursting periodic orbits for the full system contain the
perturbation of a segment connecting $p_r$ to the stable
periodic orbits in the fast subsystem at $y=y_r$.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.97\textwidth]{fig8}
\end{center}
\caption{\label{fig:porbits}Continuation of periodic orbits generated
in the Hopf bifurcation for $y=y_H$. The smaller inner orbits
(in green) are unstable and the larger outer orbits (in blue)
are stable. Stability changes at a saddle-node of limit cycles
(LPC, limit point of cycles, in red). The thick middle curve
indicates part of $C_u$. Note that only a discrete number of
periodic orbits are shown from the numerical calculation.}
\end{figure}
We shall not discuss how the periodic orbits connect back to $C_b$
(see \cite{GuckenheimerKuehn2}) but note that a compact set $N$
that is a potential singular isolating neighbourhood can have fast
subsystem periodic orbits in $\partial N$. We shall focus on
\[
N:=\{(x_1,x_2,y)\in\mathbb{R}^3:(x_1,x_2)\in K,-0.04\leq y\leq 0.084\}
\]
for a suitably chosen compact rectangle $K\subset \mathbb{R}^2$ so
that $N$ contains $p_l$ and $p_r$. Note that this choice of $N$ is
only a first step in the analysis of bursting orbits and has to be
refined to prove their existence; see the technical problems
discussed in \cite{ConleyFastSlow1,ConleyFastSlow2}. The key point
is that $\partial N\cap \{y=0.084\}$ contains two periodic
orbits $\gamma_{1,2}$ for the fast subsystem.
Figure \ref{fig:integrals}(a) shows the two periodic orbits.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.9\textwidth]{fig9}
\end{center}
\caption{\label{fig:integrals} (a) Two periodic orbits in the
fast subsystem at $y=0.084$. (b) Integrals given in
\eqref{eq:int_terman} for the two periodic orbits depending on
the parameter $k$.}
\end{figure}
From Proposition \ref{prop:porbit} we deduce that points on
$\gamma_j$ for $j=1,2$ are slow exit/entry points depending on
the sign of the integral
\begin{equation} \label{eq:int_terman}
I_j:=\int_0^{T_j} g(\gamma_j(s),0)ds=\int_0^{T_j} k-(\gamma_j(s))_1ds
\end{equation}
where $T_j$ is the period of $\gamma_j$. Figure \ref{fig:integrals}(b)
shows the value of $I_j$ for different values of $k\in[-0.3,-0.1]$
which are typical values within which bursting periodic orbits
occur \cite{GuckenheimerKuehn2}. We find that in this parameter
range the outer cycle $\gamma_1$ consists of slow entry points
while the inner cycle $\gamma_2$ consists of slow exit points.
Note that $\partial N\cap \{y=0.084\}$ also contains a point
in $C_0$ at which the slow flow is transverse to $\partial N$.
Furthermore we can choose the compact rectangle $K$ in the
definition of $N$ so that there is only one more point in
$\partial N\cap C_0$ lying on $C_b$. At this point the slow flow
is again transverse to $\partial N$. By Theorem \ref{thm:eepts}
both points are slow exit or entrance points. Next, we apply
Theorem \ref{thm:sinbhd} to conclude that $N$ is a singular
isolating neighbourhood in this case.
\emph{Remark:} Note that it is much more complicated to construct
a singular index pair due to the presence of the Hopf bifurcation
point of the fast subsystem on $C_0$. We postpone this question
on how to modify $N$ to future work.
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\end{document}