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\markboth{\hfil Nonautonomous attractors of skew-product flows \hfil
EJDE--2001/58}
{EJDE--2001/58\hfil R. A. Johnson \& P. E. Kloeden \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2001}(2001), No. 58, pp. 1--16. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
 Nonautonomous attractors of skew-product flows with  digitized driving systems
 %
\thanks{ {\em Mathematics Subject Classifications:} 34D35, 37B55.
\hfil\break\indent
{\em Key words:} pullback attractor, digitization, Bebutov flow.
\hfil\break\indent
\copyright 2001 Southwest Texas State University. \hfil\break\indent
Submitted July 05, 2001. Published August 29, 2001. 
\hfil\break\indent
Supported by the GNAFA and the DFG Forschungsschwerpunkt ``Ergodentheorie,
\hfil\break\indent Analysis und effiziente Simulation dynamischer Systeme"
} }
\date{}
%
\author{R. A. Johnson \& P. E. Kloeden}
\maketitle

\begin{abstract}
 The upper semicontinuity and continuity properties of
 pullback attractors for nonautonomous differential
 equations are investigated when the driving system of
 the generated skew-product flow is digitized.
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
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\section{Introduction}\label{intro}

The objective of  this paper is to study the semicontinuity and
continuity properties of pullback attractors   for nonautonomous
differential equations
\begin{equation}\label{eq1}
 x' =  f(t,x), \quad x  \in \mathbb{R}^d, t \in \mathbb{R},
\end{equation}
under  perturbation of the driving system  through a digitization
procedure.  We will formulate  this problem in concrete terms
using the language  of skew-product  flows \cite{CKS,S,SY}, in
particular,   the Bebutov approach to the  skew-product flow
concept.  Thus  let $\mathcal{F}$ be some  topological  vector
space of mappings $f:\mathbb{R} \times \mathbb{R}^d \to
\mathbb{R}^d$ and, for each $t\in\mathbb{R}$, let
$\theta_t:\mathcal{F}\to\mathcal{F}$ be the translation operator
defined by $\theta_t(f)(s,x):= f(t+s,x)$  for all $s\in\mathbb{R}$
and $x\in\mathbb{R}^d$. Now let $P\subset \mathcal{F}$ be a
metrizable compact set which is invariant in the sense that, if
$p\in P$, then $\theta_t(p)\in P$  for all $t\in \mathbb{R}$.
Suppose further that the mapping $\mathbb{R} \times P\to P$
defined by $(t,p)\mapsto\theta_t(p)$ is continuous. This just says
that $(P,\{\theta_t : t \in \mathbb{R}\})$ is   a topological flow
or autonomous dynamical system.

Now  consider the family of differential equations
\begin{equation}\label{eq2}
 x' = p(t,x).
\end{equation}
We  assume,  for  each $x_0\in\mathbb{R}^d$ and $p \in P$, that
the solution  $x(t)=\phi(t,x_0,p)$ of
(\ref{eq2}) satisfying $x(0)=x_0$  exists locally and is
unique.  We also assume that, if $t\in\mathbb{R}$,
$x_0\in \mathbb{R}^d$ and $p\in P$, then the
correspondence $(t,x_0,p)\mapsto (\phi(t,x_0,p),\theta_t(p))$
is continuous on its domain of
definition $V\subset \mathbb{R} \times \mathbb{R}^d \times P$.
This correspondence $(t,x_0,p) \mapsto (\phi(t,x_0,p),\theta_t(p)):
V\to\mathbb{R}^d \times P$ is the skew product (local) flow on
$\mathbb{R}^d \times P$ defined by the family of equations (\ref{eq2}).
The flow $(P,\{\theta_t : t \in\mathbb{R} \})$   is referred to as the
\textbf{driving system} of equations  (\ref{eq2}).  For each
$p\in P$,  the solutions $x(t)$  of  (\ref{eq2})  can be
viewed as projections to $\mathbb{R}^d$  of trajectories of the
above skew-product flow.

When the equations (\ref{eq2})  satisfy an appropriate
dissipativity condition, they induce  a global pullback  attractor
${\bf A}=\bigcup_{p \in P} (A_p \times \{p\})
\subset\mathbb{R}^d\times P$ , where each fiber $A_p$ is
defined by  a procedure which  amounts to a nonlinear version of
the classical Weyl limit-point construction.  The set
${\bf A}$ is compact and is invariant with respect to the above
skew-product system on $\mathbb{R}^d \times P$ defined
by the solutions of the equations (\ref{eq2}).   For each
$p\in P$,  one can think of $A_p$ as a ``pointwise" pullback
attractor;   see Section \ref{prel}  for details.  The problem  we
pose is that of studying the semicontinuity, resp. continuity,
properties of ${\bf A}$  and  the individual fibers $A_p$, as the
compact translation-invariant  set $P$ is varied within
$\mathcal{F}$.  For technical simplicity, throughout the paper,  we
assume that the equations (\ref{eq2}) contract a fixed large ball in
$\mathbb{R}^d$. In this way the concepts of local pullback attractor
and global  pullback attractor are made equivalent.

This problem has been considered by Kloeden and Kozyakin
\cite{KK2001,KK2002}, who, in particular,  studied  the upper
semicontinuity in the Hausdorff sense of the $A_p$ under
perturbation in $P$ when the driving system $(P,\{\theta_t:t
\in\mathbb{R}\})$ has the shadowing property.  Here we will
not assume that shadowing holds.  Rather, we will instead study
perturbations  in certain spaces  $\mathcal{F}$ of a general
compact, metrizable, Bebutov-invariant  subset  $P$.  Among
these perturbations are those determined by
\textbf{digitizing}  a given time-varying  vector field  $f(t,x)$.
By this,   we mean the following:  the time-axis  is decomposed
into half-open intervals of lengths, say, between $\delta/2$ and
$\delta$  for each $\delta>0$.  On each such interval,  the
time-varying   vector field  $f(t,x)$ is  replaced by an autonomous
vector field$\bar{f}(x)$ (which usually depends on the particular
interval). For  example, one may choose $\bar{f}(x)$ to be the
time-average of $f(t,x)$ over the given (or previous)  interval, or
as some particular value $f(t_{*},x)$, or in some other way as
well. If now there is a compact, metrizable, Bebutov-invariant
subset  $P$ of $\mathcal{F}$ such that $f\in P$, then we can
identify (\ref{eq1}) as one equation in the family (\ref{eq2}). In
this way one is led to identify appropriate perturbations
$P_{\delta}\subset\mathcal{F}$ of $P$.

We will formulate and prove results to the effect that if
$${\bf A}^{\delta}=\bigcup\left(A_{p_{\delta}} \times
\{p_{\delta}\} :  p_{\delta} \in P_{\delta}\right\}$$ is the
pullback  attractor for the skew-product dynamical system
defined by  $(P_{\delta},\{\theta_t : t \in\mathbb{R} \})$,
and if $p_{\delta}\in P_{\delta}$ corresponds to a digitized
version of equation (\ref{eq1}), then $A^{\delta}_{p_{\delta}}$
converges  upper semicontinuously to $A_f$ as  $\delta \to 0$,
with respect to  the Hausdorff metric on compact subsets of
$\mathbb{R}^d$.  We will also provide sufficient conditions that
$A^{\delta}_{p_{\delta}}$ converges  continuously to $A_f$  as
$\delta\to0$,  with respect to the Hausdorff metric.

We will  make use of two distinct sets of methods in our study
study ${\bf A}$ and its  fibers $A_p$.  The first  is drawn from the
area of topological dynamics and dynamical systems. In applying
these methods, we will   make systematic use of the the
skew-product local flow on $\mathbb{R}^d \times P$ induced by
the solutions of equations (\ref{eq2}).   The second is taken from
the  fixed point theory  of nonlinear mappings on Banach spaces.
Fixed point theory can be applied here for the following reason
(among others):  the set $A_p$   is equal, under our hypotheses,
to the set of initial conditions $x_0\in\mathbb{R}^d$ for
which the corresponding solution $x(t)=\phi(t, p,x_0)$  of
(\ref{eq2})   exists   and is  bounded  for all $t\in \mathbb{R}$.
These solutions can be viewed as the   fixed points
of an appropriate nonlinear mapping.

In order to illustrate the points made above, we consider the
example
\begin{equation}\label{eq3}
x' = f(t,x) = - x + h(t,x),  \quad  x  \in \mathbb{R}, t \in
\mathbb{R},
\end{equation}
where $h$ is uniformly continuous and uniformly Lipschitz in
$x$ on each  subset of the form $\mathbb{R} \times K$ where
$K\subset \mathbb{R}^d$ is compact.  Assume that there
are  positive numbers  $a$ and   $\sigma$,   such that for  each
$t\in\mathbb{R}$ one has
$$
a+h(t,-a) \geq \sigma, \quad  -a+h(t,a) \leq - \sigma.
$$
Let $f_{\delta}(t,x)$ be obtained by digitizing $f$: thus, for
example,  we might  set
$$
f_{\delta}(t,x) = -x + \frac{1}{\delta} \int_{n\delta}^{(n+1)\delta}
h(s,x) ds, \quad  t \in [n\delta,(n+1)\delta)
$$
for each  $n\in\mathbb{Z}$.  Using methods of dynamical
systems, we will first show  that $A^{\delta}_{f_{\delta}}$ exists
and tends upper semicontinuously to $A_f$ as $\delta \to 0$,
and moreover that this convergence is uniform in $t$  when
$f$ is replaced by $\theta_t(f)$ for any $t\in\mathbb{R}$.

On the other hand, a bounded solution of (\ref{eq3}) is
expressible as a fixed point of the mapping $T$ defined for each
$t\in\mathbb{R}$ by
$$
T[x](t) = -  \int_t^{\infty} e^{(t-s)} f(s,x(s)) \, ds.
$$
This mapping is defined and  continuous  on the Banach space
$C_b$ of bounded, continuous real-valued functions  on
$\mathbb{R}$.  If  $|h(t,x)|\leq a$ whenever $|x|\leq a$ and
$t\in\mathbb{R}$, and if $\left|\frac{\partial
h}{\partial x}\right|\leq\alpha<1$ for all $t$ $\in$
$\mathbb{R}$ and $|x|\leq a$, then  $T$ is  a contraction
on  the ball $\{x(\cdot) \in C_b :\|x\|_{\infty} \leq a\}$.   One
has that $A_f=\{x_0 \in \mathbb{R} : \mbox{the solution
$x(\cdot)$ of  (\ref{eq3})  with $x(0)=x_0$ is bounded on
$(-\infty,\infty)$} \}$,  and that the set of solutions $x(\cdot)$ of
(\ref{eq3})  which are  bounded on $(-\infty,\infty)$ is the  fixed
point  set of $T$.  We will show that  the digitization gives rise to
continuous convergence in the Hausdorff sense
$A^{\delta}_{f_{\delta}}$ to $A_f$ as $\delta\to0$, and
moreover the convergence is uniform in $t$ when $f$ is replaced
by $\theta_t(f)$  for any $t\in\mathbb{R}$.

\section{Preliminaries} \label{prel}

In this section, we formulate  the concept of driving system in a
way  which is convenient for present purposes. We define the
notion of  pullback attractor when the driving system is compact
and when a  uniform dissipativity condition is valid.  Finally we
introduce various   basic definitions which will be needed later on.

Let $\mathcal{F}$ denote the vector space of all functions
(time-varying vector fields) $f:\mathbb{R}\times
\mathbb{R}^d\to\mathbb{R}^d$  which satisfy the
following properties:
\begin{enumerate}
\item[i)]  For each compact set $K\subset\mathbb{R}^d$,
$f$ is  uniformly continuous on $\mathbb{R} \times K$;

\item[ii)]  For each compact set $K\subset\mathbb{R}^d$, there
exists a constant $L_K$ (also depending on $f$)  so that
$$
\|f(t,x) - f(t,y)\| \leq L_K \, \|x-y\| \quad \mbox{for all   }  x,y \in
K, \ t \in \mathbb{R}.
$$
\end{enumerate}
The second condition states that $f$ is uniformly Lipschitz in $x$
on  each set of the form $\mathbb{R}\times  K$ where
$K\subset \mathbb{R}^d$ is compact.

It will be clear that all of our results can be formulated and
proved when the vector fields $f$ in question satisfy less stringent
conditions. However,  Conditions i) and ii) are not particularly
restrictive and  they permit a simple exposition of the facts we
wish to present.

We put the topology of uniform convergence on compact sets on
$\mathcal{F}$. Thus a sequence $\{f_n\}$ of elements of
$\mathcal{F}$ converges to $f\in\mathcal{F}$ if and only
if $f_n(t,x)\to f(t,x)$  uniformly on each set $D\times
K\subset\mathbb{R}\times\mathbb{R}^d$ when $D$ and $K$
are compact. This topology is metrizable, but not complete. There
is a natural flow (Bebutov flow)  $\{\theta_t: t \in \mathbb{R}
\}$ defined on $\mathcal{F}$ by  translation  of the $t$-variable,
specifically $\theta_t(f)(s,x)$ $:=$ $f(t+s,x)$  for all  $f$ $\in$
$\mathcal{F}$,  $x\in \mathbb{R}^d$ and $s$, $t$  in $\mathbb{R}$.

A simple but basic observation can now be made: if $f\in \mathcal{F}$
(i.e.,  if $f$ satisfies the conditions i) and ii)),
then the orbit closure $P:=\rm{cls} \{\theta_t(f): t \in
\mathbb{R} \}\subset \mathcal{F}$ is compact. This
follows from the uniform continuity conditions on $f$. Moreover,
condition ii) holds for each $p\in P$ with the same set of
Lipschitz constants $\{L_K\}$.  See \cite{S} for further details.

Consider the family of differential equations
\begin{equation}\label{eq4}
 x' = p(t,x),
\end{equation}
where $p$ ranges over $P$. If $x_0\in\mathbb{R}^d$ and
$p\in P$, then equation (\ref{eq4}) admits a unique
maximally defined solution   $x(t)=\phi(t,x_0,p)$ satisfying
$x(0)=x_0$. We define a local flow  $\{\pi_t: t \in
\mathbb{R} \}$ on $\mathbb{R}^d \times P$ by setting
$\pi_t(x_0,p)=(\phi(t,x_0,p),\theta_t(p))$ for all triples
$(t,x_0,p)$ such that the right-hand side is well-defined. This
local flow is said to be of skew-product type because the
component $\theta_t(p)$ does not depend on $x_0$. As noted in
the Introduction, the flow $(P,\{\theta_t: t \in
\mathbb{R}\})$  is referred to as the \textbf{driving system}
for the family (\ref{eq4}).

Of course,  a skew-product local flow on $\mathbb{R}^d \times P$
can be constructed for any compact, translation-invariant subset
of $\mathcal{F}$; it is not required that $P$ be generated as
above by the translates of a fixed vector field $f$ (when $P$
is so generated, it is referred to as the
\textbf{hull} of $f$).


Next we turn to the concept of pullback attractor. Let us begin
with a fixed vector field $f\in\mathcal{F}$ and the
corresponding  differential equation
\begin{equation}\label{eq5}
 x' = f(t,x).
\end{equation}
To simplify matters, we assume that there exists $R>0$
such that
\begin{equation}\label{eq6}
\left<f(t,x),x\right> < 0
\end{equation}
for all $t\in\mathbb{R}$ and $x$ $\in$  $\mathbb{R}^d$
satisfying $\|x\|$ $\geq$ $R$, where $\left<\cdot,\cdot\right>$
denotes the standard inner product on  $\mathbb{R}^d$. Using
the uniform continuity properties of $f$, one sees that condition
(\ref{eq6}) actually implies that, for some $\eta>0$,
$$
\left<f(t,x),x\right>  \leq - \eta
$$
for all $t\in\mathbb{R}$ and $x$ $\in$  $\mathbb{R}^d$
satisfying  $\|x\|=R$.  Passing to the hull $P$ of $f$, we
see that, for each $p\in P$, the closed  ball
$B_R=\{ x\in  \mathbb{R}^d:\|x\|\leq R\}$ is positively invariant
with respect to the solutions of the differential equation
(\ref{eq4}). In particular,   if $p\in P$ and  $x_0\in
B_R$, then the solution $x(t)$ of  (\ref{eq4})  satisfying
$x(0)=x_0$ exists and lies in $B_R$ for all $t\geq 0$.

Let us write $\phi(t,x_0,p)$ for the solution $x(t)$ of  (\ref{eq4})
satisfying $x(0)=x_0$. If $D\subset \mathbb{R}^d$,
we write $\phi(t,D,p)=\{\phi(t,x_0,p) : x_0 \in D\}$.
Define
\begin{equation}\label{eq7}
A_p = \bigcap_{t\geq 0} \phi\left(t,B_R,\theta_{-t}(p)\right),
\end{equation}
then put ${\bf A}=\bigcup_{p \in P} \left(A_p \times \{p\}
\right)\subset\mathbb{R}^d \times P$. We refer to $A_p$
as the pullback attractor for the single equation (\ref{eq4}), and
to ${\bf A}$ as the pullback attractor for the family (\ref{eq4}),
or more simply as  the global pullback attractor. Clearly, the sets
$A_p$ are all nonempty and compact, and $\phi$-invariant in
the sense that $\phi\left(t,A_p,p\right)=A_{\theta_t(p)}$.

We will consider the sets $A_p$ and ${\bf A}$ in more detail in
the following sections.  In the remainder of the present  section
we briefly recall  two basic definitions, namely those of
exponential dichotomy and of the Hausdorff distance.

Let $P$ be a compact, translation-invariant subset of
$\mathcal{F}$. Let  us suppose that $P$ consists entirely of
linear vector fields: $p(t,x)=P(t)x$, where we permit an
abuse
of notation. The function $P(\cdot)$ takes values in the set
$\mathcal{M}_d$ of $d\times d$ real matrices and is uniformly
continuous on $\mathbb{R}$. Let $\Psi_p(t)$ be the fundamental
matrix of  the linear ordinary differential equation
\begin{equation}\label{eq8}
 x' = P(t) x ;
\end{equation}
thus $\Psi_p(t)$ is the $d\times d$-matrix solution of (\ref{eq8})
such that  $\Psi_p(0)=I$, the $d\times d$  identity matrix.
Let $\mathcal{Q}$ be the set of linear projections
$Q: \mathbb{R}^d\to\mathbb{R}^d$;  this set has finitely
many connected components determined by the possible
dimensions of the image of $Q$.

\paragraph{Definition}
Say that the family (\ref{eq8}) has an \textbf{exponential
dichotomy} (ED) over $P$ if there are  positive constants
$\gamma$, $L$ and a continuous projection-valued function $Q$
defined by  $p \mapsto Q_p\in\mathcal{Q}$ such
that for all $p\in P$,
\begin{gather*}
\left\| \Psi_p(t) Q_p \Psi_p(s)^{-1} \right\|
 \leq  L \exp^{-\gamma (t-s)} \quad (t \geq s), \\
\left\| \Psi_p(t) (I-Q_p) \Psi_p(s)^{-1} \right\|
 \leq  L \exp^{\gamma (t-s)} \quad (t \leq s)\,.
\end{gather*}


Now let $(\mathcal{X},d)$ be a metric space and let $A$ and $B$
be nonempty compact subsets of $\mathcal{X}$. Define $\mathop{\rm
dist}(a,B):=\min_{b\in B} d(a,b)$ and then define the
Hausdorff  semi-distance
$$
H^{*}(A,B) := \max_{a\in A} \mathop{\rm dist}(a,B).
$$
Thus, if $\epsilon>0$ and $H^{*}(A,B)<\epsilon$,
then for each $a\in A$ there exists $b\in B$ such that
$d(a,b)<\epsilon$.  Finally,  define the Hausdorff distance
$H(A,B)$  between $A$ and $B$  as follows:
$$
H(A,B) := \max \left\{H^{*}(A,B), H^{*}(B,A)\right\}.
$$
Note that if $B$ happens to be a singleton, $B=\{b\}$,
then $H^{*}(A,B) =H(A,B)$.

\section{Semicontinuity results}\label{USC}

We begin by fixing a compact, translation-invariant subset  $P$
of the topological vector space $\mathcal{F}$ described in the
preceding section. We assume that there exist numbers $R>0$, $\eta>0$
so that, if $\|x\|=R$ and $p\in P$, then
\begin{equation}\label{eq9}
\left<p(t,x),x\right> \leq  - \eta.
\end{equation}
Let $B_R=\left\{x \in \mathbb{R}^d:\|x\| \leq R \right\}$
be the closed ball in $\mathbb{R}^d$ centered at the origin with
radius $R$.

We make some remarks about the pullback attractor $A_p$ and
the  global pullback attractor ${\bf A}$, first when $P$ is held
fixed and then when it is varied in some systematic way. First of
all, it follows from (\ref{eq9}) that, if $t>s$, then
$\phi\left(t,B_R,\theta_{-t}(p)\right) \subset
\phi\left(s,B_R,\theta_{-s}(p)\right)$. Thus the intersection in
(\ref{eq7}) is over a decreasing  collection of sets. Using the
continuity property of the reduced \v{C}ech homology functor
\v{H} in the category of compact spaces together with the fact
that each set  $\phi\left(t,B_R,\theta_{-t}(p)\right)$ is
homeomorphic to a ball, we have  \v{H}$(A_p)=0$. In
particular, we have:

\begin{proposition} \label{prop1}
For each $p\in P$, the space $A_p$ is connected; in fact
 $A_p$ is $\infty$-proximally connected in the sense of
\cite{Gorn}.
\end{proposition}

We record a second fact which also follows quickly from
(\ref{eq9})  and from the definition  of $A_p$.
\begin{proposition} \label{prop2}
For each $p\in P$,  one has $A_p=\{x_0 \in
\mathbb{R}$:   the solution $x(\cdot)$ of (\ref{eq2})
with $x(0)=x_0$  is defined on the entire real axis
and satisfies $\|x(t)\|\leq R$ for all $t$ in
$\mathbb{R}\}$.
\end{proposition}

\paragraph{Proof.} Let $x_0\in A_p$. Then for each $t<0$, there
exists $\bar{x}\in B_R$ such that
$\phi\left(t,\bar{x},\theta_{-t}(p)\right)=x_0$, and one
has $x(t)=\bar{x}$. It follows that $x(t)$ is defined and
satisfies  $\|x(t)\|\leq R$ for all $t\in\mathbb{R}$.
It is equally  easy to see that,   if the solution $x(t)$ of (\ref{eq2})
satisfying $x(0)=x_0$  exists and is bounded on
$\mathbb{R}$, then $x_0\in A_p$. \hfil$\Box$ \medskip

It follows from Proposition \ref{prop2} that
${\bf A}\subset \mathbb{R}^d \times P$ is compact, and from this one sees
that $\check{H}({\bf A})= \check{H}(P)$ because of the continuity of the
\v{C}ech homology functor on compact spaces.  One also sees
that, if $\mathcal{K}(\mathbb{R}^d)$ is the space of all nonempty
compact subsets of $\mathbb{R}^d$, then the mapping
$p\mapsto A_p:P\to\mathcal{K}(\mathbb{R}^d)$
is upper semicontinuous in the sense that (using the notation of
Section \ref{prel}):
$$
H^{*}\left(A_{p_n},A_p\right) \to 0  \quad \mbox{whenever}
\quad  p_n \to p   \mbox{ in }  P.
$$

Next let $f\in\mathcal{F}$ be a vector field satisfying
condition (\ref{eq6}). We want to consider the upper
semicontinuity properties of the pullback attractor $A_f$ when
$f$ is digitized.   It will be convenient and informative to study
the upper semicontinuity properties of the pullback attractor
$A_f$ using the language of skew-product flows.

First we introduce some terminology. By a \textbf{digitization}
we mean a procedure which, to each $f\in\mathcal{F}$ and
each real number $\delta>0$, assigns the following data
with the indicated properties:
\begin{enumerate}
\item[I)]  There is a collection $\mathcal{I}^{\delta}$ $=$
$\{I^{\delta}_j \ : \ j \in \mathbb{Z}\}$ of nonempty half-open
intervals in $\mathbb{R}$ such that $\cup_{j=-\infty}^{\infty}
I^{\delta}_j=\mathbb{R}$, and such that each interval
$I^{\delta}_j$ has length $\leq$ $\delta$ and (say) $\geq$
$\delta/2$.

\item[II)]  To each $f\in\mathcal{F}$ there is associated a
collection $\{f_{\delta}^j : \delta > 0, j \in \mathbb{Z}\}$ of
autonomous vector fields. There is a positive function
$\omega=\omega(\epsilon)$,  defined for positive  values of
$\epsilon$ and tending to zero as $\epsilon\to0+$, such that
for each interval $I^{\delta}_j \in \mathcal{I}^{\delta}$ and
each $x\in\mathbb{R}^d$ the  following property holds:
if $\epsilon_x=\sup \left\{ \|f(r,x) - f(s,x)\|
: r,s \in I^{\delta}_j \right\}$, then
$$
\left\|f_{\delta}^j (x) - f(t,x)\right\| \leq \omega(\epsilon_x),
\quad   t  \in I^{\delta}_j.
$$

\item[III)]  There is a positive function  $\omega_1=\omega_1(M)$,
which is   defined for positive  values of $M$
and which depends only on $M$, such  that, if $x$, $y$ in
$\mathbb{R}^d$ satisfy $\|f(t,x)-f(t,y)\|\leq M$ for all
$t$ in some interval $I^{\delta}_j$, then
$$
\left\|f_{\delta}^j (x) -f_{\delta}^j (y)\right\| \leq \omega_1(M)
\|x-y\|
$$
for all $\delta>0$.

\item[IV)]  There is a positive function  $\omega_2=\omega_2(\eta)$,
defined for positive  values of $\eta$ and tending
to zero as $\eta\to0+$, such that, if $J\subset\mathbb{R}$
is an interval and if $x\in\mathbb{R}^d$ is a point, and if $f$,
$\tilde{f}\in \mathcal{F}$ satisfy
$\|f(t,x)-\tilde{f}(t,x)\|\leq \eta$ for all $t\in J$, then
$$
\left\|f_{\delta}^j (x) -\tilde{f}_{\delta}^j (x)\right\| \leq
\omega_2(\eta)
$$
for all $\delta>0$ and all $j$ such that $I^{\delta}_j\subset J$.

Although these properties are cumbersome to state, they are
reasonable requirements on a digitization scheme. Now let
$f\in\mathcal{F}$ be a vector field satisfying (\ref{eq6}).
Put $f_{\delta}(t,x)=f_{\delta}^j(x)$ for $t\in I^{\delta}_j$,
$j\in\mathbb{Z}$. Abusing language
slightly, we call $\{f_{\delta}:\delta >0\}$ a digitization of $f$.
The vector fields $f_{\delta}$ discussed in the Introduction are
obtained by procedures  for which I)--IV) are satisfied, so these
$f_{\delta}$ are digitizations in our sense.  In fact , the subintervals
$\mathcal{I}^{\delta}$ in I) for each fixed $\delta>0$ of such
digitizations often also satisfies the following recurrence condition,
in which case it  will be called a \textbf{recurrent digitization}.

\item[V)]  Fix  $\delta>0$. To each  $\eta>0$ there corresponds
a number $T$ (which may depend on $\delta$  as well as on $\eta$) so that
each interval $[a,a+T] \subset\mathbb{R}$  contains a number
$s$ such that
$\mathop{\rm dist} (\mathcal{I}^{\delta},\mathcal{I}^{\delta}+s)<\eta$.
Here  $\mathcal{I}^{\delta}$ $+$ $s$ is the
$s$-translate of  $\mathcal{I}^{\delta}$ and  $\mathop{\rm dist}$ is the
Hausdorff distance on $\mathbb{R}$.
\end{enumerate}
%
Now consider the differential equation
\begin{equation}\label{eq10}
 x' =  f_{\delta}(t,x)
\end{equation}
for each $\delta>0$. Though $f_{\delta}$ is only piecewise
continuous in $t$, it  nevertheless admits a unique local solution
$x(t,x_0)$ for each initial condition $x(0,x_0)=x_0\in \mathbb{R}^d$;
moreover $x(t,x_0)$ is jointly continuous on its
domain of  definition. Using property II) and condition (\ref{eq6})
on $f$, we see that $f_{\delta}$ also satisfies condition (\ref{eq6})
for small $\delta>0$. It follows that the pullback attractor
$A_{f_{\delta}} \subset \mathbb{R}^d$ of  the equation
(\ref{eq10}), which is defined by the formula (\ref{eq7}), is
contained in $B_R$ and is compact for small $\delta>0$.

For each $\delta>0$ and $t\in\mathbb{R}$, let
$\left(\theta_t(f)\right)_{\delta}$ be the digitization of the
$t$-translate of $f$ (we remark parenthetically that
$\theta_t(f_{\delta})\neq\left(\theta_t(f)\right)_{\delta}$
in general). We want to prove that
$H^{*}\left(A_{\left(\theta_t(f)\right)_{\delta}},
A_{\theta_t(f)}\right)$ converges to zero as $\delta
\to 0$, uniformly in $t\in\mathbb{R}$. That is, we
want to prove that $A_{\left(\theta_t(f)\right)_{\delta}}$ tends
to $A_{\theta_t(f)}$ upper semicontinuously, uniformly in
$t\in\mathbb{R}$. Actually we will prove more. Let
$P\subset\mathcal{F}$ be the hull of $f$ and let $p_{\delta}$
be the digitization of $p$ for each $p\in P$; then
$H^{*}\left(A_{p_{\delta}},A_{p}\right)$ tends to zero as
$\delta\to0$, uniformly in $p\in P$.

To prove this, it will be convenient to work in an enlarged
topological vector space $\mathcal{G}$ which contains
$\mathcal{F}$ together with the (in general, temporally
discontinuous) vector fields $p_{\delta}$. We define
$\mathcal{G}$  to be the class of jointly  Lebesgue measurable
mappings $g\in\mathbb{R} \times \mathbb{R}^d\to
\mathbb{R}^d$ which satisfy the following conditions:

\begin{enumerate}
\item[a)]  For each compact set $K\subset\mathbb{R}^d$, one
has
$$
\sup_{x \in K} \sup_{t \in \mathbb{R} } \int_t^{t+1} \|g(s,x)\| \,
ds < \infty ;
$$

\item[b)] For each compact set $K\subset\mathbb{R}^d$  there
is  a constant $L_K$ (depending on $g$) so that, for almost all $t$
$\in$ $\mathbb{R}$:
$$
 \|g(t,x)-g(t,y)\|  \leq L_K \, \|x-y\|, \quad x,y \in K.
$$
\end{enumerate}
  Now, for each $r=1$, $2$, $3$, $\ldots$  and  each $N=1$,
$2$, $3$, $\ldots$  introduce a pseudo-metric $d_{r,N}$ on $\mathcal{G}$:
$$
d_{r,N}(g_1,g_2) = \sup_{\|x\| \leq r}
\int_{-N}^{N} \|g_1(s,x)-g_2(s,x)\| \, ds.
$$
Then put
$$
d_r(g_1,g_2) = \sum_{N=1}^{\infty} \frac{1}{2^N}
\frac{d_{r,N}(g_1,g_2)}{1+d_{r,N}(g_1,g_2)},
$$
and finally  set
$$
d(g_1,g_2) = \sum_{r=1}^{\infty} \frac{1}{2^r}
\frac{d_r(g_1,g_2)}{1+d_r(g_1,g_2)}.
$$
We identify two elements of  $\mathcal{G}$  if their $d$-distance
is zero, thereby obtaining a metric space which we also call
$\mathcal{G}$.

Observe that, if $g\in\mathcal{G}$, then the Cauchy
problem
\begin{equation}\label{eq11}
 x' =  g(t,x), \quad x(0) = x_0
\end{equation}
admits a unique, maximally-defined local solution $x(t,x_0)$
for each $x_0\in\mathbb{R}^d$;  moreover,  $x(t,x_0)$
depends continuously on $(t,x_0)$  on its domain of definition.
This can be proved using the standard Picard iteration method to
solve (\ref{eq11}). We will write
$$
x(t,x_0) = \phi(t,x_0,g)
$$
to maintain consistency with notation used previously.

Observe further that, for each  $t\in\mathbb{R}$, the
translation $\theta_t:\mathcal{G}\to\mathcal{G}$,
i.e., $\theta_t(g)(s,x)=g(t+s,x)$ is well-defined.   Let
$\mathcal{G}_1\subset\mathcal{G}$  be a translation
invariant subset such that the supremum in a) and the constants
$L_K$ in b) of the definition of $\mathcal{G}$ are uniform in $g$ $\in$
$\mathcal{G}_1$, for each compact  $K\subset\mathbb{R}^d$.
Then $(t,g)\to\theta_t(g):
\mathbb{R} \times  \mathcal{G}_1\to\mathcal{G}_1$ is
continuous.

Next let $\delta_0>0$.  We will show that the set
$\mathcal{U}=P \cup \{ p_{\delta}:p \in P, 0 < \delta
\leq\delta_0\}\subset \mathcal{G}$  is equi-uniformly
continuous in the sense that, to each $\epsilon>0$, there
corresponds $\eta>0$ such that, if $|t-s|<\eta$, then
$d(\theta_t(p),\theta_s(p))<2\epsilon$ and
$d(\theta_t(p_{\delta}),\theta_s(p_{\delta}))<2\epsilon$ for all $p$,
$p_{\delta}\in\mathcal{U}$
and for all $t$, $s\in\mathbb{R}$.

To do this,  fix $\epsilon>0$. Recall that
$P\subset\mathcal{F}\subset\mathcal{G}$  is the hull of the
uniformly continuous function $f$. Hence if $B\subset\mathbb{R}^d$ is a
ball centered at the origin and if $N\geq1$,  then we can  find $\eta_1>0$
such that, if $|t-s|<\eta_1$, then
$$
\int_{-N}^{N} \|\theta_t(p)(v,x)-\theta_s(p)(v,x)\| \, dv
< \epsilon/3
$$
for all $x\in B$. Then, taking account of the definition of
the distance $d$, we see that it is sufficient to prove that, for
some sufficiently large ball $B\subset\mathbb{R}^d$
and some sufficiently large $N$,  there exists $\eta_2\in(0,\eta_1]$
such that
\begin{equation}\label{eq12}
\sup_{x\in B} \int_{-N}^{N}
\|\theta_t(p_{\delta})(v,x)-\theta_s(p_{\delta})(v,x)\|
\,dv <  \epsilon
\end{equation}
whenever $|t-s|<\eta_2$, $0<\delta\leq \delta_0$. Let us write
$d_B(\theta_t(p_{\delta}),\theta_s(p_{\delta}))$ for the quantity on the
left hand side of  (\ref{eq12}).

To prove (\ref{eq12}), we use the properties I)--IV) of a  recurrent
digitization. Choose $\epsilon_1>0$ so that
$\omega(\epsilon_1)<\epsilon/3$, then choose
$\delta_1$ so that, if $0<\delta\leq \delta_1$
then $\epsilon_x\leq \epsilon_1/(2N)$  for all $x\in B$.
Using property II), we see that, if $0<\delta\leq \delta_1$ and $p\in P$, then
\begin{eqnarray*}
d_B(\theta_t(p_{\delta}),\theta_s(p_{\delta}))
& \leq & d_B(\theta_t(p_{\delta}),\theta_t(p)) +
d_B(\theta_t(p),\theta_s(p)) +
d_B(\theta_s(p),\theta_s(p_{\delta}))
\\
& < & 3 \cdot \epsilon/3 = \epsilon
\end{eqnarray*}
whenever $|t-s|<\eta$.

If  $\delta_1\geq\delta_0$, we set $\eta=\eta_1$
and stop. If $\delta_1<\delta_0$ and if $\delta_1<\delta\leq \delta_0$,
we first choose $\eta_2\leq \delta/100$, then note that on each interval
$[u,u+N]\subset\mathbb{R}$ of length $N$, the difference
$p_{\delta}(t,x)-p_{\delta}(s,x)$ is zero except on at most
$\left[2N/\delta\right]+1$ subintervals of length $2\eta_2$,
where $[\cdot]$ denotes the integer part of a positive number.
Using the uniform boundedness of the vector fields $p_{\delta}$
$\in$ $\mathcal{U}$ on $\mathbb{R}\times B$, we can determine
$\eta_3\leq \eta_2$ so that, if $|t-s|<\eta_3$, then
$d_B(\theta_t(p_{\delta}),\theta_s(p_{\delta}))<\epsilon$.
So if $\eta=\min \{\eta_1,\eta_3\}$ we obtain  (\ref{eq12})
for all $p$,  $p_{\delta}\in\mathcal{U}$.

Now let  $\delta\in(0,\delta_0]$. For each $p\in P$,
let $P_{\delta}(p)=\rm{cls} \{\theta_t(p_{\delta}): t \in
\mathbb{R} \}$. Then $P_{\delta}(p)$ is compact (this uses the
recurrence  condition  V) of a recurrent digitization) and
translation invariant in $\mathcal{G}$. Moreover, using
property III) of a digitization and a Gronwall-type argument, one
shows that the map  $(t,x_0,g)$ $\mapsto$ $(\phi(t,x_0,g),
\theta_t(g))$  defines a (continuous)  skew-product flow on
$\mathbb{R}^d \times P_{\delta}(p)$.

Choose $\delta_0$ so that each $p_{\delta}$ satisfies (\ref{eq6})
for all $p\in P$ and $0<\delta\leq \delta_0$.
Then the pullback attractor $A_{p_{\delta}}$ exists and equals
$\{x_0 \in \mathbb{R}^d$ $:$  $\phi(t,x_0,p_{\delta})$ is defined
on all of $\mathbb{R}$  and satisfies $\|\phi(t,x_0,p_{\delta})\|\leq R$ $\}$;
see Proposition  \ref{prop2}. In fact,
$A_{p_{\delta}}$ is then the $p_{\delta}$-fiber of a global
pullback attractor ${\bf A}^{\delta}\subset\mathbb{R}^d\times P_{\delta}(p)$.
Now $p_{\delta} \to p$ in $\mathcal{G}$  as $\delta\to0$, so using property
III) of a digitization and a Gronwall argument, together with the
characterization of $A_{p_{\delta}}$ in terms of bounded
solutions of $x'=p_{\delta}(t,x)$, one shows that
$H^{*}(A_{p_{\delta}},A_p)\to0$ as $\delta\to0$.
However, more is true. Using property IV) of a digitization one
has the following: if $p_n\top\in P$ and if $\delta_n\to 0$, then
$d(p_{(n,\delta_n)},p)\to0$. Again using
II) together with a Gronwall argument and the above
characterization of $A_{p_{(n,\delta_n)}}$, one sees that
$H^{*}(A_{(p_{n,\delta_n)}},A_p)\to0$ as $n\to \infty$.
This is a strong uniformity statement, and implies

\begin{proposition}\label{prop3}
$H^{*}(A_{p_{\delta}},A_p)\to0$ as $\delta\to0$,
uniformly in $p\in P$. In particular,
$$
H^{*}(A_{(\theta_t(f)_{\delta}},A_{\theta_t(f)}) \to 0 \quad
\mbox{as }  \delta \to 0,
$$
uniformly in $t\in\mathbb{R}$.
\end{proposition}

We use the recurrence condition V) to prove that the sets $P_{\delta}(p)$
are compact.  This would seem to be a basic requirement to be satisfied
when one  sets about computing the pullback attractor,  because otherwise
the  convergence in (\ref{eq7}) of the intersection to $A_{p_{\delta}}$
cannot be expected to have any uniformity properties.  We note, however,
that Proposition \ref{prop3} could be proved without asumptions ensuring
that the sets $P_{\delta}(p)$  are compact; one only needs the uniform
continuity in $t$ on $\mathbb{R}$ (uniform on compact $x$  subsets of
$\mathbb{R}^d$) of the vector field $f(t,x)$ and the continuity of the Bebutov
flow on $(\mathcal{G},d)$.

\section{Continuity results}

In this section, we continue to investigate  the perturbation
properties of pullback attractors, this time with the goal of
giving a sufficient condition for the Hausdorff continuity (and
not just upper semicontinuity) of the sets $A_p$, resp. ${\bf A}$,
as the base space $P$ is varied in some functional space.

To be specific, let $\mathcal{F}$ be the topological space
introduced in Section \ref{prel}.  Let $P$ be a compact,
translation-invariant subset of $\mathcal{F}$ (which need not
be the hull of any one element $f\in\mathcal{F}$). Let us
assume that each $p\in P$ can be written in the form
$$
p(t,x) = L_p(t) x + h_p(t,x),
$$
where $L_p(\cdot)$ is a uniformly continuous function with
values
in the set $\mathcal{M}_d$ of real $d\times d$ matrices. We
assume that the mappings $(p,t,x)\mapsto L_p(t) x$ and
$(p,t,x)\mapsto h_p(t,x)$  are uniformly continuous on
compact subsets of $P\times \mathbb{R} \times \mathbb{R}^d$.
A sufficient condition  that this is the case is the following.
Consider the metric space  $P$; put $F(p,x)=p(0,x)$ for
each $p\in P$ and $x\in\mathbb{R}^d$. Note that
$F(\theta_t(p),x)=p(t,x)$ for all $t\in\mathbb{R}$,
$p\in P$. Suppose that the Jacobian $\frac{\partial
F}{\partial x}(p,0)$ is continuous as a function of $p$.   Then,
setting
$$
L_p(t) x = \frac{\partial F}{\partial x}(\theta_t(p),0) x, \quad
h_p(t,x) = p(t,x) - L_p(t) x,
$$
we obtain  the desired decomposition.

We now impose the following hypothesys.
\begin{enumerate}
\item[(H1)] The family of linear systems
$$
x' = L_p(t) x, \quad p \in P,
$$
admits an exponential dichotomy over  $P$ with
constants  $L>0$, $\gamma>0$  and
continuous family  of projections   $\{Q_p:p \in P\}$.
\end{enumerate}

Let $C_b=C_b(\mathbb{R},\mathbb{R}^d)$ be  the Banach
space of bounded continuous functions
$x:\mathbb{R}\to \mathbb{R}^d$ with the norm
$\|x\|_{\infty}=\sup_{t \in \mathbb{R}}\|x(t)\|$.
For each  $p\in P$,
define a nonlinear operator $T_p:C_b\to C_b$ as
follows:
\begin{eqnarray*}
T_p[x](t) &=& \int_t^{\infty} \Psi_p(t) Q_p \Psi_p(s)^{-1} h_p(s,x(s))
\, ds \\
&&+ \int_{-\infty}^t \Psi_p(t) (Q_p-I) \Psi_p(s)^{-1} h_p(s,x(s)),
\, ds
\end{eqnarray*}
where $\Psi_p(t)$ is the fundamental matrix with initial value
$\Psi_p(0)=I$ (identity matrix)  of the linear equation
$x'=L_p(t) x$.

Assume from now on that condition (\ref{eq6}) holds for all
$p\in P$. We further impose a condition of uniform
contractivity.
\begin{enumerate}
\item[(H2)] For all  $p\in P$  and for all   $x$, $y$ in $B_R$,
one has
$$
\left\|h_p(t,x) -h_p(t,x) \right\| \leq k \|x-y\|,
$$
where  $k<{\gamma}/{2L}$.
\end{enumerate}
To simplify the analysis, we now modify each $h_p(t,x)$ outside
the ball $B_R$ so that $h_p$ satisfies Hypothesis  (H2) for all
$x \in \mathbb{R}^d$ and so that $h_p(t,x)=0$ whenever
$\|x\|\geq R+1$.  This means that condition  (\ref{eq6}) does
not hold if $\|x\|$ $\geq$ $R+1$, but it will clear that this will have
no effect on our analysis of the pullback attractors of the equations
(\ref{eq13}).

\begin{proposition} \label{prop4}  For each $p\in P$,  the
equation
\begin{equation}\label{eq13}
 x' = p(t,x),
\end{equation}
admits a unique solution $x_p(t)$ which is bounded on all of
$\mathbb{R}$.
\end{proposition}

\paragraph{Proof.}   The argument is standard (see, e.g., Fink \cite{Fink}).
The operator $T_p$ is a  contraction on $C_b$ and hence admits
a unique fixed point $x_p(\cdot)$, which is a bounded solution
of (\ref{eq13}). Since each  fixed point of $T_p$ in $C_b$ is a
bounded  solution of (\ref{eq13}) the proposition is proved.
\hfil$\Box$ \smallskip

Using Propositions \ref{prop2} and \ref{prop4}, we see that, for
each $p\in P$, the pullback attractor $A_p$ $=$
$\{x_p(0)\}$, i.e.,  each $A_p$ contains exactly one point. Then
from continuity with respect to parameters of the fixed point of
a contractive mapping, we see that  ${\bf A}=\{(x_p(0),p):
p\in P\}\subset\mathbb{R}^d \times P$ is compact. One
verifies that ${\bf A}\subset\mathbb{R}^d \times P$  is the
global pullback attractor for the family (\ref{eq13}).

Now, if $\{x_0\}$ is a singleton subset of a metric space
$\mathcal{X}$, and if $B\subset\mathcal{X}$ is compact,
then $H^{*}(B,\{x_0\})$ coincides with the Hausdorff
distance $H(B,\{x_0\})$. This fact will allow us to prove that,
if $p\in P$, then $A_p$ is a point of Hausdorff continuity
for   the pullback attractors $A_{\tilde{p}}$ of equations
$x'=\tilde{p}(t,x)$ obtained by appropriate perturbations
$\tilde{p}$ of $p$.  We will formulate a fairly general continuity
result whose hypotheses are satisfied in particular by the
digitizations of Section \ref{USC}.

We view the compact metric space $P$ as a subset  of
$\mathcal{G}$. Choose fixed values  for the suprema in a) and
for the constants $L_K$ in b) of the definition of $\mathcal{G}$,
and let $\mathcal{G}_1\subset\mathcal{G}$ be the set of
all $g\in\mathcal{G}$ for which a) and b) hold with these
fixed values. As in Section \ref{USC}, write $\theta_t(g)(s,x)=
g(t+s,x)$, and  let $\phi(t,x_0,g)$ denote the solution of the
Cauchy problem  (\ref{eq11}) for each $g$ $\in$
$\mathcal{G}_1$. Using a Gronwall-type inequality, one verifies
that the mapping $(t,x_0,g)$ $\mapsto$ $(\phi(t,x_0,g),
\theta_t(g))$ is continuous on its domain of definition
$V\subset\mathbb{R} \times \mathbb{R}^d \times
\mathcal{G}_1$, and defines a continuous local skew-product
flow on $\mathbb{R}^d \times\mathcal{G}_1$.

Next let $\tilde{P}\in\mathcal{G}_1$ be a
\textbf{compact} translation-invariant set. The Hausdorff
semi-metric $H^{*}(\tilde{P},P)$ is defined relative to the metric
$d$ in $\mathcal{G}_1$. Let $\eta_{*}>0$ be a constant so
that
\begin{equation}\label{eq14}
 \left<p(t,x), x\right>  \leq - \eta_{*}
\end{equation}
for all $p\in P$, $t\in\mathbb{R}$, and
$x\in\mathbb{R}^d$ with $\|x\|=R$. One can show
that there exists $\delta>0$ so that, if $H^{*}(\tilde{P},P)<\delta$,
and if  $\tilde{p}\in\tilde{P}$, then for each
$x_0\in\mathbb{R}^d$ with $\|x_0\|=R$, the
solution $x(t)$ of the Cauchy problem
$$
x' = \tilde{p}(t,x), \quad x(0) = x_0,
$$
satisfies $\|x(t)\|<R$ for all $t>0$. The proof uses
the continuity of the local skew-product flow on $\mathbb{R}^d
\times\mathcal{G}_1$. This means that the family
$\{\mbox{(\ref{eq13})}: \ p \in P\}$  admits a global pullback
attractor $\tilde{{\bf A}}$, which lies in $B_R\times \tilde{P}$.


Since $A_p$  is a singleton set, we have
$H^{*}(A_p,A_{p_{\delta}})\leq H^{*}(A_{p_{\delta}},A_p)$,
whether the sets $A_{p_{\delta}}$
are singleton sets or not.  Thus in Proposition \ref{prop3} we actually
have continuous convergence in this case.

\begin{proposition}\label{prop5}
 For each  each $\delta>0$, let $\{p_{\delta}: p \in P\}$ be a
subset of  $\mathcal{G}$ such that $d(p_{\delta},p)\to 0$ as
$\delta\to 0$, uniformly with respect to $p\in P$.
Then $H\left(A_{p_{\delta}},A_p\right)\to0$ as $\delta \to 0$,
uniformly in $p\in P$.
\end{proposition}

Note that the equations $x'=p_{\delta}(t,x)$ need not give
rise to contractions in $C_b$, so this result cannot be proved
using the continuity properties of fixed points of contraction
mappings.

\section{Example}
We give an example to illustrate the strength of Proposition
\ref{prop5}. We will work with \textbf{quasi-periodic} vector
fields.   Let  $\mathbb{T}^k$  be the $k$-torus, $k\geq 2$,
and let $\gamma=(\gamma_1,\ldots,\gamma_k)$  be a
rationally independent vector in  $\mathbb{R}^k$. Let
$(\phi_1$, $\ldots$, $\phi_k)$ be angular coordinates
$\mod 2\pi$ on $\mathbb{T}^k$.  Introduce the corresponding
Kronecker flow $\{\theta_t:  t \in\mathbb{R} \}$  on
$\mathbb{T}^k$ by setting $\theta_t(\phi_1, \ldots, \phi_k)
=(\phi_1+\gamma_1 t , \ldots, \phi_k +\gamma_k t)$. For
brevity we set $\phi=(\phi_1, \ldots, \phi_k)$ and
 $\theta_t(\phi)=\phi+\gamma t$.

Let $F:\mathbb{T}^k\times \mathbb{R}^d   \to
\mathbb{R}^d$   be a  continuous function such that the
Jacobian $\frac{\partial F}{\partial x}(\phi,0)$ is defined and
is a continuous function of  $\phi\in\mathbb{T}^k$.   Let
$L_{\phi}(t)=\frac{\partial F}{\partial x}(\theta_t(\phi),0)$
and $h_{\phi}(t,x)=F(\theta_t(\phi),x)-L_{\phi}(t)x$.
Suppose that the family of linear equations
$$
x' = L_{\phi}(t) x, \quad \phi \in \mathbb{T}^k,
$$
admits an exponential dichotomy over $\mathbb{T}^k$ with
constants $L>0$, $\gamma > 0$ and a continuous
family of projections $\{Q_{\phi}:  \phi \in \mathbb{T}^k \}$.
Suppose further that there exists $R>0$ so that
$\left<F(\phi,x),x\right><0$ for all $\phi\in\mathbb{T}^k$
and $x \in \mathbb{R}^d$  with   $\|x\|\geq R$.
Suppose finally  that there is a constant $k<\gamma/(2L)$ such that
$$
\|h_{\phi}(t,x) - h_{\phi}(t,y)\| \leq  k  \|x-y\|
$$ for all $\phi\in\mathbb{T}^k$ and $x$, $y$ in
$\mathbb{R}^d$  with   $\|x-y\|\leq R$.

Let $G_n=G_n(\phi,x)$ be any sequence of continuous
functions on  $\mathbb{T}^k\times \mathbb{R}^d $  with values
in $\mathbb{R}^d$   such that
\begin{enumerate}

\item[1)]  $G_n\to0$ uniformly on $\mathbb{T}^k\times B_R$
as $n\to\infty$;

\item[2)] There is a real number $M$ such that
$$
\|G_n(\phi,x) - G_n(\phi,y)\|     \leq  M  \|x-y\|
$$
for all $\phi\in\mathbb{T}^k$ and $x$, $y\in B_R$
and $n$ $\geq$ $1$.
\end{enumerate}
We do not assume that $\frac{\partial G_n}{\partial x}(\phi,0)$
exists, nor that $M$ is small.  Hence, if we consider the vector
functions
$$
F_n(\phi,x) = F(\phi,x) + G_n(\phi,x),
$$
it may not be the case that the family of equations
\begin{equation}\label{eq15}
x' = F_n(\theta_t(\phi),x), \quad  \phi \in \mathbb{T}^k,
\end{equation}
generates a family of fixed-point mappings $\{T_{\phi}\}$, and
even if it does, there is no guarantee that any $T_{\phi}$ is a
contraction on $C_b$.

The $F_n$ are of course perturbations of $F$.  Let us now
introduce a further perturbative element. Namely, let
$\gamma^{(n)}=(\gamma^{(n)}_1,\ldots,\gamma^{(n)}_k)$ be a sequence of
frequency  vectors such that $\gamma^{(n)}  \to  \gamma$
in $\mathbb{R}^k$.   Of course, it is not assumed that the
$\gamma^{(n)}$   are rationally  independent.  Write
$\theta^{(n)}_t(\phi_1, \ldots, \phi_k)=
(\phi_1+\gamma^{(n)}_1 t , \ldots, \phi_k +\gamma^{(n)}_k t)$
for the corresponding Kronecker flow with $n=1,2,3,\ldots$.

Now let
\begin{gather*}
P  =    \left\{ p \in \mathcal{F}:  \ p(t,x) = F(\theta_t(\phi),x)
\mbox{ for some } \phi \in \mathbb{T}^k \right\}, \\
P^{(n)}  =    \left\{p \in \mathcal{F}:  \ p(t,x) =
F_n(\theta^{(n)}_t(\phi),x)
\mbox{ for some }  \phi \in \mathbb{T}^k \right\}, \quad
n \geq 1.
\end{gather*}
These are all compact translation-invariant subsets of
$\mathcal{F}$.  One can verify that $H(P^{(n)},P)\to 0$ as
$n\to\infty$;  this is true even though the frequency vector
has been perturbed. Since condition  (\ref{eq6}) is satisfied by the
family (\ref{eq15}) for all sufficiently large $n$, there are
pullback attractors ${\bf A}^{(n)}\subset \mathbb{R}^d\times P^{(n)}$
for each such $n$ defined by
the respective families of equations (\ref{eq15}). Using the
arguments preceding  Proposition  \ref{prop5}, one has that
 $H({\bf A}^{(n)},{\bf A})\to0$ as $n \to  \infty$,
where ${\bf A}$ is the pullback attractor defined by the equations
$$
x' = F(\theta_t(\phi),x), \quad  \phi \in \mathbb{T}^k.
$$
Let us now use this example to illustrate how information can be
obtained concerning  convergence of pullback attractors under
digitization.  Let the letters $F$, $F_n$, $G_n$, $\gamma^{(n)}$
have the significance attributed to them above. Suppose we are
given   a digitization scheme satisfying  the conditions I)--IV) of
Section \ref{USC}.

Let $P=\{ p \in \mathcal{G}:p(t,x)=F(\theta_t(\phi),x)$
 for some $\phi\in\mathbb{T}^k\}$;
thus $P$ is the same as before except that now  it is viewed as a
subset  of $\mathcal{G}$. Let $\{\delta_n\}$ be a sequence of
positive numbers which converges  to zero. For each
$\phi \in \mathbb{T}^k$, let $p_{\phi}^{(n)}$  be the
$\delta_n$-digitization of the time-varying vector field
$(t,x)\to F_n(\phi+\gamma^{(n)} t,x)$.

Now, for large $n$, each $p_{\phi}^{(n)}$  ($\phi$ in $\mathbb{T}^k$)
satisfies  condition (\ref{eq6}), so the pullback
attractor $A^{(n)}_{\phi}$ defined by the equation  $x'=
p_{\phi}^{(n)}(t,x)$   is contained in $B_R$.
Let ${\bf A}\subset\mathbb{R}^d\times P$  be the global pullback
attractor defined by the family of equations (\ref{eq2}). Each
$p\in P$ corresponds to (at least) one  point $\phi\in\mathbb{T}^k$.
Let us write  $p=p_{\phi}$ if $p$
corresponds to $\phi$, then write $A_{\phi}$ instead of $A_p$
for the fiber of ${\bf A}$ at $p$. Each fiber $A_{\phi}$  is a
singleton subset of $\mathbb{R}^d$.

Finally, arguing as in Section \ref{USC}, we can show that
$H(A^{(n)}_{\phi},A_{\phi})\to0$ as $n \to \infty$.
In fact, the convergence here is uniform  with respect  to $\phi
\in\mathbb{T}^k$.

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\noindent\textsc{R. A. Johnson}\\
Dipartimento di Sistemi e Informatica \\
Universit\`{a} di Firenze \\
I-50139 Firenze, Italy \\
e-mail: \texttt{johnson@ingfi1.ing.unifi.it} \medskip

\noindent\textsc{P. E. Kloeden}\\
Fachbereich Mathematik\\
 Johann Wolfgang Goethe Universit\"at \\
D-60054 Frankfurt am Main, Germany \\
e-mail: \texttt{kloeden@math.uni-frankfurt.de}

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