diff options
Diffstat (limited to 'macros/latex/contrib/springer/svjour/cvs/example.tex')
-rw-r--r-- | macros/latex/contrib/springer/svjour/cvs/example.tex | 647 |
1 files changed, 647 insertions, 0 deletions
diff --git a/macros/latex/contrib/springer/svjour/cvs/example.tex b/macros/latex/contrib/springer/svjour/cvs/example.tex new file mode 100644 index 0000000000..e1a474c2ce --- /dev/null +++ b/macros/latex/contrib/springer/svjour/cvs/example.tex @@ -0,0 +1,647 @@ +\documentclass[cvs,envcountsect]{svjour} +\usepackage{amsfonts} +\usepackage{psfig} + +\begin{document} + + +\newcommand{\Ref}[1]{(\ref{#1})} +\newcommand{\R}{{\mathbb R}} +\newcommand{\N}{{\mathbb N}} +\newcommand{\cB}{{\cal B}} +\newcommand{\cM}{{\cal M}} +\newcommand{\diam}{\mathop{\rm diam}\nolimits} + +\renewcommand{\labelenumi}{(\roman{enumi})} + + +\title{An adaptive subdivision technique + for the approximation\\ of + attractors and invariant measures} + +\author{Michael Dellnitz\thanks{Research of the authors is partly supported + by the Deutsche Forschungsgemeinschaft under Grant De 448/5-2 + and by the Konrad-Zuse-Zentrum f\"ur Informationstechnik Berlin.}, + Oliver Junge} + +\institute{Mathematisches Institut, Universit\"at Bayreuth, + D-95440 Bayreuth, Germany\\ + {\scriptsize\tt + (http://www.uni-bayreuth.de/departments/math/\~{}mdellnitz; + http://www.uni-bayreuth.de/departments/math/\~{}ojunge)}} + +\date{Received: 13 January 1997 / Accepted: 29 September 1997 \\[1em] + Communicated by: P. Deuflhard} + +\maketitle + +\begin{abstract} +Recently subdivision techniques have been introduced in the +numerical investigation of the temporal behavior of +dynamical systems. In this article we intertwine the subdivision +process with the computation of invariant measures and propose an +adaptive scheme for the box refinement which is based on the +combination of these methods. Using this new algorithm the numerical +effort for the computation of box coverings is in general significantly +reduced, and we illustrate this fact by several numerical examples. +\end{abstract} + +\section{Introduction} +% +Recently subdivision methods have been successfully applied to the +numerical analysis of complex dynamical behavior +(e.g.\ \cite{Eiden,DH1,DH2,DJ:96,DJchua:97}). These methods can be used for two +essentially different purposes: the first is to +understand the geometric structure of an underlying attractor. +Secondly the goal may be to approximate the observable dynamical behavior +of the underlying system in a specific region of state space by +the computation of invariant measures. This paper +concerns the second possibility, and we propose an adaptive scheme +incorporated into the subdivision technique which allows to +reduce the numerical effort significantly. + +Obviously the dynamical behavior +just needs to be approximated on the support of a certain invariant measure. +Indeed, the idea for the adaptive principle stated here +is to intertwine the subdivision techniques with the computation of a +natural invariant measure, an {\em SBR-measure}, say. Roughly speaking, +the size of the covering boxes is reduced in those parts of +state space where the natural invariant measure $\mu$ is concentrated, and, +on the other hand, boxes are not subdivided in areas which have +$\mu$-measure zero. + +The main goal of this article is to illustrate the efficiency of +the new method by numerical examples. For that purpose we +consider several dynamical systems +for which the SBR-measure is known analytically since this allows us to +compare the numerical results obtained by the {\em adaptive\/} +subdivision algorithm to those obtained by the {\em standard\/} +subdivision procedure. The adaptive algorithm is essentially based +on the combination of two existing methods for which +convergence results are known. However, this fact does not +immediately imply convergence of the adaptive method as well. +Rather this theoretical but relevant problem is +currently under investigation, and the results will be published elsewhere. + +An outline of the paper is as follows: in Sect.\,\ref{sec:CSA} we +recall the standard subdivision technique from \cite{DH1}. The numerical method +for the approximation of SBR-measures is described in Sect.\,\ref{sec:fp}. +Then, in Sect.\,\ref{sec:ASA}, we present our adaptive subdivision technique, and +the efficiency of this method is illustrated by several examples in +Sect.\,\ref{sec:num}. + + +\section{The standard subdivision algorithm} +\label{sec:CSA} +% +The purpose is to approximate invariant sets of discrete +dynamical systems of the form +\[ +x_{j+1} = f(x_j),\quad j=0,1,\ldots, +\] +where $f$ is a continuous mapping on $\R^n$. +The central object which is approximated by the subdivision algorithm +developed in \cite{DH1} is the so-called {\em relative global attractor}, +\begin{equation} +\label{eq:relativAttractor2} +A_Q=\bigcap_{j\geq0}f^j(Q), +\end{equation} +where $Q\subset\R^n$ is a compact subset. Roughly speaking, +the set $A_Q$ should be viewed as the {\em union of unstable +manifolds of invariant objects inside $Q$}. In particular, +$A_Q$ may contain subsets of $Q$ which cannot be approximated +by direct simulation. + +The subdivision algorithm for the approximation of $A_Q$ +generates a sequence $\cB_0,\cB_1,\cB_2,\ldots$ of finite +collections of boxes with the property that for all +integers $k$ the set $Q_k=\bigcup_{B\in\cB_k}B$ +is a covering of the relative global attractor under consideration. +Moreover the sequence of coverings +is constructed in such a way that the diameter of the boxes, +\[ +\diam(\cB_k) = \max_{B\in\cB_k}\diam(B) +\] +converges to zero for $k\rightarrow\infty$. + +Given an initial collection $\cB_0$, one inductively obtains $\cB_k$ from +$\cB_{k-1}$ for $k=1,2,\ldots$ in two steps. +\begin{enumerate} +\item {\em Subdivision:} Construct a new collection $\hat\cB_k$ such that +\begin{eqnarray*} +\bigcup_{B\in\hat\cB_k}B &=& \bigcup_{B\in\cB_{k-1}}B \\ +\mbox{and}\quad +\diam(\hat\cB_k) &\leq& \theta\diam(\cB_{k-1}) +\end{eqnarray*} +for some $0<\theta<1$. +\item {\em Selection:} Define the new collection $\cB_k$ by +\[ +\cB_k=\left\{B\in\hat\cB_k \!:\! f^{-1}(B)\!\cap\!\hat B\!\ne\!\emptyset +\!\!\quad\mbox{for some}\!\!\quad\hat B\!\in\!\hat\cB_k\right\}. +\] +\end{enumerate} + +The following proposition establishes a general convergence property of +this algorithm. + +\begin{proposition}[\cite{DH1}] +\label{prop:sdconv} +Let $A_Q$ be the global attractor relative to the compact set $Q$, +and let $\cB_0$ be a finite collection of closed subsets with +$Q_0=Q$. Then +\[ +\lim\limits_{k\rightarrow\infty} +h\left( A_Q, Q_k \right)=0, +\] +where we denote by $h(B,C)$ the usual Hausdorff distance between two +compact subsets $B,C\subset\R^n$. +\end{proposition} + +\setcounter{equation}{0} +\section{Approximation of SBR-measures} +\label{sec:fp} +% +Recently it has been shown in \cite{DJ:96} how to compute numerically +approximations of an {\em SBR-measure\/} supported on a hyperbolic +invariant set. Since we want to use this method in our adaptive scheme +we now sketch the main ingredients of this algorithm. +To make the ideas more transparent we simplify the description +drastically by avoiding all technical details +concerning the underlying mathematical foundation in Ergodic Theory. + +The crucial observation is that the calculation of +invariant measures can be viewed as a fixed point problem. +Let ${\cal M}$ be the set of probability measures on $\R^n$. +Then $\mu\in{\cal M}$ is invariant if and +only if it is a fixed point of the {\em Frobenius-Perron operator\/} +$P:{\cal M}\to {\cal M}$, +\begin{equation} +\label{eq:fp-fixed} +(P\mu)(B)=\mu(f^{-1}(B))\quad\mbox{for all measurable $B\subset\R^n$}. +\end{equation} + +For a discretization of the operator $P:\cM\rightarrow\cM$ we +replace $\cM$ by a finite dimensional +set $\cM_k$: let $B_i\in\cB_k$, $i=1,\ldots,N$, +denote the boxes in the covering obtained after $k$ steps in the subdivision +algorithm and set as before $Q_k=\bigcup_{B\in\cB_k}B$. +We choose $\cM_k$ to be the set of ``discrete probability +measures'' on $\cB_k$, that is, +\[ +\cM_k = \left\{ u:\cB_k\to [0,1] \quad\Bigg| +\quad \sum_{i=1}^N u(B_i)=1\right\}. +\] +Assuming that $f(Q_k)\subset Q_k$ the discretized +Frobenius-Perron operator $P_k:\cM_k\rightarrow\cM_k$ is given by +\begin{eqnarray}\label{eq:FP} +v= P_k u,\quad v(B_i)=\sum_{j=1}^N +\frac{m(f^{-1}(B_i)\cap B_j)}{m(B_j)}u(B_j),\\ +\hfill i=1,\ldots,N,\nonumber +\end{eqnarray} +where $m$ denotes Lebesgue measure. Now a fixed point $u=P_k u$ of +$P_k$ provides an approximation to an invariant measure of $f$. +\begin{remark} +For the mathematically precise statement on the convergence of this +method one would have to introduce the concept of {\em small random +perturbations}. The reason is that this allows one to use a result of +Yu.\ Kifer on the convergence of invariant measures in the perturbed +systems to the SBR-measure (\cite{Kifer:86}). +However, the purpose of this article is to develop and to test an adaptive +scheme for the box refinement in the subdivision algorithm rather than to +explain the theoretical background concerning the computation of +SBR-measures. Therefore the +reader is referred to \cite{DJ:96} for the rigorous mathematical +treatment. +\end{remark} + + +\section{The adaptive subdivision algorithm} +\label{sec:ASA} +% +As mentioned above, the standard subdivision algorithm may approximate +a part of the global attractor which is dynamically irrelevant in the +sense that no invariant measure has support on this subset. +The reason is that {\em each\/} box is subdivided in a step of the +subdivision algorithm regardless of any information on the dynamical +behavior. In particular, also those subsets of the relative global +attractor corresponding to unstable or transient dynamical behavior +are approximated by the standard procedure. + +On the other hand, if one is mainly interested in the approximation of +the support of the (natural) invariant measure rather than in the precise +geometric structure of the global attractor then this strategy may lead to +unnecessary high storage and computation requirements. In the following +we present a modified subdivision strategy which avoids this drawback: +roughly speaking, +\begin{itemize} +\item[--] in the subdivision step we use +the information on the actual approximation of the +invariant measure to decide whether or not a box should be subdivided; +\item[--] in the selection step we keep only those boxes +which have a nonempty intersection with the support of the invariant measure. +\end{itemize} + +To be more precise, let $\{\delta_k\}$ be a sequence of positive +real numbers such that $\delta_k\to 0$ for $k\to\infty$. The algorithm +generates a sequence of pairs +\[ +(\cB_0, u_0),(\cB_1, u_1), (\cB_2, u_2), \ldots +\] +where the $\cB_k$'s are finite collections of compact subsets of +$\R^n$ and the discrete measures $u_k:\cB_k\to[0,1]$ can be interpreted +as approximations to the SBR-measure $\mu_{SBR}$: +\[ +u_k(B) \approx \mu_{SBR}(B)\quad\mbox{for all $B\in\cB_k$}. +\] + +Given an initial pair $(\cB_0, u_0)$, one inductively +obtains $(\cB_k, u_k)$ from $(\cB_{k-1}, u_{k-1})$ for +$k=1,2,\ldots$ in three steps: +\begin{enumerate} +\item {\em Subdivision:} Define +\begin{eqnarray*} +\cB_{k-1}^- &=&\{B\in\cB_{k-1}: u_{k-1}(B) < \delta_{k-1}\} \\ +\mbox{and}\qquad && \\ \cB_{k-1}^+ &=& \cB_{k-1}\backslash\cB_{k-1}^-. +\end{eqnarray*} +Construct a new (sub-)collection $\hat\cB_k^+$ such that +\[ +\bigcup_{B\in \hat\cB_k^+} B = \bigcup_{B\in \cB_{k-1}^+} B +\] +where +\[ +\diam(\hat\cB_k^+) \leq \theta\diam(\cB_{k-1}^+) +\] +for some $0 < \theta < 1$. +\item {\em Calculation of the invariant measure:} +Set +\[ +\hat \cB_k = \cB_{k-1}^-\cup\hat\cB_k^+. +\] +For the collection $\hat \cB_k$ calculate the approximating +invariant measure as the fixed point $\hat u_k$ of the discretized +Frobenius-Perron operator defined by \Ref{eq:FP}. +\item {\em Selection:} Set +\[ +\cB_k = \{ B\in \hat \cB_k: \hat u_k(B) > 0\} +\] +and +\[ +u_k = \hat u_k|_{\cB_k}. +\] +\end{enumerate} + +\begin{remark}\label{rmk:real} +\begin{itemize} +\item[(a)] In the realization of the algorithm we typically subdivide the +boxes in the collection $\cB_k^+$ by bisection. This guarantees that +the number of boxes is not growing too fast. For the details concerning the +implementation the reader is again referred to \cite{DH1,DJ:96}. +\item[(b)] In principle there is some freedom in choosing the sequence +$\{\delta_k\}$ of positive numbers used in the subdivision step. Note however +that this sequence determines the number of boxes which will be subdivided +and hence it has a significant influence on the storage requirement. +In the computations it turned out to be quite efficient to choose the +average +\[ +\delta_k = {1\over N_k}\sum_{B\in\cB_k} u_k(B) = {1\over N_k}, +\] +where $N_k$ is the number of boxes in $\cB_k$. +\item[(c)] In the numerical realization of the selection step (iii) +we check whether $\hat u_k(B) > \epsilon$ where $\epsilon>0$ is chosen +sufficiently small with respect to the machine precision. +\end{itemize} +\end{remark} + + +\section{Numerical examples} +\label{sec:num} +% +In this section we illustrate the efficiency of the adaptive scheme by +several numerical examples. First we consider three one-dimensional +mappings for which the SBR-mea\-sures are known analytically. For these +cases we +will see that, as expected, the new technique is particularly useful +if the underlying invariant density has singularities. Additionally we +consider the H\'{e}non map as a two-dimensional +example and show the box refinement produced +by the adaptive subdivision algorithm at a certain step. + +Before proceeding let us indicate some details concerning the implementation +of the adaptive subdivision algorithm: +\begin{itemize} +\item[(a)] The subdivision is always done by bisection and the boxes +are stored in a binary tree. This way we keep the storage requirement +at a low level. +\item[(b)] For the computation of the transition probabilities +\linebreak $m(f^{-1}(B_i)\cap B_j)$ in \Ref{eq:FP} +we use an exhaustion technique as described in \cite{GDK}. This method +is particularly useful when -- as in our examples -- local Lipschitz constants +are available for the underlying dynamical system. +\item[(c)] The computation of the discrete measures is done by an inverse +power method. In the solution of the corresponding linear systems +the fact is taken into account that the discretized +Frobenius-Perron operator is extremely sparse. +\end{itemize} +The adaptive algorithm is integrated into the C++ code {\sf GAIO} +({\bf G}lobal {\bf A}nalysis of {\bf I}nvariant {\bf O}bjects). +A link to a detailed description of {\sf GAIO} can be found on +the homepages of the authors. + + +\subsection*{Three one-dimensional examples} +\label{subsec:1D} +% +Motivated by the numerical investigations in \cite{DingDuLi:93} +we apply the adaptive subdivision algorithm to three different +one-dimensional dynamical systems on the interval $[0,1]$. In each +case we have chosen the initial collection $\cB_0=\{[0,1]\}$. + +\begin{itemize} +\item[{\it 1.}] +As a first example we consider the Logistic Map $f_1:[0,1]\to [0,1]$, +\[ +f_1(x)=\lambda x(1-x) +\] +for $\lambda=4$. The unique absolutely continuous invariant measure $\mu$ +of $f_1$ has the density +\[ +h_1(x)={1\over \pi \sqrt{x(1-x)}} +\] +(see e.g.\ \cite{Lasota:94}). Using the standard and the adaptive +algorithm we have approximated this density on several levels +and the results are shown in Table 1. We remark that even +the computation for $\ell=20$ only takes about 50 sec +on an MIPS R4400 cpu. + +\begin{table} \label{tab:log} +\caption{Comparison between the standard and the adaptive subdivision +algorithm for the Logistic Map. The minimal box volume in each row +is $2^{-\ell}$} +\begin{tabular}{ l l l l l } +\hline +$\ell$ & +\multicolumn{2}{c}{number of boxes} & \multicolumn{2}{c}{$L^1$-error} \\ + & standard & adaptive & standard & adaptive \\ \hline +6 & 64 & 17 & 0.1390 & 0.1431 \\ +8 & 256 & 34 & 0.0670 & 0.0765 \\ +12 & 4096 & 151 & 0.0210 & 0.0258 \\ +16 & 65536 & 679 & 0.0064 & 0.0073 \\ +20 & $2^{20}$ & 2831 & - & 0.0021 \\ +\hline +\end{tabular} +\end{table} + +In Fig.~\ref{fig:log} we illustrate the fact that the size of the boxes +is much smaller for those which are close to zero or one. Indeed, this is +what we expect since the density has singularities in these points. + +\begin{figure*} +\vbox{\hbox to\textwidth{\psfig{figure=cvs21.f1a,width=6cm}\hspace{5mm} +\psfig{figure=cvs21.f1b,width=6cm}\hfill \parbox[b]{50mm}{% +\caption[]{Illustration of the relation between the density and the + actual box refinement produced by the adaptive subdivision + algorithm for $\ell =10$: {\bf a} the density $h_1$; + {\bf b} the radii versus the midpoints of +boxes}\vspace{2mm}}}} +\label{fig:log} +\end{figure*} + +\item[{\it 2.}] We consider the map $f_2:[0,1]\to [0,1]$, +\renewcommand{\arraystretch}{2.1} +\[ +f_2(x)=\left\{ +\begin{array}{ll} +\displaystyle{2x\over 1-x^2} & \mbox{for}\quad 0\le x< \sqrt{2}-1, \\ +\displaystyle{1-x^2\over 2x} & \mbox{for}\quad \sqrt{2}-1 \le x \le 1. +\end{array} +\right. +\] +\renewcommand{\arraystretch}{1.0} +Its invariant density is +\[ +h_2(x)= {4\over \pi (1+x^2)}\; , +\] +see Fig.\,\ref{fig:tent}. + +\begin{figure*} +\vbox{\hbox to\textwidth{\psfig{figure=cvs21.f2a,width=6cm}\hspace{5mm}% +\psfig{figure=cvs21.f2b,width=6cm}\hfill +\parbox[b]{50mm}{% +\caption[]{{\bf a} The map $f_2$; and {\bf b} its invariant density + $h_2$}\vspace{1mm}}}} +\label{fig:tent} +\end{figure*} + +In Table \ref{tab:tent} we present the numerical results +for this case. As expected the application of the adaptive +subdivision algorithm is not more efficient than the +standard one since the invariant measure is quite close to Lebesgue +measure. + +\begin{table} +\caption[]{Comparison of the numerical results for $f_2$ + ($\ell$ as in Table~1)} +\label{tab:tent} +\begin{tabular}{ l l l l l } +\hline +$\ell$ & +\multicolumn{2}{c}{number of boxes} & \multicolumn{2}{c}{$L^1$-error} \\ + & standard & adaptive & standard & adaptive \\ \hline +6 & 64 & 45 & 0.0027 & 0.0053 \\ +8 & 256 & 187 & $6.5\cdot 10^{-4}$ & 0.0013 \\ +10 & 1024 & 759 & $1.7\cdot 10^{-4}$ & $3.2\cdot 10^{-4}$ \\ +12 & 4096 & 3047 & $4.6\cdot 10^{-5}$ & $7.8\cdot 10^{-5}$ \\ +\hline +\end{tabular} +\end{table} + +\item[{\it 3.}] Finally we consider the map $f_3:[0,1]\to [0,1]$, +\[ +f_3(x) = \left({1\over 8} -2 \left|x-{1\over 2}\right|^3\right)^{1\over 3} ++ {1\over 2}, +\] +with the invariant density +\[ +h_3(x) = 12\left(x-{1\over 2}\right)^2. +\] +The graphs of $f_3$ and $h_3$ are shown in Fig.\,\ref{fig:stahl}. + +\begin{figure*} +\vbox{\hbox to\textwidth{\psfig{figure=cvs21.f3a,width=6cm}\hspace{5mm}% +\psfig{figure=cvs21.f3b,width=6cm}\hfill +\parbox[b]{50mm}{% +\caption[]{{\bf a} The map $f_3$; and {\bf b} its invariant density + $h_3$}\vspace{2mm}}}} +\label{fig:stahl} +\end{figure*} + +\begin{figure*} +\vbox{\hbox to\textwidth{\psfig{figure=cvs21.f4a,height=5.3cm}\hspace{5mm}% +\psfig{figure=cvs21.f4b,height=5.3cm}\hfill}} +\caption[]{{\bf a} A tiling of the square $[-2,2]^2$ obtained by the + adaptive subdivision algorithm; and {\bf b} the subcollection + $\tilde{\cB}$ of boxes with discrete density bigger than + $0.35$ (see also \protect\Ref{eq:dm})} +\label{fig:henon} +\end{figure*} + +We now discuss the numerical results presented in Table~\ref{tab:stahl}. +Note that the derivative of $f_3$ has singularities at two points +inside $[0,1]$. This is the reason why several boxes get lost in +the realization of the selection step in the standard subdivision +algorithm. Consequently the computation of the invariant measure +does not lead to satisfying results. In contrast to this no boxes are lost +in the application of the adaptive subdivision technique, and accordingly +the $L^1$-error is decreasing with an increasing number of subdivision steps. + +\begin{table} +\caption[]{Comparison of the numerical results for $f_3$ + ($\ell$ as in Table~1)} +\label{tab:stahl} +\begin{tabular}{ l l l l l } +\hline +$\ell$ & +\multicolumn{2}{c}{number of boxes} & \multicolumn{2}{c}{$L^1$-error} \\ + & standard & adaptive & standard & adaptive \\ \hline +6 & 63 & 11 & 0.0260 & 0.2931 \\ +8 & 249 & 30 & 0.0105 & 0.2583 \\ +10 & 993 & 189 & 0.0110 & 0.0435 \\ +12 & 3967 & 816 & 0.0133 & 0.0065 \\ \hline +\end{tabular} +\end{table} +\end{itemize} + + +\subsection*{The H\'enon map} +\label{subsec:henon} +% +We apply the adaptive subdivision algorithm to a two-di\-men\-sio\-nal +example, namely a scaled version of the well known {\em H\'enon map} +\[ +f:\R^2\to \R^2,\quad +f(x,y) = (1-ax^2+y/5,\; 5bx). +\] +In the computations we have fixed the parameters by $a=1.2$, $b=0.2$, +and we have chosen $\cB_0=\{[-2,2]^2\}$. + +In Fig.\,\ref{fig:henon} we present a tiling of the square $[-2,2]^2$ +obtained by the adaptive subdivision algorithm after several subdivision +steps. The resulting box-collection $\cB$ consists of the +grey boxes shown in part (a) of this figure. +We expect that due to the numerical approximation some boxes have +positive {\em discrete\/} measure although they do not intersect the +support of the {\em real\/} natural invariant measure. +Having this in mind we neglect those boxes with very small discrete +measure and show in Fig.\,\ref{fig:henon}b a subcollection +$\tilde\cB\subset\cB$ with the property that +\begin{equation} +\label{eq:dm} +\sum_{B\in\tilde\cB} u(B) \approx 0.99 +\end{equation} +(see also Remark~\ref{rmk:real}(c)). + +\begin{figure} +\vbox{\hbox to\textwidth{\psfig{figure=cvs21.f5,width=8.6cm}\hfill}} +\caption[]{Illustration of the (natural) invariant measure for the + H\'enon map. The picture shows the density of the discrete + measure on $\tilde\cB$, see \protect\Ref{eq:dm}} +\label{fig:hen2} +\end{figure} + +\begin{remark} +For our choice of the parameter values we cannot explicitly +write down a natural invariant measure. Hence it is impossible +to compare our numerical results using analytical ones. +Moreover, it is not even known for an arbitrary choice of the parameter +values whether or not the H\'enon map possesses an SBR-measure. +However, recently it was proved by M.\ Benedicks and L.-S.\ Young that +the H\'enon map indeed has an SBR-measure for a +``large'' set of parameter values, see \cite{BenYoung}. +\end{remark} + + +Finally we apply the numerical techniques described in +\cite{DJ:96} to determine the essential dynamical behavior of the +H\'enon map for this choice of parameter values. An approximation +of the (natural) invariant measure obtained by the adaptive +subdivision algorithm is shown in Fig.\,\ref{fig:hen2}. This +computation is based on the total number of 1514 boxes inside the +square $[-2,2]^2$, whereas the support of the +invariant measure is covered by 1442 boxes. +The results indicate that the H\'enon map exhibits complicated dynamical +behavior. + +Moreover it can be shown that the areas +where the density is colored red resp.\ blue +are permuted cyclically by the mapping. Hence altogether +we may conclude that for these parameter values the H\'enon +map exhibits a two-cycle (the ``macro-dynamics'') in addition to +unpredictable (chaotic) behavior. This fact is also demonstrated +by a {\sf Java}-animation for which a link can be found on the +homepages of the authors. \thisbottomragged + + + +\begin{thebibliography}{10.} + +\bibitem{BenYoung} +M.\ Benedicks, L.-S.\ Young. +Sinai-\protect{Bowen}-\protect{Ruelle} measures for certain + \protect{Henon} maps. +{Invent.\ math.}, {\bf 112}:541--576, 1993 + +\bibitem{DH2} +M.\ Dellnitz, A.\ Hohmann. +The computation of unstable manifolds using subdivision and + continuation. +In H.W. Broer, S.A. van Gils, I.~Hoveijn, F.~Takens (eds), + {Nonlinear Dynamical Systems and Chaos}, pages 449--459. Birkh\"auser, + {PNLDE} {\bf 19}, 1996 + +\bibitem{DH1} +M.\ Dellnitz, A.\ Hohmann. +A subdivision algorithm for the computation of unstable manifolds and + global attractors. +{Numerische Mathematik}, {\bf 75}:293--317, 1997 + +\bibitem{DJ:96} +M.\ Dellnitz, O.\ Junge. +On the approximation of complicated dynamical behavior. +To appear in SIAM J. on Numerical Analysis, 1998 + +\bibitem{DJchua:97} +M.\ Dellnitz, O.\ Junge. +Almost invariant sets in \protect{Chua's} circuit. +To appear in {Int.\ J.\ Bif.\ and Chaos}, 1997 + +\bibitem{DingDuLi:93} +J.~Ding, Q.~Du,, T.~Y. Li. +High order approximation of the {Frobenius}-{Perron} operator. +{Appl.\ Math.\ Comp.}, {\bf 53}:151--171, 1993 + +\bibitem{Eiden} +M.\ Eidenschink. +{Exploring Global Dynamics: A Numerical Algorithm Based on the + Conley Index Theory}. +PhD Thesis, Georgia Institute of Technology, 1995 +\newpage + +\bibitem{GDK} +R.\ Guder, M.\ Dellnitz,, E.\ Kreuzer. +An adaptive method for the approximation of the generalized cell + mapping. +{Chaos, Solitons and Fractals}, {\bf 8}(4):525--534, 1997 +\bibitem{Kifer:86} +Yu.\ Kifer. +General random perturbations of hyperbolic and expanding + transformations. +{J.\ Analyse Math.}, {\bf 47}:111--150, 1986 + +\bibitem{Lasota:94} +A.\ Lasota, M.C.\ Mackey. +{Chaos, Fractals and Noise}. +Springer, 1994 + +\end{thebibliography} + +\end{document} |