%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\schrift{12pt}
\input format

\def\baselinestretch{1.1}

\begin{document}

\Blatt{3}{WS 2008/2009, 6.11.2008}
         {Theoretische Biophysik I}
         {Prof. J.~L.~van~Hemmen}

\pagestyle{empty}

\begin{deflist}{20.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\item[3.]
{\bf Logistische Abbildung II}
\par\smallskip
Betrachten Sie die logistische Abbildung
\begin{equation}
  x_{t+1} = f_r(x_t) := r x_t ( 1-x_t ) \quad\mbox{für $r \in ]0,4]$, $x_t \ge 0$}
\end{equation}
aus Aufgabe 1, Blatt 1.
\begin{deflist}{(0)}
\item[(a)] Erzeugen Sie numerisch das Bifurkationsdiagramm. Das
  Bifurkationsdiagramm ist die Vereinigungsmenge aller Grenzzyklen in
  Abhängigkeit des Parameters $r$.
\item[(b)] Zeigen Sie, dass die Abbildung für $r=4$ chaotisch ist.\\
Anleitung: Transformieren Sie $f_4(x_t)$ auf die Zeltabbildung
\begin{equation}
  x_{t+1} = \Delta(x_t) = \left\{
    \begin{array}{ll}
      2 x_t       & \mbox{für $0\le x_t < \frac{1}{2}$} \\
      2 ( 1- x_t) & \mbox{für $\frac{1}{2} \le x_t < 1$} 
    \end{array}\right.
\end{equation}
und zeigen Sie, dass diese den Ljapunow-Exponenten $\lambda = \ln 2$
hat. \\
Hinweis: Die benötigte Transformation lautet $h(x)=\frac{2}{\pi}
\arcsin \sqrt{x}$.
\end{deflist}




\par\smallskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{deflist}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vfill 

\footnotesize
\rule[.5mm]{\textwidth}{0.1mm}\\
Besprechung der Übungen am Donnerstag 13.11.2008 um 8.30 Uhr im
Raum PH 227. \\
Übungsleitung: Dr. Moritz Franosch, mail@Franosch.org,
http://www.t35.ph.tum.de.

\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
