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\Blatt{1}{WS 2008/2009, 24.10.2008}
         {Theoretische Biophysik I}
         {Prof. J.~L.~van~Hemmen}

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\item[1.]
{\bf Logistische Abbildung}
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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}
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\item[(a)] Bestimmen Sie die Fixpunkte von $f_r(x_t)$.
\item[(b)] Diskutieren Sie die Stabilität der Fixpunkte. 

  Hinweis: Betrachten Sie zur Stabilitätsanalyse die führenden Terme
  einer Taylor-Ent\-wick\-lung um den Fixpunkt und überlegen Sie, ob
  sich der Betrag einer kleinen Abweichung vom Fixpunkt mit der Zeit
  $t$ vergrößert oder verringert. In Grenzfällen ($r=1$, $r=3$) sind
  höhere Terme der Taylor-Entwicklung heranzuziehen. Warum macht $r=3$
  trotzdem Probleme? Ziehen Sie für $r=3$ auch die Ergebnisse von
  Aufgabe (c) zu Rate!
\item[(c)] Bestimmen Sie die Grenzzyklen der Länge 2 und diskutieren
  Sie ihre Stabilität.
\item[(d)] Berechnen Sie $x_t$, wobei sich die Startwerte $x_0$ in der
  Nähe der Fixpunkte oder Grenzzyklen befinden, numerisch für
  repräsentative Werte von $r$ und überprüfen Sie Ihre Aussagen
  bezüglich der Stabilität der Fixpunkte und Grenzzyklen anhand der
  numerischen Simulation.
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\footnotesize
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Besprechung der Übungen am Donnerstag 30.10.2008 um 8.30 Uhr im
Raum PH 227. \\
Übungsleitung: Dr. Moritz Franosch, mail@Franosch.org,
http://www.t35.ph.tum.de.

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