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\begin{document}

\title{Numerical Methods I}
\author{Lab Session 2}
\date{September 18, 2003}
\maketitle

\noindent
The sequence of simple iteratates of the \emph{logistic map} 
\begin{gather}
  g(x) = 1 - m \, x^2
\end{gather}
converges for some values of $m$, for others it displays periodic or
non-periodic, but bounded behavior.  Investigate the behavior of this
sequence as a function of $m \in [0,2]$ as follows.

\begin{enumerate}

\item For each value of $m$, compute a number of \emph{pre-iterations}
which you discard.

\item Store the next few iterations.

\item Plot $m$ vs.\ these stored values.

\item Repeat for different values of $m$.  How do you interpret the
final plot?

\end{enumerate}
\emph{Note:} This type of plot is called a bifurcation diagram.  For
more information, see the section on the logistic map in S\"uli and
Mayers, Chapter 1.

\end{document}

