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\title{General Mathematics and Computational Science I}
\author{Exercise 17}
\date{November 28, 2006}
\maketitle

\begin{enumerate}

\item On the previous homework, you have shown that the logistic map
\begin{equation*}
  x_{n+1} = \mu \, x_n \, (1-x_n)
\end{equation*}
has a $2$-cycle whenever $\mu>3$ with
\begin{gather*}
  a = \frac{1+\mu - \sqrt{(\mu-3)(\mu+1)}}{2 \, \mu} \,, \\
  b = \frac{1+\mu + \sqrt{(\mu-3)(\mu+1)}}{2 \, \mu} \,. \\
\end{gather*}
Use the result from Exercise 15, Question~3 to prove that this
$2$-cycle is asymptotically stable for $3 < \mu < 1 + \sqrt 6$ and
unstable for $\mu > 1 + \sqrt 6$.

\item Solve the so-called \emph{Pielou logistic equation}
\[
  x_{n+1} = \frac{\alpha \, x_n}{1 + \beta \, x_n} \,,
\]
find its equilibrium points and determine their stability.

\emph{Hint:} To solve the equation, substitute $x_n=1/y_n$.
\end{enumerate}
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