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\title{General Mathematics and Computational Science I}
\author{Exercise 16}
\date{November 23, 2006}
\maketitle

\begin{enumerate}

\item The population of some species is modeled by
\begin{equation}
  x_{n+1} = \mu \, x_n \, \mathrm{e}^{-x_n} \,,
\end{equation}
with $\mu>0$.
\begin{enumerate}
\item Prove that $x^*=\ln \mu$ is the only equilibrium point. 
\item For which values of $\mu$ is the equilibrium point
asymptotically stable, for which values is it unstable?
\end{enumerate}

\item For a given continuously differentiable function $g$, consider
the difference equation
\begin{equation*}
x_{n+1} = x_n+h \, g(x_n) \,.
\end{equation*}
Determine conditions on $h$ for which an equqilibrium point is
asymptotically stable, respectively unstable.

\emph{Remark:} This difference equation is in fact the so-called Euler
method for solving the differential equation $\dot y = g(y)$.
However, this is not relevant for answering this question.

\item Show 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*}

\end{enumerate}
\end{document}
\item Find the general solution of
\begin{equation*}
x_n-x_{n-1}-6x_{n-2} = -36n
\end{equation*}

\textbf{Hint} First find the general solution $x_n^h$ of the
homogeneous equation
\begin{equation*}
x_n-x_{n-1}-6x_{n-2} = 0
\end{equation*}
using generating functions. Then find a particular solution $x_n^p$
of the original equation; try looking for solutions of the form
$an+b$. The general solution will then be $x_n =x_n^h+x_n^p$.

\item Consider the iterated map:
\begin{equation*}
x_{n+1} = rx_n(1-x_n^2)
\end{equation*}
Show that if $r<1$ then the only fixed point in $[0,1]$ is zero, and
that this is stable. When $r>1$ there is another fixed point in
$[0,1]$. Find the value of this fixed point (as a function or $r$)
and for which values of $r$ is it stable, respectively unstable.