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\begin{document}

\title{Numerical Methods I}
\author{Lab Session 9}
\date{November 27, 2003}
\maketitle

\noindent
Solve the following systems of differential equations, for example by
using a fourth order explicit Runge--Kutta method,
\begin{align*}
  \k_1 & = \f(t_n, \y_n) \,, \\
  \k_2 & = \f(t_n + \tfrac12 h, \y_n + \tfrac12 h \, \k_1) \,, \\
  \k_3 & = \f(t_n + \tfrac12 h, \y_n + \tfrac12 h \, \k_2) \,, \\
  \k_4 & = \f(t_{n+1}, \y_n + h \, \k_3) \,, \\
  \y_{n+1} & = y_n
             + \tfrac16 h \, (\k_1 + 2\,\k_2 + 2\,\k_3 + \k_4) \,,  
\end{align*}
or any of the other methods from Homework~10.

\begin{enumerate}
\item The \emph{R\"ossler system},
\begin{align*}
    \dot x &= -y-z \\
    \dot y &= x + \tfrac15 \, y \\
    \dot z &= \tfrac15 + x \, z - \mu \, z
\end{align*}
with any initial condition.  Solve the system over a long interval of
time and plot the trajectory.  What happens if you increase $\mu$ from
$2$ to $6$?

Hint: You can produce a parametric 3D plot as follows:
\begin{verbatim}
gset parametric
gsplot y'
\end{verbatim}

\item The \emph{Brusselator equation},
\begin{align*}
  \dot x & = 1 - (b+1)\, x + a \, x^2 \, y \,, \\
  \dot y & = b \, x - a \, x^2 \, y \,.
\end{align*}
Again, pick your initial data as you like, and study the behavior as
$a=1$ and $b$ is increased from $1$ to $3$.
\end{enumerate}




\end{document}
