\section{$D_4$ Symmetric Vector Field}

A dissipative vector field which is symmetric
under $D_4$, the symmetry group of the square, generated by
the transformations
\begin{eqnarray*}
	(x,y,z,w) &\to& (z,w,x,y) \\
	(x,y,z,w) &\to& (-x,-y,z,w).
\end{eqnarray*}
        The vector field on $R^4 = {(x,y,z,w)}$ is given by
\begin{eqnarray*}
                \dot{x} &=& y \\
                \dot{y} &=& x(\mu-(x^2 + z^2)) + \delta x z^2
                       + \epsilon( -y (x^2 + z^2) + \nu y
                       + A_{xzw} x(x y + z w) + A_{yz2} y z^2 \\
                \dot{z} &=& w \\
                \dot{w} &=& w(\mu-(x^2 + z^2)) + \delta z x^2
                       + \epsilon(-w (x^2 + z^2) + \nu w
                       + A_{xzw} z (x y + z w) + A_{yz2} w x^2
\end{eqnarray*}

For $\epsilon=0$ the system is Hamiltonian so we include two auxiliary
functions:
\begin{eqnarray*}
        \mbox{Energy} &=& (y^2 + w^2)/2 -  \mu(x^2+z^2)/2 + 
                    (x^2 + z^2)^2/4 - \delta x^2 z^2/2 \\
        \mbox{Angular~Momentum} &=& yz - xw
\end{eqnarray*}

        Note that in general the energy and angular momentum are not
preserved. The system presented here is the special case $a= -1, c=1$
given \cite{AGK}, to which the reader should refer for further details.

The file {\tt d41.dat} contains values of the parameters which produce a
chaotic attractor.  The file {\tt d42.dat} contains the same parameter
values, but also sets up a Poincar\'e map using the surface of section
$y=0$.
