\section{$D_3$-Symmetric Mapping}
This is a mapping from the complex plane to itself which is symmetric
under $D_3$, the symmetry group of the triangle.  Let {\it Re} and
{\it Im} denote the operators which take the real and imaginary part of a
complex number.  Let $z=x+iy$ be complex and let 
\[ v = \alpha |z| + \lambda + \beta \mbox{\it Re}(z^3).
\]
  Then this map is defined by
\begin{eqnarray*}
	x &\to& v x+\gamma \mbox{\it Re}(z^2) \\
	y &\to& v y-\gamma \mbox{\it Im}(z^2)
\end{eqnarray*}

There is also an auxiliary function, $|z|$.  This system was studied
by M. Golubitsky and his colleagues in \cite{golubitsky1} and \cite{golubitsky2}.

The default parameter settings are such that the system has a global
attractor with three ``leaves.''  Orbits jump from leaf to leaf of the
attractor.  If, however, the parameter $\lambda$ is decrease (say, to
1.5) then the leaves separate and the system now has three attractors.

