\section{Lorenz System}

\noindent Introduced in 1963 by E. Lorenz \cite{lorenz1} as a simple model for
atmospheric convection, the three-dimensional system
\begin{eqnarray*}
    \dot{x} & = & \sigma (y - x) \\
    \dot{y} & = & r x - y - x z \\
    \dot{z} & = & xy - bz 
\end{eqnarray*}
persists as a standard illustration of how complicated dynamics may arise in
deterministic systems.   As the three parameters $(\sigma,r,b)$ are varied, the
behavior of the Lorenz system undergoes a variety of complicated changes       
including supercritial Hopf bifurcation, period doubling bifurcations and
the creation of a strange attractor through homoclinic `explosion'.  Moreover, 
the Lorenz system given above is symmetric with respect to the reflection
$(x,y,z) \rightarrow (-x,-y,z)$, and variations which break this symmetry property
are of current research interest.
%\cite{}.
Finally, sensitive dependence upon initial
conditions which serves as a working description of {\em chaos} is easily observed
in this system, a characteristic which is caused by the violent stretching and
folding of trajectories as they pass near the saddle point present at the origin
(for some parameter values).
See the treatise by Sparrow \cite{sparrow1} for additional details. 
\medskip

\noindent File {\bf lorenz1.dat} clearly illustrates this stretching and folding
process.  To see this effect, press the FORWARDS button on the Multiple Orbits Panel.
A square set of initial condition will be simultaneously propagated forward in time
and, as it passes near the origin,  the flow is stretched along the unstable manifold
of the equilibrium.  
