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        {\LARGE\bf The Lorenz Model }
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        <center><img src=../../gif_files/GRAD_BLUE_LINE_f.gif></center>
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{\large \bf \em The model}

~~~

The Lorenz Model was put forward by Lorenz in 1963 as 
a simplified model of the convective movements in the 
atmosphere in the context of weather forecasting.  This model was
the first model that exhibited chaotic type motion and 
initiated a new era of studies in nonlinear dynamics.  
The model made it very evident that the solutions 
that exhibit chaos depends sensitively on 
the initial conditions as it happens in weather forecasting.

The model is very nicely described in the book {\em NonLinear 
Dynamics: A Two-Way Trip from Physics to Math} by Solari, Natielo 
and Mindlin.  

The model is summarized by three coupled first order 
ordinary differential 
equations in a three dimensional phase space 
labeled $x(t)$, $y(t)$, $z(t)$.  The equations 
governing this system 
are the following:

\begin{eqnarray*}
\frac{dx}{dt} & = & - \sigma x + \sigma y \\
\frac{dy}{dt} & = & R x - y - x z \\
\frac{dz}{dt} & = & - B z  + x y 
\end{eqnarray*}


From our point of view of solving ODEs this problem is 
quite simple.  We use the  Runge-Kutta fourth order to solve the equations
subject to given initial conditions.  What is remarkable 
about this system is the complexity of the solution that follows. 

The code 
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  <a href=../lorenz.c>lorenz.c</a>
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solves the Lorentz model.

~~~

{\large \bf \em The Lorenz Attractor }

~~~

The values of the parameters, $\sigma = 10$, $B = 8/3$ and
$R = 28 $,  given as default in the code above,
were found by Lorenz in 1963 to generate a chaotic solution.  
Plotting $x(t)$, $y(t)$, $z(t)$ in 3-dimension 
(using the  gnuplot command {\em splot})
displays the very famous  butterfly picture of the Lorenz attractor
that you find in every textbook.  The two-dimensional projection plots,
Y vs. X, Z vs. X, and Z vs. Y, are also illustrative of the attractor.

Note that the chaos is exhibited by the solution in going from 
one spiral to the other one in the butterfly figure. 
 
Plot $x(t)$ vs. $t$.  You will see a very rapid motion from 
left  to right on the attractor as exhibited by the curve going 
from an upper value to a lower value.  There is no repetitive 
pattern to be encountered in this time series.

The Lorenz attractor is a chaotic attractor; it is  a fractal.  
Being an attractor implies that initial trajectories starting 
from initial conditions in different locations in phase space all 
converge to it.  This is why this arbitrary initial condition 
that you have seen in the image above leads to the chaotic attractor.  
The trajectory follows a transient then latches onto the chaotic attractor.  

Here is how we can illustrate that and at the same time form a much 
better image of the Lorenz attractor.  Integrate the Lorenz model 
for a scale time of 100.  Store the trajectory into a file.  
Restart the Lorenz integrator with the {\em - i} option to
 specify X, Y, Z initial as the last X, Y, Z values generated 
in the previous run over a 100 time scale unit.  
Run this trajectory for another scaled time of 200 units. 
Plotting the data in three dimension will produce a sharp 
image of the Lorenz attractor.

There exist unstable periodic orbits within the Lorentz chaotic attractor.  
Naturally these are hard to find since they are unstable.  
We saw in class that the procedure by Henry, Watt, Wearne 
that uses a lattice refinement scheme to find periodic orbits 
could be applied to the Lorenz model.  Let's illustrate the point 
that there exists such orbits in a chaotic attractor.  

\begin{itemize}

\item      Integrate the Lorenz model for a scale time of 10 units 
starting with the initial condition 
( x, y, z ) = ( 2.147367 -2.078048 27.0 ) and store the data in a file.
                
\item Integrate the Lorenz model for scale time interval of 
10 units starting from the initial condition  
( x, y, z ) = ( 4.250440 1.008479 27.0 ) and store the data.

\item  Plot  the two trajectories in three dimensional phase space.

\item   Plot the Lorenz attractor together with the two trajectories 
that you just generated.  Note that these two trajectories are lying
on the attractor.

\end{itemize}

This illustrates the fact that these trajectories which are 
periodic unstable orbits are part of the dynamics that drive 
the establishment of a chaotic attractor.

~~~



{\large \bf \em  plot3d.c }

~~~~

The limit cycles, the Lorenz attractor and the unstable periodic 
orbits within it are all objects in three dimensions. Wouldn't it
be nice to rotate these objects in space to better appreciate 
their structures. Or equivalently we could rotate the observation point.

The code 
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  <a href=../plot3d.c>plot3d.c</a>
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uses gnuplot to display three dimensional data sets from
a rotating observation point.  Download the code,
run {\em ./plot3d -h} and use the three hills potential
used in our scattering studies,
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  <a href=../potential.dat>potential.dat</a>
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, as a sample data set to practice with.

Use the code to display the three dimensional Lorenz
attractor and various periodic orbits.

~~~~

~~~~~


{\large \bf \em A movie! }

~~~~~

The Lorenz attractor is an extremely complicated object.  
To further develop one's instinct about how this attractor is built, 
it is interesting to follow graphically the time evolution of a 
trajectory that spans the attractor.  This is illustrated 
in the two methods that follow.  
Note the syntax and the use of LINUX utilities
in these two approaches.. 
 In both cases, the trajectory is re-drawn for an ever 
increasing number of points as generated by our solution of the 
Lorenz ODE's via rk4.  These graphs form a movie when seen in succession.

~~~


{\bf \em  Method 1 }

The scripts   {\em gnuplot\_script\_1 } and 
{\em gnuplot\_script\_2 } are two GNUPLOT scripts.  
They assume that the data to span the Lorenz attractor is 
in the file {\em butterfly}.  The GNUPLOT graphs are 
generated by "loading" {\em gnuplot\_script\_1} via the command
{\em load gnuplot\_script\_1} from GNUPLOT.
The looping over the movie frames is generated by
{\em gnuplot\_script\_2} when it loads  itself 
again and again in GNUPLOT, tthereby ``replotting'' the
trajectory with an increasing No of points.

The scripts are in 
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  <a href=../movie_gnuplot.tar>movie_gnuplot.tar</a>
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{\bf \em Method 2}

A mix of a {\em bash} shell script and a GNUPLOT script is used to 
generate a GIF animated ``movie'' from frames containing GNUPLOT graphs. 
The method reads the trajectory data from the file {\em butterfly}.
It requires an empty sub-directory {\em movie\_frames}.  
It is launched via running {\em bash\_script}.  The script
then writes a GNUPLOT script to generate the movie frames in 
{\em movie\_frames} in PNG format, converts these to GIF format
using the command {\em convert}, removes the PNG files
 and finally (after looping over the frammes)
 uses {\em gifsicle} to 
merge these GIF files in an animated GIF file in the file 
{\em anim.gif}.  The latter can be viewed via ImageMagik commands
or via Firefox.



The scripts are in 
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  <a href=../movie_ImageMagik.tar>movie_ImageMagik.tar</a>
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{\large \bf \em References }


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<p>
<a href="http://en.wikipedia.org/wiki/Lorenz_attractor">Wikipedia</a>
</p>

<p>
Serious General <a href="http://books.google.com/books?id=-SPPDBaJPgoC&printsec=frontcover&dq=Solari+Mindlin&source=bl&ots=3TU7stItPT&sig=-OKYlgXig9zUKsxMCGPXnlsPaoY&hl=en&ei=VslyS5P7C4rT8Aan5ozICw&sa=X&oi=book_result&ct=result&resnum=2&ved=0CAwQ6AEwAQ">Non-Linear Dynamics book</a> by Solari, Natiello and Mindlin
</p>

<p>
<a href=http://www.math.uni.wroc.pl/~karch/B2/lorenz-system.pdf>Paper</a>
( <a href=../lorenz-system-2.pdf>local copy</a> )
 by Nicholas Report</a>
</p>



<p>
Nice <a href="http://mathforum.org/library/drmath/view/61791.html">history</a> of the model
</P>

<p>
<a href="http://www.google.com/images?hl=en&sugexp=nsp7&xhr=t&q=lorenz+attractor&cp=3&qe=TG9y&qesig=Qp8TxVGLTwgsWB6FOPSO0Q&pkc=AFgZ2tmZWBQdwHIH7QgXAtg7G1QGK2p7phd2ZJ506o_ZF0OhgjeNeibzctuy6cUJFU7w8BcewAnoSmsSY6TcVlgyN6D_EmrVmw&bav=on.1,or.&um=1&ie=UTF-8&source=univ&ei=DLRdTdrFDcH58AbAusDkCg&sa=X&oi=image_result_group&ct=title&resnum=3&sqi=2&ved=0CEQQsAQwAg&biw=1186&bih=974">Amazing Pictures</a>
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