This page leads you through the main features of the applet. You can also review the background and mathematical details.

The parameters of the Lorenz Model are

The "standard" value of *a* is *a=28*. Start the applet
at these default values by hitting **Reset** and then
**Start**. You will see a plot of *Z(t)* against *X(t)*
as *t* varies. The familiar Lorenz butterfly appears. Notice
both the complexity of the plot (the "randomness" or
"unpredictability") but also the definite structure - this
combination is characteristic of chaos.

A number of parameters can be changed to give the best animated
plot, depending on the speed of the computer etc. The variable
**dt** gives the time stepping in the numerical integration. Small
values may make the animation smoother but slower. Beware: too large
a value will give inaccurate answers - there is currently no accuracy
check in the numerical scheme! The variable **tran** sets a number
of time steps over which the equations are evolved first to eliminate
transients and then to construct a "phantom image", which also sets
the scale of the plot to eliminate the continual rescaling at early
times. You can set *tr an=0* to eliminate this effect. The
**speed** scrollbar sets the speed of the animation (roughly,
updates per second), and can be tuned to smooth the animation.

The full dynamics needs to show the three variables *X,Y,Z*
as they vary in time. On the computer screen it is easiest to show
various *projections* such as the *Z-X* projection just
seen. The two variables plotted are chosen by the *x-ax* and
*y-ax* parameters with *1* standing for *X*, *2*
for *Y* and *3* for *Z*. Try out various combinations
to look at different aspects of the full 3-d plot. (Remember to hit
**Enter** or the **Reset** button after changing a value.)

Choosing *x-ax=0* plots the time variable along the x-axis.
This can be used to look directly at *X(t)* etc.

The starting values of the three variables are set by *X0, Y0,
Z0*. Try out different values. If you have *tr an=0* you will
see different behavior at early times (the transient). However the
dynamics evolves towards the same butterfly structure, which is
therefore called an *attractor*. The dynamics on the attractor
will be similar in outline, but different in details depending on the
initial conditions, i.e. chaotic. The attractor is therefore called a
*strange* attractor.

Perhaps the defining property of chaos is the *sensitive
dependence on initial conditions* or poetically the
butterfly effect. This is demonstrated
by plotting two solutions together starting from very slightly
different initial values. The small difference between the initial
values is set by *dX0, dY0, dZ0*.

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Last modified 18 August, 2009

Michael Cross