The butterfly effect is shown by simultaneously plotting two different solutions started from

First, reset the parameters to their default values by quitting and restarting
the applet. Now
set, for example, *dX0=0.004*, i.e. prepare two
initial conditions that differ by one part in a thousand.

Start the
evolution, plotting *Z(t)* against *X(t)* (i.e. set *x-ax=1* and
*y-ax=3*).

Initially the two
solutions track each other so closely that only one orbit appears on the
screen. By a time as short as *t=10* the difference has grown large
enough to be visible, and by time *t=15* the trajectories no longer
track each other at all.
The behavior is particularly apparent in a direct
*X(t)* plot (set *x-ax=0*). In only 15 characteristic time units
for the dynamics, we have completely lost the ability to forecast the
behavior
if the "error" in our initial conditions is only one part in a thousand!

Study the existence of the butterfly effect for different values of *a*.

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Last modified 18 August, 2009

Michael Cross