The Butterfly Effect - Demonstration

This page leads you through a demonstration illustrating the butterfly effect: sensitive dependence on initial conditions.
The butterfly effect is shown by simultaneously plotting two different solutions started from slightly different initial values. The small difference between the initial values is set by dX0, dY0, dZ0.

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.

[Return to Home Page]
Last modified 18 August, 2009
Michael Cross