What is a Chaotic System?

This intro to chaos theory with Java applets was written by Ben Fisch, as part of a 2009 summer project at PPPL working with Luc Peterson and Greg Hammett, based on a chaos tutorial by Charles Karney.
(There are some bugs: Some of the control boxes don't appear until you click on a figure or in some of the empty space around a figure.)

Consider frictionless balls in a square box:

One Ball, One Box

In this first demonstration, a single ball with some constant velocity, mass, and radius will move around inside the box and bounce off the walls.

This is an example of regular motion. The ball's motion is restricted to 4 different directions. The ball's position at any given time is easily predicted.

Two Balls, One Box

Now we place two balls in the box, very close to each other.
We give the second ball a slightly different velocity. (30.5 compared to 30).
You can see the separation of the balls grow slowly. (WE IGNORE COLLISIONS BETWEEN THE TWO BALLS. THEY PASS THROUGH EACH OTHER.)

This is still regular motion.

Characteristics of Regular Motion:

1.) Motion of each ball is restricted.
2.) Errors (i.e. imprecisions in ball velocity) grow slowly.
3.) Even given errors, it is possible to predict the balls' trajectories and positions for a fairly long time.

Round One Corner

Now we round one corner, expanding the possible directions of motion.

This is no longer regular motion, it is irregular motion.

Characteristics of Irregular Motion:

1.) No restriction on motion. (It is possible for every point to be visited at every angle).
2.) Errors now grow exponentially making long term prediction impossible.

Watch what happens when there are two balls in this system:

If we were to allow collisions between the balls as well, errors would grow even more rapidly.

Chaotic System: Smiley Face

Demonstration of a simple chaotic system:
These 18 balls are arranged into a smiley face. Click "Start." It will run for a bit then freeze. Click "Reverse" and "Start" again. The smiley face should reform.

Now click "Reset" and "Start" again. Wait for the balls to freeze, and click "Stop."
The positions and velocities of the balls are displayed in the text boxes on the right.
Try changing just one of the 5 values for one of the 18 balls by the slightest amount (this is your "error").
Do you still get your smiley face back?

The "Butterfly Effect" and Ed

This sensitivity to initial conditions is also commonly called the "Butterfly Effect": the idea that a butterfly flapping its wings in Brazil will cause a tornado in Texas.
This term was first popularized by meteorologist Edward Lorenz. Ed Lorenz became one of the first contributors to chaos theory when, in 1961, he saw his computer
weather patterns diverge significantly due to the slightest difference in input value. (Figure from James Gleick's book, Chaos.)