#8 Lorenz Finds Chaos and Chaos is Beautiful

The legend: In the early 1970s Edward Lorenz was working on a computer-calculated weather model.  He was using multiple machines to do different parts of the work and at several points had sent the data-states of the model between the computers and then back again. 
But he was experiencing an unusual glitch.  Even though he had previously run this model with exactly the same conditions, when he removed and then reinserted the data mid-way through the model's progress, the model would rapidly predict entirely different results! Investigating this further, Lorenz found that the only difference possible was in the very very small round-off error that occurred when transferring the data.  But a small difference in input is supposed to result in a small difference in result so long as all of the modeling functions are nice and smooth - right?  That's the case in physics most of the time - unless you're specifically looking at the critical point of a function, the behavior is locally predictable.  A very small change in the input creates a fairly small change in the output.  In particular, a /small enough/ change can ensure that your output is as close to the original as you'd like.  Lorenz, however, had discovered a class of models for which this fact is NOT true - quite the opposite actually.  And thus began "Chaos Theory."


In 1972, Lorenz published what is thought of as the first paper in modern Chaos Theory.  It was entitled: "Predictability: Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas?"  In other words - can a difference in initial conditions as small as weather or not a butterfly moves its wings make the difference in weather or not a tornado is brewed across the world?  But Chaos isn't only about unpredictability.  While in the vernacular, "Chaos" means "complete disorde r" or "a confusing mess" mathematically, the word means a much more structured kind of unpredictability and randomness.  Rigorously, mathematical chaos has three components:

1) Sensitivity to Initial Conditions: arbitrarily close to every state S1 of the system, there is a state S2 whose future eventually is significantly different from that of S1.
2) Dense Periodic Points: arbitrarily close to every state S1 of the system, there is a state S2 whose future behavior eventually returns exactly to S2.
3) Mixing: given any two states S1 and S2, the futures of some states near S1 eventually become near S2.

The first of these components alone is what Lorenz experienced with his Weather model.  It's commonly known as "The Butterfly Effect" - that "Even the tiniest change can alter the future in ways you can't imagine."  Adding other two conditions completes the definition by requiring that the system's range defines a phase-space (a space of possible states of the system) within which, so long as even the slightest imprecision exists in the start state, any state is an eventual possible outcome, no matter where you start.  The image to the left is of the Lorenz Attractor: a rigorously chaotic system - all three of the above conditions hold.  However, in the past century, Chaos Theory has expanded to include far more than weather models and similar systems: Conway's Game of Life and the behavior of Fractals (below) exhibit behavior closely related to the chaos Lorenz found in the weather.

Exercises:
1) Use Your Calculator to Create Chaos
2) The Dragon Curve - the Chaos of Iterative Folding
3) An introduction to Fractals
4) Chaos in Conway's Game of Life
5) How Chaos creates the "Entropy" of Physics

References:
The Mathematical Definition of Chaos
Wikipedia Article on Chaos Theory
Lorenz Obituary
Yale's Fractals Mainpage
Conway's Game of Life: The Basics
Golly Download of Conway's Game of Life


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