What is Chaos theory?


We can illustrate chaos theory in various ways- one way being the idea of two pendulums being connected end to end as this gif from Wikipedia illustrates. If the starting position of the pendulums was slightly different, then we would get an entirely different result. So chaos theory basically is the idea that tiny changes to a system can result in huge changes over time, most popularly illustrated by the idea that a butterfly's wings can cause enough current to completely change the weather on the other side of the world. It is actually pretty accurate- this is why even the most precise computers can't predict the weather for more than a few days as tiny changes can completely change the outcome.

However, interestingly, chaos theory isn't necessarily part of quantum mechanics as you might expect- the two can be confused. Chaos theory is a classical theory because it is actually hypothetically possible to predict the outcomes of a system if you could measure precisely enough, whereas as discussed before, it is a fundamental part of quantum mechanics that you can't measure a particle's position and velocity. Usually with chaos theory the outcome can be predicted up to a point but after that appears random, because of the small change.

So is there a definition for when a system can be called chaotic?
Not a universally accepted one. However, Robert L. Devaney formulated three principles a system must have to be chaotic: 
 1. It must be sensitive to original conditions.
 2. It must be topologically mixing.
 3. It must have dense periodic orbits.

What on Earth does that mean?
Good question. I don't truly understand myself, however in rough terms:
 1. Small changes to the original conditions make a major impact.
 2. No matter how far away a given pair of particles are, eventually they will get close to each other.
 3. A particle, having passed through a point, (let's call it P), will eventually come close to point P again (though not necessarily touching P, just going extremely close).

I can't pretend that I'm sure of the accuracy of this as I'm trying to simplify concepts I haven't fully understood yet so I suggest this article which is part of a series of others which are linked to at the bottom of his post on chaos theory.

So what are some examples of chaotic systems?
Well weather is the obvious one. Others include the solar system.

A nice definition of a chaotic system from Edward Lorenz, a chaos theory pioneer:

“Chaos: When the present determines the future, but the approximate present does not approximately determine the future.”



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