Chaos

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Lorenz Attractor
Henon Attractor
Blue Fern
Strange Attractors
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Logistic Bifurcation

Dynamical systems are called chaotic when it is impossible to predict what the system will be like at some future time. A mathematically precise definition is given in the next section.

To understand chaos, we first take an example of a non-chaotic system: a cannon shooting cannonballs. If we shoot a cannonball of certain weight, with a certain amount of gun powder, and at a certain angle, it would land at a specific distance. If we then shoot a cannonball of slightly different weight, slightly different amount of gun powder, and slightly different angle, then we would expect it to land at a slightly different distance than the first cannonball.

On the other hand, weather would be an example of a chaotic system. If we have some initial state of certain humidity, temperature, and wind speed, then a certain weather state would be produced two weeks later. If we started with some initial state of very slightly different values of humidity, temperature, and wind speed, the weather state produced two weeks later would be completely different from the the first. While this seems hard to believe, try to remember the last time meteorologists predicted a hurricane or tornado two weeks beforehand. They cannot because weather is a chaotic system.

It is important to note that chaotic systems are still deterministic. That means if it were possible to have infinitely precise data about all of the variables involved and how they changed in time, and we had an infinitely powerful computer, all future outcomes could be calculated precisely.

A weather forecaster can only ever give you an estimate of who likely it is to rain--they simply cannot tell you how many rain drops will fall on your head. This example also illustrates another common property of chaotic systems: while we can't say anything exact about where the system will be in the future, we can make general statements about how likely certain outcomes are.

While not all chaotic systems can be analyzed this way, describing probabilities can be a powerful tool for analyzing many systems.

Examples of chaotic systems include the Lorenz Attractor and the Henon Attractor.

A More Mathematical Definition

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A chaotic system exhibits three main properties:

1. The system has a dense set of periodic points.

Despite the seemingly erratic behavior of chaotic systems, some states (called periodic points) move in cycles over time. This first condition guarantees that every point is arbitrarily close to a periodic point.

2. The system displays a sensitive dependence on initial conditions

This means that two initial states very close to each other will eventually get more and more different. Think about placing a perfectly smooth marble on top of a small perfectly smooth dome. The marble will fall roll off to the side, and if you try to place it back again in the exact same spot, chances are it will roll of in a different direction.

3. The system is transitive

This is the least intuitive condition chaotic behavior, but it guarantees that when states evolve, they jump around near all of the possible states.

Check out Wikipedia and MathWorld for some more information, but be aware both sites use highly technical language.