Chaos is normally considered to be about disorder or confusion. However, in science it describes an important conceptual paradox which has a precise mathematical meaning:

A chaotic system is a deterministic system which is difficult to predict.

We normally think that a deterministic system is completely predictable, and this is a mathematical truth. By definition, a deterministic system is one whose state at one time completely determines its state for all future times. Chaos, however, is not a mathematical contradiction, it is a conceptual contradiction with practical consequences. The real question is how accurately can you predict over what length of time given a certain amount of information. The reason a deterministic system can be difficult to predict is that what happens in the future can be very sensitive to its current state. This property, called "sensitivity to initial conditions," has been described as the Butterfly Effect: the possibility that a large storm in New England may be due to a butterfly wing flap in China.

A technical way to describe sensitivity is through the divergence of trajectories of the system. Over time, a system starting from one state becomes less and less similar (farther and farther away in state space) to a system which starts out in a similar, but not exactly the same, state. It is worth emphasizing that this means that the more accurately the initial state is known, the more accurate can be a prediction. The problem with prediction is that the degree of accuracy needed in many practical cases is likely to be impossible to obtain.

The paradox of chaos strikes at the roots of traditional concepts of science which suggest that increasing knowledge will lead to predictability.

Chaos is not the only source of unpredictability of a system's behavior. Conceptually, there are three sources for the lack of predictability. The first is the influence of random noise, the second is the effect of the environment on the system, and the third is lack of knowledge of the initial conditions. The third is the one dealt with by studies of chaos.

Chaos and fractals have an underlying relationship to each other, however, what is called fractal dynamics is not the same as chaos.

.

 

Copyright © 2000 Yaneer Bar-Yam All rights reserved.