Exploring Chaos Theory: Can Future Systems Be Predicted?

[ion]

I am not too sure where to post this,and so have decided on the general category.
My question is on the Chaos theory. A very basic question as I have only recently come across this subject. Am I right in assuming that:

The future of a system is not predictable if the initial variables(that determine an event) exceed a certain finite number.

If this is true could someone please enlarge on the matter.If it is wrong please educate me. I am quite green on this subject.

Thank you.

 

[mathman]

Typical examples of chaos are not concerned particularly with the size of the input variables. Rather, a chaotic process is typically one in which a small change in input results in a very large, usually unpredictable change in output.

 

[zoobyshoe]

Originally posted by ion
The future of a system is not predictable if the initial variables(that determine an event) exceed a certain finite number.

I am also green, being only about a third of the way into James Gleicks Chaos but your statement may be correct in certain instances of Chaos. He gives the example of the "logistic difference equation" used by ecologists, x_next=rx[1-x], which works well so long as the variable r (rate of population growth) has a relatively low value. Above a certain value r the formula refuses to arrive at an equilibrium.
x = population
r = rate of growth
x_next = next years populationThe chaos exhibited by fluid flow, to cite another example, seems only to come into play when one of the variables: speed of flow, exceed a certain rate, after which the fluid begins to behave "chaotically".

However, your statement is somewhat ambiguous and what I said may be off the mark if I misunderstood what you meant.

If you meant that some formulas and dynamic systems become chaotic when the value of a specific, important variable exceeds a certain threshold, then it would be true. If you meant that a formula yields chaotic results when the number of variables in the formula exceeds a certain number, then I wouldn't think so.

-Zooby

 

[selfAdjoint]

It's not the number of variables that deterimes chos, but the sensitivity to initial conditions. Specifically if the rate of separation of nearby varianles grows exponentially as a function of the time and their intitial separation. This is the point of the butterfly's wing story. The tiny puff of the wing's effect on the atmosphere can grow exponentially and become huge after a while.

 

[zoobyshoe]

Noticing, and paying attention to, the butterfly effect was the first step in the science of Chaos, but all Chaos can, by no means, be boiled down to the butterfly effect. That would be akin to explaining Relativity as the photoelectric effect.

On page 306 of Gleick's Chaos he lists a few attempts at a definition of the new science by some of the men involved in it:

"The complicated, aperiodic, attracting orbits of certain (usually low-dimensional) dynamical systems."

"A kind of order without periodicity."

"Apparently random recurrent behavior in a simple deterministic (clockwork-like) system."

"The irregular, unpredictable behavior of deterministic, nonlinear dynamical systems."

"The translation from mathese is: behavior that produces information (amplifies small uncertainties), but is not utterly unpredictable."

What makes a system chaotic is not its sensitive dependence on initial conditions (when that is the case, which it isn't for all chaotic systems), but its refusal to settle down and stabilize. Chaotic systems are dynamic, yet they won't repeat. Not only that, they are sometimes subject to complete, unexpected reversals.

 

[FZ+]

The future of a system is not predictable if the initial variables(that determine an event) exceed a certain finite number.

It is true that most chaotic systems are not always chaotic, and that they frequently show a gradual approach to chaos based on several bifurcations determined by the Feigenbaum constant. So I guess you are pretty right, but that is not the definition of chaos itself. Chaos is really not a type of system, but a description of a state of a system.

And chaos can be predicted to a reasonable degree of accuracy, for a while. We still have weather forecasts, don't we? However, our error increases exponentially with time.

 

[BasketDaN]

It is a theory developed on the observation of how easily a chain reaction can occurr, yet ignores the tendency of the vastness of the world to "absorb."