pariah

I heard river say it, and have a vague idea of what it is, but can someone please explain it to me, or provide me with some quality literature about it?

Thanks.

Kat_Somm_Faen

Well its not really a theory - its more of a metaphor

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Edit:

Chaotic system is a system where a small difference of the input can produce a disproportional response in the output. The sort of butterfly flapping his wings can cause a tornado. A chaotic system is one that exhibits extreme sensitivity to the starting conditions.


So River is trying to say that the energy for all these otherworldly claims of stars falling ond other stuff can come from very small discrete sources.

NialScorva

The only thing I know is that any problems with Chaos Theory can easily be resolved by installing Linux.

Abel Stable

Chaos Theory- This page offers a pretty good introduction I think. He introduces the "Butterfly Effect" with a medieval (I think) song:
And, appropriately I guess, the graph arranged to describe this is shaped like a butterfly. Sadly, most of this is incomprehensible to me, but I’m sure others will appreciate.

pariah

Is this a significant theory? The site says:

Maybe I'm stupid--exactly _how_ would we benefit from this...? Predicting the future he says. So we could predict the outcome of random systems? What random systems are we currently having problems with?

Lobstrosity

The picture of the Lorentz attractor should help you. The problem is that nonlinear systems are very sensitive to initial conditions and can diverge rapidly for similar but non-identical initial conditions. To what they diverge, however, is a function of the system. Chaos does not necessarily imply boundless--it is not fair to say that in a chaotic system any result is possible. All one can say is that it is impossible to predict the future with great accuracy because initial uncertainties are magnified with time. The Lorentz attractor is a great example of a bounded system whose behavior never repeats. Similarly, our solar system is chaotic, but this doesn't mean the planetary orbits are unstable. It is not fair to say that one day this chaos could lead to the ejection of a planet from the solar system, to a planet's falling into the sun, or to a reversal of a planet's orbit. Another example of a chaotic system is a double pendulum (one pendulum hanging off the end of another).

pariah

Lobstrosity: In your pedulum example, it is possible to predict where the second pedulum goes using physics isn't it? The reason this becomes a chaos effect is that you are able to predict the second pedulums movement from the first, and the firsts from known movement at its base. Of course you can rule out the pedulum leaving certain areas.

RIght? That's pretty simple.

So what systems can we NOT guess the outcome of yet, that would benefit us to be able to?

And, an astronomy question, how is our solar system chaotic? I could understand the universe being chaotic in expansion (another area I don't understand well), but...

Kat_Somm_Faen

Pariah, you have to spin a double pendulum to see it

Once you do so you will have a much better visual undestanding of "chaos".

The base movement is not exactly the same and even the slightest difference in starting conditions can lead to drastically different conditions. Weather is a good example except there are a lot of variables to consider.

If you have ever seen a double pendulum in action its movement is quite unpredictalble.

Normal linear system a sligth change of variable x ( say for example x' = .000005% different ) produces similar change in the output ( y' = .000005% different ) so the system responds to changing initial conditions quite timid.

In a chaotic system a small change in initial condition will result in large differences in the output.

A single pendulum just swings back and forth while a double one twists and turns in very different ways so that it is impossible to predict or replicate the output ie. the pendulum oscillations after it has been set in motion.

Same effect can be found in turbulent flow in fluid dynamics or a score of "caotic systems". For example the present structure of the Universe is considered greatly due to extremely small perturbations in the beggining after the Big Bang. the CBR is extremely uniform when you "look" at the night sky.

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Edit: Don't worry about Rivers calling upon chaos theory - it is just grasping at straws and as much of his calling upon science - a veryyyyyy big stretch indeed. About as far as deriving Big Bang theory from "he is the Originator" or string theory from "space is a tapestry" verses!

Lobstrosity

In order to describe the system you end up with a series of nonlinear differential equations that do not have a closed-form solution. You can try to solve these equations numerically, but you note that solutions will diverge over time for initial conditions that are not precisely identical. This is what makes it chaotic. There's no real way to say where the pendulums will be at time t given a set of initial conditions unless (a) your initial conditions are 100% accurate and precise and (b) your numerical solving technique introduces no error (which of course it will through rounding errors in an attempt to tackle a continuous problem through discrete analysis). The short answer is that you can't predict where the second pendulum goes because its motion is affected by the first pendulum. The first pendulum's motion is affected by the motion of the seconds. The two are coupled and the coupling is nonlinear.


I'm not really sure how to answer this. In reality pretty much everything is probably chaotic to a certain extent, especially when one takes into account quantum uncertainty. I would expect that the ability to make predictions would arise from an understanding of the limits of the system such that statistical analysis could be performed.


It's chaotic because it's a many-body problem. You don't just have a planet orbiting a star, you have a bunch of planets orbiting a star. Each planet exerts a gravitational influence on every other planet, which acts as a slight perturbation factor. These perturbations are enough to destroy linearity and result in a very subtle chaos. Obviously the gravitational force on Earth is slightly different for the portion of our orbit that takes us closest to Jupiter than it is for the portion that takes us farthest away. This results in small deviations from the perfect elipses one might naively expect. The planetary orbits are not unstable in any way, it's just that no two orbits of a given planet will be exactly alike. The perturbations are so weak, however, that over short timescales this would be very difficult to notice. Furthermore, I would expect that they would average out so as to produce no significant deviations (even over long periods of time) from the near-perfect elipses we already trace out. The chaos would probably be easier to observe with regards to moons orbiting in close proximity around something like Saturn or Jupiter.

Bag of Ass

There should be a double pendulum simulator somewhere on the internet. Hey look, here's one.

Addendum: Now with more interactive goodness!