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Carbonated_Brainss · 06-03-2004, 03:17 PM

What you just described is a linear system.

You can't "bottle" a chaotic system and treat it as periodic; if you somehow did that, it wouldn't be a chaotic system.

A chaotic system is dynamical, in the sense that it's a model of an actual system whose behavior varies with time.

The idea of chaos theory doesn't mesh with the very linear scientific practice of putting stuff in a box and pretending it has no interactions with the outside world. Frictionless surfaces, perfect springs, perpetual pendulums and such are all fine and good for approximations of LINEAR systems, but you can't take a CHAOTIC system (which is based on the tiniest preturbations of the initial conditions) and pretend it's going to react in a similar way if you stick it in a box.

But now that I think of it, Lorenz stuck his model of the atmosphere in a box, and still got chaotic results. Check this out, I believe this diagram refutes your claim that ONE variable cannot affect the outcome.

haha, now that I look again, this is the best diagram yet. It perfectly rebuts the idea that small changes cannot individually have an impact.

Stay with me here:

If you take the Lorenz equations and use:

= 10
b = 2.6667
r = 20
you get this:

This is the same graph with X plotted against time:

The BLACK line in the above picture is if you make your initial X = 5, instead of 1 (the red line assumes X = 1) See how the black line follows the red line, and near the end they're pretty much equal?

Seems to suggest a little change results in not much difference of outcome...
BUT WAIT!

What if we use the same conditions as in the first graph, except we change r = 28? So:

= 10
b = 2.6667
r = 28

we get:

Wow, that's crazy different. The thing no longer spins around a single point in the x-y plane, but it flips randomly between 2 points. Shall we plot it on the X vs Time graph again?

NOW! Here's the kicker! Suppose we take the value of X, which is 1.0000, and change it to 1.0001! We'll superimpose the black line on the red line.

For a while they're the same, but then black diverges and takes an entirely different random path! In fact, the futher you run the experiment the weirder black is, in relation to red!

Absolute graphical mathematic proof that if you take the initial condition x = 1.0000 , and change it to x = 1.0001, you get an ENTIRELY different end result, given time!

Are you convinced yet? Please? My fingers are blistering!

06-03-2004, 03:17 PM	#64
Carbonated_Brainss Recruit or Something Join Date: Jun 2004 Location: Between the smoky layers of a prosciutto sandwich! Posts: 4	What you just described is a linear system. You can't "bottle" a chaotic system and treat it as periodic; if you somehow did that, it wouldn't be a chaotic system. A chaotic system is dynamical, in the sense that it's a model of an actual system whose behavior varies with time. The idea of chaos theory doesn't mesh with the very linear scientific practice of putting stuff in a box and pretending it has no interactions with the outside world. Frictionless surfaces, perfect springs, perpetual pendulums and such are all fine and good for approximations of LINEAR systems, but you can't take a CHAOTIC system (which is based on the tiniest preturbations of the initial conditions) and pretend it's going to react in a similar way if you stick it in a box. But now that I think of it, Lorenz stuck his model of the atmosphere in a box, and still got chaotic results. Check this out, I believe this diagram refutes your claim that ONE variable cannot affect the outcome. haha, now that I look again, this is the best diagram yet. It perfectly rebuts the idea that small changes cannot individually have an impact. Stay with me here: If you take the Lorenz equations and use: = 10 b = 2.6667 r = 20 you get this: This is the same graph with X plotted against time: The BLACK line in the above picture is if you make your initial X = 5, instead of 1 (the red line assumes X = 1) See how the black line follows the red line, and near the end they're pretty much equal? Seems to suggest a little change results in not much difference of outcome... BUT WAIT! What if we use the same conditions as in the first graph, except we change r = 28? So: = 10 b = 2.6667 r = 28 we get: Wow, that's crazy different. The thing no longer spins around a single point in the x-y plane, but it flips randomly between 2 points. Shall we plot it on the X vs Time graph again? NOW! Here's the kicker! Suppose we take the value of X, which is 1.0000, and change it to 1.0001! We'll superimpose the black line on the red line. For a while they're the same, but then black diverges and takes an entirely different random path! In fact, the futher you run the experiment the weirder black is, in relation to red! Absolute graphical mathematic proof that if you take the initial condition x = 1.0000 , and change it to x = 1.0001, you get an ENTIRELY different end result, given time! Are you convinced yet? Please? My fingers are blistering!