Mathematics? It's all just life. - Page 5

lumberjim · 06-03-2004, 01:36 PM

why would you get banned......? there wasn;t even a poll.

i suspect your pesky work computer again.

Carbonated_Brainss · 06-03-2004, 01:39 PM

Yet I can post with this account and not the other!

Beestie · 06-03-2004, 02:16 PM

I think Adam Smith (an economist) said it long before these guys when he postulated his Invisible Hand theory.

Your post was very illuminating and I will be taking a closer look. The Smith peice pretty much sums up where I'm coming from - that chaotic systems behaviour is the sum of countless other small scale actions acting independently. No single action or lack of action dictates the outcome.

To use your stock market analogy, no single small investor moves the market - its the sum of ALL the small investors (forget the big boys - mutual fund managers, Warren Buffet, etc. for the moment).

Put another way, here is an experiment that I propose: Take a chaotic system and "bottle it." Now, put it through a process so as to generate an outcome. Now, replicate that process over and over again but each time, remove ONE variable and put back the previously removed variable until the process has been run without the contribution of each variable in the system at least once. That is to say if there are ten variables, remove one and run it. Then put it back, remove a different one and run it again. When you are done, you will have run the process 10 times: once with 10 variables and 9x with 9 but a different 9 each time. Obviously, the values of the variables must be held constant.

My hypothesis is that the outcomes would never change because no single variable can change the outcome. If a single variable could influence the outcome then the system wouldn't be chaotic. It if for this (hypothesized) reason that I cannot accept the butterfly example.

Even if I am wrong (very possible), at least I hope I have explained my objections clearly.

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!

Carbonated_Brainss · 06-03-2004, 03:20 PM

btw, the change from x = 1.0000 to x = 1.0001 is analagous to using a thermometer which isn't accurate to the fourth decimal place. Changing that TEN THOUSANDTH little bit was enough to blow up the end result. If we went 12 more decimal places down the line (far down enough to be in the realm of measurement error!) we'd still see the same end result; crazy differences.

Carbonated_Brains · 06-03-2004, 03:23 PM

My old nickname works! Sweet!

Slartibartfast · 06-03-2004, 04:29 PM

Kneel. Kneel before Professor Chaos!

MWAHAHAHAHA!!!!!!

Carbonated_Brains · 06-03-2004, 04:35 PM

btw, is anybody reading this hooey besides the 5 or so people who have posted?

lumberjim · 06-03-2004, 04:37 PM

dude, i tried.

i just feel really stupid now.

somebody pull my finger!

perth · 06-03-2004, 04:39 PM

I am, but I confess that I barely understand a lot of it.

Carbonated_Brains · 06-03-2004, 04:45 PM

I just remember reading all of it from those oatmeal packets I ate as a kid.

Those "daily facts" are detailed as hell.

jinx · 06-03-2004, 04:45 PM

Same here. I'm pretty sure it's very interesting though.

Slartibartfast · 06-03-2004, 05:13 PM

Don't stop just yet!

Carbonated, I don't think there is a problem in 'bottling' chaos. Computers can create chaotic results from very simple formulas. The problem is in predicting chaotic behavior, not creating it.

Okay, Beestie, one chart demonstration of the butterfly effect. This is the granddaddy example from Lorenz himself who stumbled onto these results.

Lorenz had a weather simulator on his computer. It would spit out a row of numbers for each day. It would take maybe a minute to simulate each day. One day, Lorenz was looking at old data, and he decided he wanted to run that particular simulation over again. To save time, he decided he wanted to start the simulation 'from the middle' rather than from the beginning. This seemed easy enough, he input as his initial conditions the numbers from some day in the middle of the data he had on the printout. Now here's the catch - the data on the printout were printed with three decimal places. The computer really kept track of things up to six decimals. In other words, the numbers that Lorenz input were just a tiny tiny bit off from what they had been in with the simulation first time around.

But this tiny difference shouldn't make a difference, right?

But these were his results:

Time is the X axis, and the results from both simulations are plotted on the same graph. Notice how on the left, the beginning of the simulation, there is one solid line. That means both simulations were spitting out almost identical results. But now look as how the two lines diverge over time. By the end of the graph, the two lines are totally apart. That means that with the weather simulation Lorenz was using, tiny differences in initial conditions will result in totally different outcomes. Maybe not at first, but after some time.

lumberjim · 06-03-2004, 05:18 PM

test your chaos theories:

HP linked this originally for the artistic potential, but i think it might apply

The Scribbler!!

Beestie · 06-03-2004, 10:56 PM

Quote:

Originally posted by Carbonated_Brains
My old nickname works! Sweet!

Old nickname, my ass. You went and drug some poor grad student out of the research lab at MIT and made him post all that stuff under your "so-called" alias

Actually, the Lorenz graphs you posted were new to me and quite persuasive - particularly the last one. I also liked Slarti's graph (which, seemed to be making the same point). In particular, I thought the idea of "starting in the middle" to be significant since, although we haven't talked about it, chaotic systems don't really have a discreet beginning or end. Its not like dropping a bowling ball and measuring the dent in the ground. Its more like jumping on a moving freight train, taking some measurements and jumping off again.

What I don't get is how such a small change can pick up strength as it ripples through the system over time. While Bruce provided a metaphorical scenario in which it was possible, it required a "parting of the Red Sea" w/r to all other variables - a state I considered so unlikely that it was hardly worth considering. At this point, I'll accept that my position was not correct -the graphs do a good job of explaining the increasing divergence. But now I'm really curious to understand how and why.

Oh, and thanks for taking the time to completely deconstruct my mental model for understanding the universe. Now that I understand what causes hurricanes, I'm going to go butterfly hunting - the world can thank me later. After that, I want to hunt down the cockroach that's going to cause the next earthquake - or is it prevent the next earthquake - I forget

06-03-2004, 01:36 PM	#61
lumberjim I can hear my ears Join Date: Oct 2003 Posts: 25,571	why would you get banned......? there wasn;t even a poll. i suspect your pesky work computer again. __________________ This body holding me reminds me of my own mortality Embrace this moment, remember We are eternal, all this pain is an illusion ~MJKeenan

06-03-2004, 02:16 PM	#63
Beestie -◊\|≡·∙■·∙≡\|◊- Join Date: Feb 2003 Location: Parts unknown. Posts: 4,081	I think Adam Smith (an economist) said it long before these guys when he postulated his Invisible Hand theory. Your post was very illuminating and I will be taking a closer look. The Smith peice pretty much sums up where I'm coming from - that chaotic systems behaviour is the sum of countless other small scale actions acting independently. No single action or lack of action dictates the outcome. To use your stock market analogy, no single small investor moves the market - its the sum of ALL the small investors (forget the big boys - mutual fund managers, Warren Buffet, etc. for the moment). Put another way, here is an experiment that I propose: Take a chaotic system and "bottle it." Now, put it through a process so as to generate an outcome. Now, replicate that process over and over again but each time, remove ONE variable and put back the previously removed variable until the process has been run without the contribution of each variable in the system at least once. That is to say if there are ten variables, remove one and run it. Then put it back, remove a different one and run it again. When you are done, you will have run the process 10 times: once with 10 variables and 9x with 9 but a different 9 each time. Obviously, the values of the variables must be held constant. My hypothesis is that the outcomes would never change because no single variable can change the outcome. If a single variable could influence the outcome then the system wouldn't be chaotic. It if for this (hypothesized) reason that I cannot accept the butterfly example. Even if I am wrong (very possible), at least I hope I have explained my objections clearly. __________________ ♠ ♥ ♣ ♦

06-03-2004, 04:37 PM	#69
lumberjim I can hear my ears Join Date: Oct 2003 Posts: 25,571	dude, i tried. i just feel really stupid now. somebody pull my finger! __________________ This body holding me reminds me of my own mortality Embrace this moment, remember We are eternal, all this pain is an illusion ~MJKeenan

06-03-2004, 04:45 PM	#72
jinx Come on, cat. Join Date: Nov 2003 Location: general vicinity of Philadelphia area Posts: 7,013	Same here. I'm pretty sure it's very interesting though. __________________ Crying won't help you, praying won't do you no good.

06-03-2004, 05:18 PM	#74
lumberjim I can hear my ears Join Date: Oct 2003 Posts: 25,571	test your chaos theories: HP linked this originally for the artistic potential, but i think it might apply The Scribbler!! __________________ This body holding me reminds me of my own mortality Embrace this moment, remember We are eternal, all this pain is an illusion ~MJKeenan

06-03-2004, 01:39 PM	#62
Carbonated_Brainss Recruit or Something Join Date: Jun 2004 Location: Between the smoky layers of a prosciutto sandwich! Posts: 4	Yet I can post with this account and not the other!

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!

06-03-2004, 03:20 PM	#65
Carbonated_Brainss Recruit or Something Join Date: Jun 2004 Location: Between the smoky layers of a prosciutto sandwich! Posts: 4	btw, the change from x = 1.0000 to x = 1.0001 is analagous to using a thermometer which isn't accurate to the fourth decimal place. Changing that TEN THOUSANDTH little bit was enough to blow up the end result. If we went 12 more decimal places down the line (far down enough to be in the realm of measurement error!) we'd still see the same end result; crazy differences.

06-03-2004, 03:23 PM	#66
Carbonated_Brains Does it show up here when I type? Join Date: Oct 2003 Location: Between the smoky layers of a prosciutto sandwich! Posts: 355	My old nickname works! Sweet!

06-03-2004, 04:29 PM	#67
Slartibartfast \|-0-\| <-0-> \|-0-\| Join Date: Dec 2003 Location: New Jersey Posts: 516	Kneel. Kneel before Professor Chaos! MWAHAHAHAHA!!!!!!

06-03-2004, 04:35 PM	#68
Carbonated_Brains Does it show up here when I type? Join Date: Oct 2003 Location: Between the smoky layers of a prosciutto sandwich! Posts: 355	btw, is anybody reading this hooey besides the 5 or so people who have posted?

06-03-2004, 04:39 PM	#70
perth Strong Silent Type Join Date: Mar 2002 Location: Fort Collins, CO Posts: 1,949	I am, but I confess that I barely understand a lot of it.

06-03-2004, 05:13 PM	#73
Slartibartfast \|-0-\| <-0-> \|-0-\| Join Date: Dec 2003 Location: New Jersey Posts: 516	Don't stop just yet! Carbonated, I don't think there is a problem in 'bottling' chaos. Computers can create chaotic results from very simple formulas. The problem is in predicting chaotic behavior, not creating it. Okay, Beestie, one chart demonstration of the butterfly effect. This is the granddaddy example from Lorenz himself who stumbled onto these results. Lorenz had a weather simulator on his computer. It would spit out a row of numbers for each day. It would take maybe a minute to simulate each day. One day, Lorenz was looking at old data, and he decided he wanted to run that particular simulation over again. To save time, he decided he wanted to start the simulation 'from the middle' rather than from the beginning. This seemed easy enough, he input as his initial conditions the numbers from some day in the middle of the data he had on the printout. Now here's the catch - the data on the printout were printed with three decimal places. The computer really kept track of things up to six decimals. In other words, the numbers that Lorenz input were just a tiny tiny bit off from what they had been in with the simulation first time around. But this tiny difference shouldn't make a difference, right? But these were his results: Time is the X axis, and the results from both simulations are plotted on the same graph. Notice how on the left, the beginning of the simulation, there is one solid line. That means both simulations were spitting out almost identical results. But now look as how the two lines diverge over time. By the end of the graph, the two lines are totally apart. That means that with the weather simulation Lorenz was using, tiny differences in initial conditions will result in totally different outcomes. Maybe not at first, but after some time.

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