Chaos theory/Butterfly Effect

I dont know if anyone has had a post on this before, and i also dont know if this fits a particular forum, so, here goes,

first off, the movie, butterfly effect, and also the actual theory, do you guy's support the theory, or what? personally, i do, and i actually do think that a butterfly's flap of its wings could cause a typhoon or tornado half way around the world, and if this is true, wouldn't it also go along with the chaos theory? cause if one thing goes against the grain of existance (for example), it would cause a chain reaction that would breed total chaotic situations. and i also suggest (if you havent before,) watch the movieSLC Punk because it gives good points supporting the chaos theory. (its also a good movie if you want to see what punk was like back in the 80's in Salt Lake City, Utah)

"Butterfly Effect" was a good movie and yes I do believe that little things happening can cause major occurences. I'm not an expert or anything on chaos theory but it sounds fascinateing.
But does that mean the butterfly's wings have to disturb the air ONLY during the" right conditions" for a tornado to occur?
What I mean is the flap does'nt cause a tornado everytime right? Only sometimes when certain conditions are around?

When I say "flap" though, I mean any"occurence " and when I say "tornado" I mean any "result". I'm not saying that all butterfly wings cause tornados. LOL.

Originally posted by elaine
What I mean is the flap does'nt cause a tornado everytime right? Only sometimes when certain conditions are around?

Right...

In a very small nutshell, chaos theory says that even the smallest and most random thing that occurs has an effect on everything and within these many many series of events, the random acts of chaos create a systematic order....chaos theory is used to control many traffic lights b/c although at first glance traffic may seem disorderly and unorganized, there is a pattern...just like the riplles in a pond - concentric circles that have equal distance that travel the length of a pond all b/c one stone was thrown

The butterfly effect is just one of the examples people use to get chaos theory across to people to prove that even the smallest action can create a large reaction given the appropriate amount of time and support....

hmm.. good question. i would think that it would cause an occurance every time, just a different one, each time. and i was also thinking that this also could go with the time travel theory of donnie darko . cause as you can see in the movie, his actions of going back in time to save gretchen causes a "Butterfly Effect", and so he dies, and never meets gretchen and never caused the end of the world and frank, appearing to donnie in that life-line is also causing a chaos effect, which is destroying existance as we know it. See what i mean?

The only way a butterflies wings could start a tornado is if it is set up like a bunch of lined up dominoes. In which case the chain reaction might have been started at any of several places and the butterfly happened to be in the right place at the right time to tip a delicate balance to a likely course.

It is the proverbial 'straw' that breaks the camels back. But it only happens because there are already many other straws on the camel's back.

The only way a butterflies wings could have more than it's innate energy release is if it's wings have some extension into the (spacial) 4th dimension, in which case you could balance an infinite number of 3D universes against them.

Chaos theory throws up a smoke screen about unpredictability. The fact is there are simply lots [too many] of unseen/unrecorded/unfactorable elements to calculate. To try to cast it in some 'mystic' way is stupid in my opinion. It is because you don't have the information on each of an astronomical number of air molecules, EM radiation, etc. Multiple sources of input to the system [EM radiation, molecular vectors of movement, internal actions of atoms within molecules, etc. mean you have some N dimensional input (mathematically speaking) which gets more difficult to calculate as the numbers of dimensions increases.

The energy that causes the tornado is already in the system. What confuses people is the fact that it is stored so complexly [diffusely, fractally] within the system that it SEEMs to appear out of almost nothing, but the root of the tornado was/is infact there whether we see [detect] it or not.

It has to do with complex storage of energy. If you balance a universe on a single quark, it is not ABSOLUTELY impossible to keep it balanced, but is SO improbable that sensibly one should probably ignore the possibility.

The butterfly may be the NECESSARY initializer in one case out of 3 trillion, but in the other 2trillion,999 billion, 999 million, 999, thousand, 999 cases the tornado is going to happen anyway, with or without the butterfly.
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Originally posted by slank
The only way a butterflies wings could start a tornado is if it is set up like a bunch of lined up dominoes. In which case the chain reaction might have been started at any of several places and the butterfly happened to be in the right place at the right time to tip a delicate balance to a likely course.

But that's exactly what chaos is...so stop right there....

Yes but it is MISLEADING. It makes it sound like a brick house is ready to fall ALL-the-time and at anytime.

Butterfly wing flap + astronomical number of preconditions = tornado

In 1 case in 3 trillion

astronomical number of preconditions with or without butterfly wing flap = tornado

In 2,999,999,999,999 cases out of 3 trillion

It's like using Einstein relativity calculations for everyday physics. Newtonian will cover virtually all everyday events.

It is just silly.

It is trying to paper over our inability to predict hugely complex systems with some kind of MYSTIC explanation.

*I hear the theme from the Twilight Zone playing softly in the background*
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Chaos theory's a fun area of math because the basics are easy to play around with yourself: unlike a lot of math-beyond-the-basics you really only need to be able to solve some algebra equations to start learning about it (although knowing some calculus doesn't hurt, either)...although, if you want to find good books on chaos theory, look under "dynamical systems," because that's what most of the mathematicians whose work gets called "chaos theory" actually call what they do.


The butterfly effect's a nice metaphor. Here's a fun way to spend an afternoon that'll give you a good feel for what the butterfly effect's a metaphor for (if you're still in school you'll probably have a ti-82/83 to do this on; a graphing calculator / nice calculator makes this a lot more fun -- and makes parts of this much less work):

the function f(x) = 4x(1-x) has a nice "hump" in the range between 0 and 1...and in fact, any value for x that's between zero and one will come out of this function and still be between 0 and 1. So, here's some questions you can try to answer:

stuff about periodic points:

a) are there any values of x for which f doesn't change the value of x? ie, are there any places for which f(x) = x? these are called "fixed points" because the value of x is "fixed" under application of f
b) what about values of x for which f(x) isn't x, but f(f(x)) = x? ie, are there points that have an "orbit" of length 2 -- they return to their initial value after you use f twice? these are called periodic points of length 2
c) what about points with orbits of length 3/4/5....?

if you're still interested, try figuring out where the periodic points are located between 0 and 1...try marking them on your graph. one way of interpreting the butterfly effect is that most of the time it's impossible to predict what's going to happen to a point without actually running the calculations. For example, if a value of x is a periodic point, it's easy to guess where it's going to be after each application of f, because it only ever visits a finite number of places....for non-periodic points, however, you can't just take that shortcut, but actually have to do the calculation to figure out where it is at each step. The butterfly effect could be interpreted as the difficulty of knowing whether a particular point is periodic or not -- if x is a periodic point, x + 0.0001 probably isn't -- and so for real systems in the real world -- where absolutely precise measurements are impossible -- our ability to predict and model them is going to be limited a lot of the time.

This was a philosophically new result, in a sesne -- in simple physical systems there is usually a way to correct for error, and small errors in your measurements lead to small errors in your predicted outcomes -- because for more complicated physical systems it's no longer true that your prediction accuracy will necessrily improve even as your measurement accuracy does.

If you're interested in learning some "chaos theory" I'd reccomend

www.amazon.com...=1094323862/sr=8-1/ref=pd_ka_1/002-9460616-0771216?v=glance&s=books&n=507846

It's pretty much the standard intro to dynamical systems and "chaos theory" and the first chapter (out of 3) only needs high-school level calculus to work through (although a course in basic real analysis will help your understanding, of course)...it shouldn't be hard to get a used copy at your local university bookstore or online, and any university library will probably have it.

slank: we crossed in passing it looks like. the analogy between newtonian / relativistic gravity isn't accurate here, though:

the point of "chaos theory" is that there are complicated systems for which your prediction error doesn't decrease as your measurement error does, ie:

if measurement error = \delta (can we get latex as a feature request on this board)
and
if prediction error = \epsilon

then

when \delta = 0, \epsilon = 0
but if we take \epsilon as a function of \delta
lim \epsilon(\delta) as \delta goes to 0 does not exist...ie, the error is no longer a decreasing continuous function of \delta.

most large physical systems like the examples you cited aren't chaotic in that sense -- which is part of why i think the "butterfly flapping it's wings in tokyo" sounds cool but is a horribly inaccurate analogy -- because your accuracy improves more or less monotonely as you add variables and information about the system you're trying to predict. a chaotic system would be one in which you could no longer count on your prediction error decreasing as your measurement error did.

and yeah, it bugs me too that people get all mystic about chaotic systems -- it reminds me of people citing quantum mechanics they don't understand to make their points -- but there's at least some justification to the attitude, b/c accurate long term prediction of a chaotic system's usually impossible (except on small subsets of the input variables).

You are sort of catching my interest. I'm not real advanced, but do like to dabble.
Question: Is there any connection between chaos theory and the butterfly effect?

Did you ever play with those two sheet of paper toys, like with trees with cutout rectangles inbetween and then you could shift the pieces of paper's position to one another to create two different scenes?

If you take it that the Universe is always in a state between instability and balance, I suppose you could see it sort of like a continuous function with local stability valleys separated by slopes between valleys. A lot of chemical reactions work that way, getting enough energy to go from one quasi stable state to another quasi stable state.

With a bowling ball you can use gross descriptions/generalizations to describe it. It is of couse actually a complex interaction of a huge number of atoms and molecules. But it is something to do with the connective nature of the atoms/molecules to create the system/ball that allows the reasonably accurate description using gross generalizations.

What I am thinking is we depend for our lives and the ways we live on those connective forces. Without those It would be virtually impossible for us to deal with the world. That is why we can use abstractions to deal with the Universe.

I'm reading what you are saying, its a little out of my depth and will take some time and attention for me to see if i comprehend it.

I think I remember reading that for some systems mathematicians could create predictive models of future events, but the probability of accuracy was a function on top of it. Like at 10:00am tommorrow event x has a 40% chance of happening and at 10:34am tommorrow event y has a 70% chance of happening, with perhaps mutual independence of x and y. And some periods of time were quite unpredictable.

Let me play with this some.

You are sort of catching my interest. I'm not real advanced, but do like to dabble.
Question: Is there any connection between chaos theory and the butterfly effect?

The "butterfly effect" i think was named as such in a popularization of chaos theory...it doesn't really have any formal definition, but in a general sense it means "sensitive dependence on initial conditions", or is at least an analogy to understand sensitive dependence on initial conditions. Here's my best try at an explanation:

Say we're in the army and we're doing ballistics calculations -- trying to figure out how to aim our cannon and how much powder to put in our cannon, say, like in that "scorched earth" videogame if you're familiar. Let's also say we're on a totally flat field, so we can ignore the shape of the ground, and we can also (for now) ignore the wind. Say we set as our "default" configuration a cannon angle of 45 degrees and 100 grams of powder, and that the cannonball lands 100 feet away if we fire it with those initial conditions.

If we experiment a bit, we'll see that changing the angle a bit changes the location a bit -- say, something like

42 degrees -> 94 ft
43 degrees -> 98 ft
44 degrees -> 99 ft
45 degrees -> 100 ft
46 degrees -> 99 ft
47 degrees -> 98 ft
48 degrees -> 94 ft,

etc., and similarly changing the amount of powder (but keeping the angle at 45) makes something like the following table:

98 grams -> 95 ft
99 grams -> 98 ft
100 grams -> 100 ft
101 grams -> 102 ft
102 grams -> 105 ft

And now let's say we have a tub of exactly 100 grams of powder and we want to predict where our soldiers are going to be able to hit. Our soldiers are good, but they're not perfect, so they could be off by a degree either way when they aim...so if they think they've set the angle to 45, it could be either 44 or 46 (it could also be 45, we're just considering the total range it could fall under). That level of accuracy would mean that we'd have to predict that the cannonball's going to travel somewhere between 99 and 100 ft -- just look at the table -- so with one degree of possible error in our measurement of the angle we wind up with at most 1 ft of possible error in our calculation.

That's not that bad for battlefield work -- although we'd be pretty sorry soldiers if we're having to calculate how to hit something 100ft away -- but what if we're trying to hit something tiny, say only an inch across (and let's say our cannonball's small, too, so we can't just get close and assume the ball's big enough to crush it). With our toy cannon the solution's pretty obvious -- we can just train the troops so they're only, say, going to get the angle wrong by at most 0.001 of a degree (or however precise they need to be) we can rest assured the cannonball's going to fall within an inch of where we need it to go.

So far so good -- this toy cannon is pretty intuitive, and as your accuracy of setup increases so does the accuracy of your predictions -- but what "chaos theory" studies is in a nutshell systems where this intuition breaks down....in a chaotic system, there's no longer a nice relationship between your measurement accuracy and your prediction accuracy. In ordinary life -- and for most simple systems and objects -- the usual intuition holds, and if your ability to set things up accurately (or measure your variables accurately) improves, your ability to predict what happens will improve as well; in a chaotic system your ability to predict what will happen won't necessarily get any better with better information -- sometimes it does, but most often it doesn't.

As an example, take the function

f(x) = 4x(1-x)

from my earlier post; it's usually called the logistic function or logistic map, because it's a pretty simple function and you can use it as a toy model of much more complex chaotic systems. For points from 0 to 1 (i'll write [0,1] to mean all points between 0 and 1, inclusive from here on out) it's very easy to tell where the point goes to under the map:

if x is my point, f takes x to 4x(1-x), which is just the definition of a function. Similarly, if I apply f twice, (call this f2 for here), f2(x) = f(x(f(x)) = 4(4x(1-x)(1 -4x(1-x)), which while more complicated is still pretty easy to calculate, and given any particular x in [0,1] all we have to do is run it through that expression to find f2(x).

So, in theory our ability to calculate fn(x) for any x in [0,1] and any n a positive integer (1,2,3,...) is straightforward, and so in theory our predictive ability for this system is perfect -- if we want to know fn(x) all we have to do is calculate it. But, what about the sequence

(x,f1(x),f2(x),f3(x),....)

which for a particular x you can think of as the "trajectory" of x under f. How well can we predict the trajectories of x?

Again in theory we should be perfectly able to do so -- just run the calculations -- but let's take a look at an example (here's where a calculator comes in handy):

if x = 1/2, then f(x) = 1/2, so the whole "trajectory" is (1/2,1/2,....), but if x = 1/2 + 0.0001, say, x is going to go all over the place....try a few examples if you've got a nice calculator, but otherwise take my word for it; 0.50001 isn't going to have a trajectory that looks at all like x = 0.5.

If this was a cannon (or a cannon-like system) you could find a small enough number - call it delta -- so that as long as x was within delta of 0.5 (ie, if delta = 0.0000001 then as long as x is in [0.499999999,0.50000001] (i might have messed up the number of digits there, but it doesn't matter)) then the trajectory would be "close enough" to the trajectory for x = 0.5....but this is actually impossible for the logistic map; you may be able to find individual points nearby to x = 0.5 such that their trajectory is similar to the one for x = 0.5, but you won't ever be able to find a range so small that every point within that range behaves similarly to x = 0.5, if you see what I'm saying.

That's what it means to be chaotic, in a nutshell: although in principal one could run all the necessary calculations to find out what's going to happen, in the real world you run into two problems:

a) your measurements won't ever be perfect
b) your ability to do calculations is limited

Non-chaotic systems are those for which a) isn't that big a problem, because it's possible to figure out "how good" your measurements have to be and as long as they're that good or better you know how good your predictions will be; in a chaotic system no matter how accurate you are you're never accurate enough, because even a tiny error in measurement can lead to a huge difference in outcome.

b) is also a problem: obviously for x = 0.5 there's a pattern to its trajectory (because for x = 0.5 f(x) = 0.5 and so it's always the same) and so if someone asked you "where will x =0.5 be after 25 applications of f" you can take a shortcut and just say " it'll still be 0.5", but for x = 0.500001 you'll have to actually do the calculation yourself. For our simple function f this isn't that big a deal, but a lot of very complicated systems are only computable if you can figure out some simplifications or shortcuts. For chaotic systems it's not generally possible to find accurate approximations to most of the system, because each simplification builds in error which causes part a) to blow up on you.

That's basically the butterfly effect right there -- for systems where even a small initial error can lead to enormous differences in outcome, the butterfly flapping its wings is supposed to be the tiny error (ie, the 0.000001 in our example) that can lead to a dramatic difference in outcome (say, the difference between rain and sunshine).


What this means in general is that the ability to predict the behavior of complicated systems accurately is surprisingly limited if one wishes to predict for the long-term, even when the short-term can be predicted very accurately indeed. You can see this with the "trajectories" from the logistic map above:

if you limit yourself to a small window -- say, x,f1,....f5 or so -- you can see that if you pick two numbers really close to each other (like, 0.50001, 0.500011, say) in that window their behavior's pretty similar, but the further outside that window you get the more the two paths diverge (which is what i'm guessing is what happens in the movie that started this thread -- slightly different beginnings leading to vastly different endings in time). So, even though at each step of the way you can predict with perfect accuracy what comes next, if you want to step back a bit it's hard to predict things accurately in the long term, which is what I think you're getting at with

think I remember reading that for some systems mathematicians could create predictive models of future events, but the probability of accuracy was a function on top of it. Like at 10:00am tommorrow event x has a 40% chance of happening and at 10:34am tommorrow event y has a 70% chance of happening, with perhaps mutual independence of x and y. And some periods of time were quite unpredictable.

The basics of dynamical systems are really straightforward and accessible if you can do algebra and especially if you know basic calc...I really do recommend the book I linked to earlier if you know calculus...it's at least worth checking out at a bookstore and/or library. But the best thing to do is just to play around with this a bit...

yea sure its possible. some guy could be like at the edge of a cliff trying to commit suicide he is thinking that there is no reason to live. but then he sees a butterfly and somehow it makes him not want to kill himself anymore. later he grows up to stop a plot to blow up the world. so im sure its possible that one small event can stop a very powerfull event from happening.

Perhaps something like f(x) = sin(1/x) ?
so the limit as x->0 doesnt exist?
The closer you get to 0 the more the oscillations of f(x) accelerate.
(you may have noticed i am trying to get this the shortcut way) (little lazy here)

Another thought about your ballistics example is that it seems to assume that once you have 45 degrees as the optimal angle for 100 grams of explosives that varying the angle in conjunction with the grams of explosives won't yield unexpected results. (I realize that 45 is the optimal angle) In any event if the number of grams of explosives wasn't a monotonic funtion and angle a parabolic you could have a whole map of peaks and valleys for the combined functions. And in fact I think if you have sufficient explosive force it would blow up the base end of the cannon and therefore not expend essentially all the explosive force on the cannon ball, because much of the force would be expended breaking the cannon. So at some point grams of powder is not a completely monotonic function. ( I wonder if at some level of explosive it would shatter the cannonball as well?)

I did plug the 4x(1-x) into my spreadsheet, how would you suggest i play with it? vary the coefficient 4? alter the one in 1-x?
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I do think that they are connected, like what if one day, a guy, just a normal average schmo, gets hit by someone, like in his car, just a little knick, but it makes him a little ticked off. then later on, other stuff, little stuff, like paper jamming, then on to bigger stuff, like boss firing him, and wife divorcing him, kids hating him, then years go buy, and woops, the domino effect got too big, and BOOM! he blows up, goes to an airport and hijacks an airplane, and crashes into las angeles, killing millions of people. do you see now how i think its connected?

Perhaps something like f(x) = sin(1/x) ?
so the limit as x->0 doesnt exist?
The closer you get to 0 the more the oscillations of f(x) accelerate.
(you may have noticed i am trying to get this the shortcut way) (little lazy here)

This is a classic example of a function for which lim x->0 doesn't exist, even though the function's continuous everywhere outsite of x = 0...it's used a lot in calc books as an example of a function with a discontinuity that's not artificial looking. It's not an example of a chaotic system, per se, but it could be used to illustrate the point:

for a non-chaotic system, as the error in measurement goes to zero so should the error in prediction; ie, if we graph the size of the measurement error on the x-axis and the size of the prediction error on the y-axis (I'm using the size so both are positive), if you zoom in around (0,0) enough the graph should look something like f(x) = x, f(x) = x^2, etc., or in other words something curving smoothly in towards zero; also, for a non-chaotic system if we pick a particular size of prediction error there should be an amount of zoom in enough around (0,0) we can do such that if we zoom that much, than every value of measurement error we see should be less than the size we picked....which is one formal way of putting the assumption you picked up on here:

Another thought about your ballistics example is that it seems to assume that once you have 45 degrees as the optimal angle for 100 grams of explosives that varying the angle in conjunction with the grams of explosives won't yield unexpected results. (I realize that 45 is the optimal angle) In any event if the number of grams of explosives wasn't a monotonic funtion and angle a parabolic you could have a whole map of peaks and valleys for the combined functions.

that small changes in the parameters lead to small changes in the results. i picked the ballistics example because it's a simple system and one for which (barring accidents like blowing up your cannon) the assumption of small change in paramters = small change in outcome holds.

for a chaotic system that's not true; in a chaotic system if we graphed measurement error and prediction error like before your graph would be similar to the graph of sin(1/x): going up and down wildly, and not approaching zero as you approached zero....so sin(1/x) isn't really chaotic but its wild oscillations near zero are a good example of what chaotic functions behave like.

here's my suggestion for your spreadsheet: put f(x) = 4x(1-x) in as a function, then set up a few columns (maybe 10 or so) so that (say we're in column B) B0 is some number between 0 and 1, and then BN = f( B(N-1)), maybe all the way to B50 or so...you can then graph the columns as a line graph (with each line = one column, not one row!!!) and check out the trajectories...

I realize I didn't explain something: the point of f(x) = 4x(1-x) isn't the graph of the function itself -- it's just a little hump of a downward-facing parabola -- but instead what happens if you think of that function as an "update rule".

Basically, real physical systems can be modeled by differential equations -- which require time to be a smooth parameter -- but sometimes it's easier to study the systems by making time "discrete" and breaking it down into finite steps...so instead of just thinking of f(x) as specifying some static graph, think of it as telling you if x is here at time N, at time N+1 x is at f(x), at time N+2 x is at f(f(x)), etc...

so the function f(x) isn't at all chaotic on its own, but if you use it to calculate the "trajectories" of points over a long period of time (like, 20-40 units) lots of chaotic behavior emerges. that's why i'm suggesting you use the spreadsheet to calculate and then graph the "trajectory" of points as line graphs, because it will make the behavior more apparent.

try this:

after you've set up your columns, try making one column start @ x = 0.5, then try out x = 0.49, x = 0.51, x = 0.499, x = 0.501, etc., and see the behavior differences between those five lines...hope this helps

This was done real well on one of those Ray Bradbury Theatre short stories a while back...

There was a time travel safari vacation that people would go on. Basically, they'd go back in time, and hunt a dino that was going to die anyways.

They had a levitating walkway, to prevent any disturbance to the ground. The rule was to retrieve any stray bullets, as well as any bullets into the dino. Such things left became OOPARTS (out of place/time artifacts).

Anyhow, one of the guys got spooked, and fell off the pathway, crushing a dragonfly. Upon their return to their own time, the Time Travel Center was run by Nazi troops!!!

Hmmmmmmmmmm. I didn't see the movie as any type of theory at all.....I just saw it in relation to choices....cause and effect...things do indeed "bounce" off of each other by the choices one makes in their life....it was a lot of bouncing, caused by different things happening.....which changed the outcome

i think i actually got to see that movie, it was great, but i forgot what the name was, can you remember? that and it was also a topic of a Simpsons episode were homer went back and time, and crushed a plant, and came back to where everyone ate doughnuts all the time. did you ever see that one? it was great

P.S - I don't know who said it first, but this guy was wise for saying: "anything that can happen, will, and absolutely has to happen sometime"