Chaos theory/Butterfly Effect

[obviously still struggling with this some]
Basically as you zoom in on a non-chaotic funtion it essentially achieves linearity. (at infinte zoom)
With a chaotic function if you zoom in more it does not change it's random unpredictable qualities? Is that like (if i remember right) brownian motion? As you zoom in it is still appears just as random and unpredictable as it did at your original view scale.
Much like the mandelbrot (sp?) set. As you zoom in you find areas that are almost impossible to distinguish from the more distant view (original scale)

Question does a chaotic funtion have to be a 'funtion' in the strict definition of function. "for every y there is only one x"
Can it be discontinuous? Can it be undefined? Should it be called an equation?

Are they generally speaking of a single funtion? Would it possibly be a complicated aggregate function?

Isn't the reason it comes up is because people try to create an encapsulating wrapper function that gives a broad out line of a complex system? But Chaos theory sort of says this, in some systems, is impossible?

Is it saying that at any scale the function is undefinable, so in a sense scale almost doesn't matter? Which would make the butterfly causing the tornado possibly fit.

[It is a real mind-bender if it is true.]

I will see if i can understand and implement what you are speaking of in the spreadsheet.
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letting my mind run a bit, for whatever it's worth.

In a universe a butterfly flaps it's wings, causing a tornado. The butterfly sees that it has caused the tornado. Decides to cause another tornado, flaps it's wings, only this time there is no tornado, leaving a frustrated Butterfly.
[footnote: beware of very determined, VERY frustrated butterflies, especially if they are carnivorous or bloodsucking or just given to malicious violence.]

What it speaks to is cause and effect. We are trained by our experiences in the Universe to expect that first you have cause, then you have effect. Now imagine an 'S' curve structure where the effect comes prior in time to the effect. This would be in the order of someone doing actual time travel within the same universe [not just parallel Universe jumping]. A future action is taken, someone decides to and travels into the past, they do something in that past that causes events to unfold from that point into subsequent events in that universe. This creates the question of inverted cause and effect making it effect then cause. Keep in mind there might be natural time 'S' curves that are not this elaborated.

It raises the original question again did the butterfly infact cause the tornado? This also is closely associated with the fundamental question of fate and will. Are we something that extends in however a minute quantity beyond the Universe [or at least some levels of the Universe] so that we can have some kind of 'free will' or are we so 'of the universe'that we are simply at the mercy of ongoing events?

I guess being humans that create all kinds of shelters, physical, economic, mental, we probably have more chance at experiencing some kind of 'free will' state. This would ofcourse be personal from individual to individual and be temporally limited for different spans of time. [sometimes, maybe always, this would be illusitory, but, functionally speaking, might for some limited time period operate as a free will.]

sorry, this is sounding pretty dribbly. When you get old your mind wanders.

Maybe we have to loosen our attachment to cause and effect and always start by initially descibing events as disassociated. The very nature of our brains is making connections, maybe we have to slow the tendency to too quickly make those connections. If nothing else i think it would allow us to see things in finer and more precise detail. Sort of like softening ones focus on the world [visual is a good place to start]. Instead of focusing on say a letter, word or the ideas being discussed, relax ones mind and see peripherally as well as that which is in front of us, softly all at once. It allows things to drift to you slowly, instead of reaching so aggressively/desperately with ones own mind. Obviously there would probably be better times and places for this than others.

anyway I'll go play in my spreadsheet.
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If it was MOTHRA it could happen !

www.kensforce.com...

www.stomptokyo.com...

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sisonek, you mean a continuously nested function where the output of the function is used as the next input?(recording each round of output)
In the manner of a pseudo-random number generator. But trying to find values that don't go asymtotic or lock onto a single value?
f(f(f(f(f(f(f(x))))))) ad nauseum.

I tried .49 in my spreadsheet with nesting and came up with a jiggy jaggy chart that fell between 0 and 1.
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13-Sept-2004
Preliminary Personal Explorations of Chaos Math

With the example presented by sisonek: 4x(1-x)
The bounding of the function was [0,1]
with flat/dead values of 0, 0.25, 0.5, 0.75, 1
0, 0.5, 1 ->(degenerated to line at) 0
0.25, 0.75 ->(degenerated to line at) 0.75

predictive pivot[axis] line was found at 0.75 as follows:
1. any current point below 0.75 is/was followed by a point greater than the current point
2. any current point above 0.75 is/was followed by a point below 0.75
Therefore slope to the next point is always determinable
and a minimum magnitude change is certain for all points above 0.75

Other Chaos functions I have found [leading to a series of Chaos functions]

1/0.1481472 * x ( 1 - x^0.5 ) with a pivot line at approximately 0.725653195
1/0.71579404 * x ( 1 - x^0.3333333 ) with a pivot line at approximately 0.71579404

general form:
1/(maximum(x-x^(1+(1/n))) * x ( 1 - x^(1/n))

I am guessing that instead of only 1/n for a fractional power of x it could be m/n where, 0 < m/n < 1

I haven't looked into any explanation or proof of my proposal about the pivot line
additionally I have not explored what flat/dead values for subsequent found chaos functions

I'm guessing this has probably been done somewhere before
on the off chance this is initial original work, this can I suppose be considered first presentation.

Since noting the actions of the logistical function about the .75 pivot line I wondered if there might be an uneven distribution of points. Because, i rationalized, there was always only one point in the function above the .75 line before it went below it perhaps there was a slightly lower statistical distribution of points above the .75 line, instead of what i would expect of 25% of the time.

Running some sample values through my spread sheet and doing a tally on them I found that without exception (on the samples i ran), more than 30% of time It was above the .75 line. And that an overall average time spent above the .75 line was right at 1/3rd of the time instead of 1/4th of the time which one would expect for a completely even distribution.

sample data:

0.3424837
0.3542484
0.3257902
0.3124183
0.3163399
0.3464052

data average: 0.3329476

*I think i can hear the Earth creaking on it's axis*
*or is that groaning?*

have found another type of what appears to be a chaos function.

a * x ( 1 - x^2 )

So i would speculate that form a * x ( 1 - x^(m/n)) will work where 0 < m/n

with this [x to a power greater than 1] it is not clear to me how to determine the a coefficient

In a * x ( 1 - x^2 ) I plugged in values at .1 intervals from 2 to 3 (why this range of coefficients worked is not apparent to me)

a < 2 ; goes linear
a = 2 - 2.2 ; rythmic bipolar oscillations
a = 2.3 ; complex rythym with whispers of chaotic imperfections
a = 2.4 ; goes chaotic, but range is limited (less than 0 to 1)
a = 2.5 ; chaotic, with increasing range
a = 2.6 ; chaotic span above zero alternating with a chaotic span below zero
a = 2.7 ; shortly stablizes to a neg-pos(x-axis) spanning rythmic
a = 2.8 - 2.9 ; neg-pos(x-axis) spanning chaos with some bits of just pos or neg
a = 3.0005 - 3001 ; somewhere in this range it goes into an 'L' shaped asymtotic

Now if the ATS administration would answer my u2u and explain why after purchasing enough download points i still can't bring in a .bmp under a 100kb I could possibly post some of these exceedingly exciting graphs.
UPDATE 22-9-04 : The entire set of values is MISLEADING the actual a*b=p for m/n=2 is 2.5980762 , I must have had the b value at something like 0.866 , which would make the above a values work

[edit on 22-9-2004 by slank]

Ah, Chaos Theory, butterfly effect, causaility without violations. The world at one point consisted of diseparate tribes scavaging for resources. The only cause and effect flaps of the butterfly wings that did occur at that time was when you spilled the milk on the ground and it effected everyone around you in a small microcosom which resulted in no cake tonight. Due to the new interconnectedness of everything, only through increased population, lost production in one part of the world effects the other. As the popuplation and technology increases, the chaotic, yet patterned events in the world is dictated by sometimes, the most suttle influences... Pluck to hard at the violin string and it produces harsh unrememberal music that is isolated to the moment. Play soft, relaxing and beautiful strokes and the effects around you seem to last forever.

[edit on 16-9-2004 by pfunkarocka]

[edit on 16-9-2004 by pfunkarocka]

ok, well, if the Butterfly Effect is just a small reality thing, then what causes the wind to blow, or other things? what if there was nothing to stirr up anything? And if is the fluttering of something like butterfly's wings that stirr's up the wind, then what makes winds on other planets?

I used the analogy of a set of lined up dominoes as the set up for the butterfly effect. I think air molecules can be thought of as dominoes that are always on edge. They spin, move (vector), and have other ancillary motions, but never 'lie down' the way dominoes would. With the possible exception of chemical reaction (mostly oxygen, nitrogen is virtually inert) they are always in play. They are in totally random positions and motions and leave open the possibility for some weird unpredictability. That combined with the vast amounts of data that can (should?) be used, in terms of humidity, microcurrents, solar heating, etc. leaves the weather something that will probably not be very predictable for a very long time.

And on a personal note, i think something may be lost when people can finally reliably predict/control(?) the weather. It's very uncontrolableness is part of it's magic. It is almost the essence of wildness. We are losing so much so fast already on the ground and in the water.

edit grammar

[edit on 17-9-2004 by slank]

in playing further with the basic logistic funtion ax(1-x^(m/n)) i replaced the 1 with b to have ax(b-x^(m/n)) and found that the product of a * b that must remain constant for optimal and presumably infinitely sustained chaotic properties of the function. For the baseline logistic [m/n=1] that product is 4, a*b=4.
examples:
a * b = product
4.0 * 1.0 = 4
2.0 * 2.0 = 4
0.8 * 5.0 = 4
8.0 * 0.5 = 4

in these examples and others it appears quite consistent and the difference in the b value establishes the range of the chaotic function.
UPDATE: 17-9-04, It would appear that the simple product rule is the case where m/n is not equal to one. I believe there might be a more complex form of the product rule for cases where m/n not equal one. Oh Well, back to the drawing board . . .
edit for clarity and corrected content

[edit on 17-9-2004 by slank]

I don't understand why it is called Chaos theory because it seems quite logical and structured to me. It is the the calculation of all possible factors evolving through time.

Update 18,9,04
The product rule actually holds up quite well for m/n = 2 ; a*b=3 (actually used 2.9999999999 for extreme values) It continued with ratios as extreme as 0.03 * 100 = 3
So after many hours playing with plugging the derivative back in inverting etc. I found that i kept having to make major adustments [read it wasn't really working], so finally i took the values i ended up hacking for a and multiplying times my b value and that gave me something like 3.0000363 and seemed to work. As i got close to the 0.05-0.03 range i had to adjust the value to 2.9999999999 . At 0.02? It simply blew up. That could be a problem with the product rule or simply the numerical accuracy.

I guess i will try hacking m/n = 3
UPDATE 22-9-04 : The above value is just plain incorrect. The a*b=p value p should have been 2.5980762 for m/n = 2 ; I must have inadvertantly had the b value set at about 0.866 which would have made 3.0 the a value.

[edit on 22-9-2004 by slank]

To get the a*b product number I set the a and b values both to one take the derivative of the function, b-(m/n+1)x^(m/n), set equal to zero (to get local maximum), x=(b/(m/n+1))^(n/m), plug that result in the original function to get the parabolic's max value, invert [1/max_value], and that is the product value, call it p. a*b=p, note:lock in p as a fixed number.
Now in my spread sheet i can play with the b value setting the a value to p/b.

I notice that as i change the b value to very low numbers it is subject to blowing up, whereas if i change the b value to the corresponding high numbers the system seems to be not subject to the same breakdown, tested to values as high as one trillion. It would seem the system is very subject/sensitive to changes in values of b and is also probably more vulnerable to discrete math errors.

I suspect getting the p value very accurately calculated would help with instabilities with low b values as well as using deeper floating point values.

results for:
m/n = 3 : a*b = p = 2.1165347 , blew up @ b

Why is it just the flap of a butterfly can create a disaster thousands of miles away? Many animals have wings so would the "Butterfly Effect" relate to them aswell?

I'm no expert, so take my comments with a grain of salt, but yes there should be any number of inputs [read astronomical] to the entire system of weather, what it implies is that it is a very sensitive system. So much so that things do not happen in the mechanistic ways our minds are conditioned to think. It is a bit more like trying to grab a handful of jello than a ball or something more structurally substantial. Only more so.
Have you ever watched smoke? Think of something like that.
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I believe i know now what was blowing up the a*b=p rule for low values of b,
There is also a seed range for a given b value
Through hacking i have a good approximate formula for seed range that will not upset the delicate balance of a chaotic function for a single m/n value.

For m/n = 2 The acceptable seed range is:
sqrt(1.3848967*b) to zero (not inclusive)
((1.3848967*b)^0.5 , 0 )

I have no idea what significance the 1.3848967 number means or is from.
Anyone with any cogent possiblities feel free to make suggestions.

Additionally this value defines the output range and therefore the input range, which in a chaotic function must be equal, because each iteration of the process takes the output range and feeds it into the input domain.

Next step: hack other m/n values and see what the formula is there and see if i can get a master formula for the general case of a chaotic function.

note: this was done at the b*a=p and not for b*a < p which might have some effect , or not.
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for m/n = 3 The acceptable seed range is:
(( 0.9999988*b)^0.33333333 , 0 ) OR (( 0.9999988)^(n/m) , 0 )

So I think the general case should be some particular constant for each (m/n), say c, times b raised to the n/m power.
((c*b)^(n/m) , 0 )

again, no clue where the constant is coming from
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If weather is created by movement of particles (sub-atomic and all) and it has to have movement of something to cause wind, rain, ect... Also, wouldn't that kind of help prove some theory's like the "Big Bang" theory, and other theory's that state that there is life on other planets? Because, as you all may know, there are winds, solar storms, dust storms and things on other planets, galaxy's and throughout the universe. Without some kind of "Big Bang" there would'nt be spinning planets, or rotating solar systems at all. But im not totally discrediting that God made the universe, maybe God created not humans, or planets, or even the "Big Bang", Maybe God just accidentally started a "Butterfly Effect"

General equation of the form ax(b-x^(m/n))

To get p, the product value of a*b that optimizes chaos we,
set a=1, b=1
Set the derivative equal to zero, 1-(m/n+1)x^(m/n)=0 this gives the function's x component of the max value point
giving (1/(m/n+1))^(n/m)
plug this value into the main function and that gives you the max height of the parabola.
Invert 1/max_height and you get the p value and a*b=p is the optimal chaotic action of the function.

If a*b > p , then it goes asymtotic for all but some specific seed values
If a*b < p , it eventually converges towards a particular value as long as the seed is within a specified domain

p & c values for some integer m/n values (note: for c at a*b=p)
m/n . . . . . a*b=p. . . . . . . . c
m/n=1 . . 4.0 . . . . . . . . . 0.99999999
m/n=2 . . 2.5980762 . . . 1.3848967
m/n=3 . . 2.1165347 . . . 0.9999988
m/n=4 . . 1.869186 . . . . 1.5349922


To retrieve the limits on the seed value, hack a optimal/max seed value then
(max_seed^(m/n)/b) = c
then the acceptable seed domain changes as b changes (b*c)^(n/m)= domain's max_seed. (note: a is taken as p/b)

Choosing a & b such that a*b < p changes the acceptable seed domain

to get a*b < p if i lower b it lowers the top input domain, if i lower a it raises the top input domain. so obviously there is no direct relationship from the top seed value to p

There is some delicate balance between a, b and seed value.

I would think there must be some comprehensive relationship between a,b,c,p and the maximum seed value.

Clyde Crashcup strikes again! Need to do more work.
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