Chaos is not unpredictable

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posted on Apr, 20 2005 @ 05:59 PM
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Using the standard statistical function 4*x(1-x) [considered to be a classic chaos function]

The function has predictability depending on which quadrant of values it falls in.

It makes a variable length cycle.

whenever the function creates a value between 0.25 and 0.75 the next value will be greater than 0.75, followed by a value less than 0.75, falling to the lowest value of that cycle additionally this is the only descending slope of the cycle/function.

If the value following 0.75 falls below 0.25 it (gradually) climbs till it surpasses 0.25, then it goes to something above 0.75 repeating the cycle.

It never goes from a value below 0.25 directly to a value above 0.75

Also the number of times the function falls between 0.25 and 0.75 is equal to the number of times the function hits above 0.75. (plus or minus one depending on seed value)

The 4th quadrant will be hit exactly the sum of hits in quadrants 3 & 2

from trials i would say hits in the 4th and 1st quadrants are about the same with a slightly higher proportion going to the 1st quadrant.

seed 0.4924 0.924 0.153
Q4 . . . 86 . . . 82 . . . 87
Q3 . . . 47 . . . 41 . . . 50
Q2 . . . 38 . . . 42 . . . 37
Q1 . . . 85 . . . 91 . . . 82

So an approximate distribution of hits by quadrant would be about
Q4 33%
Q3 17%
Q2 16%
Q1 34%

I refer to the [in this case] 0.75 line as the pivot line because both the value preceding and following a hit above 0.75 will be below 0.75.
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