03/28/2013, 12:20 AM
consider the sequence starting with x=2 then 2+ln(2) and taking x+ln(x) at every step.
Now a brute estimate of the sequence would be x + n ln(x) where x=2 and n is the n th iterate.
However that is not accurate.
First the sequence grows approximately like x + n ln(n) ln(x) on average.
Later like x + n (ln(n) + ln(ln(n))) ln(x).
And it continues like x + n (ln(n) + ln^[2](n) + ln^[3](n)) ln(x).
The pattern is clear. However it seems to be most regular for values of x near x = 2. Fascinating.
I guess there is a simple reason for this behaviour , right ?
It seems like selfreference almost.
guess this is in many dynamical systems books but maybe not.
regards
tommy1729
Now a brute estimate of the sequence would be x + n ln(x) where x=2 and n is the n th iterate.
However that is not accurate.
First the sequence grows approximately like x + n ln(n) ln(x) on average.
Later like x + n (ln(n) + ln(ln(n))) ln(x).
And it continues like x + n (ln(n) + ln^[2](n) + ln^[3](n)) ln(x).
The pattern is clear. However it seems to be most regular for values of x near x = 2. Fascinating.
I guess there is a simple reason for this behaviour , right ?
It seems like selfreference almost.
guess this is in many dynamical systems books but maybe not.
regards
tommy1729
