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