Deterministic Chaos

6.N. Period-Doubling Scaling and the Feigenbaum Constant

Here are the ratios of the differences of the first few superstable points.

(s*1 - s*0)/(s*2 - s*1) = 4.70894301
(s*2 - s*1)/(s*3 - s*2) = 4.68051916
(s*3 - s*2)/(s*4 - s*3) = 4.66429741
(s*4 - s*3)/(s*5 - s*4) = 4.66770552
(s*5 - s*4)/(s*6 - s*5) = 4.66856419
(s*6 - s*5)/(s*7 - s*6) = 4.66915718
(s*7 - s*6)/(s*8 - s*7) = 4.66919100
(s*8 - s*7)/(s*9 - s*8) = 4.66919947
(s*9 - s*8)/(s*10 - s*9) = 4.66920113
(s*10 - s*9)/(s*11 - s*10) = 4.66920151
(s*11 - s*10)/(s*12 - s*11) = 4.66920159
(s*12 - s*11)/(s*13 - s*12) = 4.66920160
(s*13 - s*12)/(s*14 - s*13) = 4.66920161

Continuing, we would find

(s*n - s*n-1)/(s*n+1 - s*n) -> 4.6692016091029...

as n -> infinity. This number is called the Feigenbaum delta constant.

Return to Period-Doubling Scaling and the Feigenbaum Constant.