Different parts of the math cannon which seem to have nothing to do with each other.....my dad's words, not mine.........

History[edit]

Feigenbaum originally related the first constant to the period-doubling bifurcations in the logistic map, but also showed it to hold for all one-dimensional maps with a single quadratic maximum. As a consequence of this generality, every chaotic system that corresponds to this description will bifurcate at the same rate. It was discovered in 1978.^[1]

The first constant[edit]

The first Feigenbaum constant is the limiting ratio of each bifurcation interval to the next between every period doubling, of a one-parameter map

${\displaystyle x_{i+1}=f(x_i),}$

where f(x) is a function parameterized by the bifurcation parameter a.

It is given by the limit^[2]

${\displaystyle \delta =\underset{n\to \mathrm{\infty }}{lim}\frac{a_{n-1}-a_{n-2}}{a_n-a_{n-1}}=4.669201609\dots ,}$

where a_n are discrete values of a at the n-th period doubling.

Here is this number to 30 decimal places (sequence A006890 in the OEIS): δ = 4.669201609102990671853203821578…