First Feigenbaum constant

List of numbers Irrational and suspected irrational numbers $γ$ $ζ$ (3) √2 √3 √5 $φ$ $ρ$ $δ$ _$S$ $e$ $π$ $δ$

In mathematics, specifically bifurcation theory, the Feigenbaum constants are two mathematical constants which both express ratios in a bifurcation diagram for a non-linear map. They are named after the mathematician Mitchell Feigenbaum.

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.

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

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

It is given by the limit

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

According to (sequence in the OEIS), this number to 30 decimal places is $δ$ = 4.669201609102990671853203821578….

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