Feigenbaum constants
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.
History
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 constant
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 an are discrete values of a at the nth period doubling.
According to (sequence A006890 in OEIS), this number to 30 decimal places is: δ = 4.669201609102990671853203821578….
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