Definition:Feigenbaum Constants/First

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Definition

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

$x_{i + 1} = f \left({x_i}\right)$

where $f \left({x}\right)$ is a function parameterized by the bifurcation parameter $a$.

It is given by the limit:

$\displaystyle \delta = \lim_{n \mathop \to \infty} \dfrac{a_ {n - 1} - a_{n - 2} } {a_n - a_{n - 1} } = 4 \cdotp 66920 \, 16091 \, 02990 \, 67185 \, 32038 \, 20466 \, 20161 \, 72 \ldots$

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


Its precise value appears to be a topic of research, as various resources quote it differently from the above, including the David Wells $1997$ source work Curious and Interesting Numbers, 2nd ed..

This sequence is A006890 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Also known as

The first Feigenbaum constant is also referred to as merely the Feigenbaum constant or Feigenbaum number.

As it was the first to be noticed and investigated, it was inevitably named before the second Feigenbaum constant was identified.


Some sources give it as the Feigenbaum bifurcation velocity.


Also see


Source of Name

This entry was named for Mitchell Jay Feigenbaum.


Sources