Definition:Feigenbaum Constants

From ProofWiki
Jump to navigation Jump to search


The Feigenbaum constants are a pair of real constants which arise in bifurcation theory.

First Feigenbaum Constant

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..

Second Feigenbaum Constant

The secondFeigenbaum constant $\alpha$ is the ratio between the width of a tine and the width of one of its two subtines (except the tine closest to the fold).

A negative sign is applied to $\alpha$ when the ratio between the lower subtine and the width of the tine is measured.

Its approximate value is given by:

$\alpha \approx 2 \cdotp 50290 \, 78750 \, 95892 \, 82228 \, 39028 \, 73218 \, 21578 \cdots$

Source of Name

This entry was named for Mitchell Jay Feigenbaum.