# Definition:Feigenbaum Constants

## Definition

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.