# Definition:Feigenbaum Constants/First

## 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

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $4 \cdotp 66920 \, 16609 \, 0 \ldots$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**Feigenbaum number**(M.J. Feigenbaum, 1979) - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Feigenbaum number**(M.J. Feigenbaum, 1979)