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:
- $x_{i + 1} = \map f {x_i}$
where $\map f x$ is a function parameterized by the bifurcation parameter $a$.
It is given by the limit:
- $\ds \delta = \lim_{n \mathop \to \infty} \dfrac{a_ {n - 1} - a_{n - 2} } {a_n - a_{n - 1} } = 4 \cdotp 66920 \, 16091 \, 02990 \, 67185 \, 32038 \, 20466 \, 20161 \, 72 \ldots$
where $a_n$ are discrete values of $a$ at the $n$th period doubling.
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.
Historical Note
The first Feigenbaum constant was the result of serendipitous observations by Mitchell Jay Feigenbaum in $1979$.
Its precise value may still be a topic of research.
Different sources quote it differently.
Watch this space.
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): Feigenbaum number (M.J. Feigenbaum, 1979)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Feigenbaum number (M.J. Feigenbaum, 1979)