The Feigenbaum constant 
is a universal constant for functions approaching chaos
via period doubling. It was discovered by Feigenbaum
in 1975 (Feigenbaum 1979) while studying the fixed points of the iterated function
 |
(1)
|
and characterizes the geometric approach of the bifurcation parameter to its limiting value as the parameter 
is increased for fixed 
. The plot above is made by iterating equation (1)
with 
several hundred times for a series of discrete but closely spaced values of 
, discarding the first hundred or so
points before the iteration has settled down to its fixed points, and then plotting
the points remaining.
A similar plot that more directly shows the cycle may be constructed by plotting 
as a function of 
.
The plot above (Trott, pers. comm.) shows the resulting curves for 
, 2, and 4.
Let 
be the point at which a period 
-cycle appears, and denote the converged value by 
. Assuming geometric convergence, the difference between
this value and 
is denoted
 |
(2)
|
where 
is a constant and 
is a constant now known as the Feigenbaum constant. Solving for 
gives
 |
(3)
|
(Rasband 1990, p. 23; Briggs 1991). An additional constant 
, defined as the separation of adjacent elements of period doubled attractors
from one double to the next, has a value
 |
(4)
|
where 
is the value of the nearest cycle element to 0 in the 
cycle (Rasband 1990, p. 37; Briggs 1991).
For equation (1) with 
, the onsets of bifurcations occur at 
, 1.25, 1.368099, 1.39405, 1.399631, ..., giving convergents
to 
for 
,
2, 3, ... of 4.23374, 4.5515, 4.64617, ....
For the logistic map,
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
(OEIS A006890, A098587, and A006891; Broadhurst 1999; Wolfram 2002,
p. 920),
where 
is known as the Feigenbaum constant and 
is the associated "reduction parameter."
Briggs (1991) calculated 
to 84 digits, Briggs (1997) to 576 decimal places (of
which 344 were correct), and Broadhurst (1999) to 1018 decimal places. It is not
known if the Feigenbaum constant 
is algebraic, or if it can be expressed in terms of other
mathematical constants (Borwein and Bailey 2003, p. 53).
Briggs (1991) calculated 
to 107 digits, Briggs (1997) to 576 decimal places (of
which 346 were correct), and Broadhurst (1999) to 1018 decimal places.
The Feigenbaum constant 
and associated reduction parameter 
describe universal scaling in period
doubling for suitable families of one-dimensional maps 
with a single locally quadratic maximum. Feigenbaum (1978) investigated this scaling, and
Lanford (1982) established rigorous results for the quadratic case 
. For other critical orders, Epstein (1985, pp. 32-33)
proved the existence of solutions of the associated Feigenbaum
function equation for integer 
from 2 through 14. This existence result does not assert that
the scaling constants are independent of 
.
A negative Schwarzian derivative is a common hypothesis in the study of one-dimensional period doubling (Tabor 1989, p. 220). The Schwarzian derivative is
![D_(Schwarzian)=(f^(''')(x))/(f^'(x))-3/2[(f^('')(x))/(f^'(x))]^2,](/images/equations/FeigenbaumConstant/NumberedEquation5.svg) |
(9)
|
where 
.
Its negativity alone does not ensure universality. The scaling constants depend on
the order of the critical point (Feigenbaum 1978,
p. 25). Examples of systems exhibiting period
doubling include the Hénon map, logistic
map, Lorenz attractor, Navier-Stokes truncations,
and sine map 
.
The value of the Feigenbaum constant can be computed explicitly using functional
group renormalization theory. The universal constant also occurs in phase transitions
in physics.
The value of 
for a universal map may be approximated from functional group renormalization theory
to the zeroth order by solving
![1-alpha^(-1)=(1-alpha^(-2))/([1-alpha^(-2)(1-alpha^(-1))]^2),](/images/equations/FeigenbaumConstant/NumberedEquation6.svg) |
(10)
|
which can be rewritten as the quintic equation
 |
(11)
|
Solving numerically for the smallest real root gives 
, only 0.7% off from the actual value (Feigenbaum
1988).
For an area-preserving two-dimensional map with
 |  |  |
(12)
|
 |  |  |
(13)
|
the Feigenbaum constant is 
(Tabor 1989, p. 225).
For a function of the form (1), the Feigenbaum constant for various 
is given in the following table (Briggs 1991, Briggs et al. 1991, Finch 2003),
which updates the values in Tabor (1989, p. 225).
 |  |  |
| 3 | 5.9679687038... | 1.9276909638... |
| 4 | 7.2846862171... | 1.6903029714... |
| 5 | 8.3494991320... | 1.5557712501... |
| 6 | 9.2962468327... | 1.4677424503... |
Broadhurst (1999) considered additional Feigenbaum constants. Let 
and 
be even functions with 
and
 |  |  |
(14)
|
 |  |  |
(15)
|
and 
as large as possible. Let 
be positive numbers with
 |
(16)
|
and 
as small as possible. Also let 
be the order of the nearest singularity, with
 |
(17)
|
as 
tends to zero. The values of these constants are summarized in the following table.
