WOLFRAM|DEMONSTRATIONS PROJECT

Estimating the Feigenbaum Constant from a One-Parameter Scaling Law

​
unimodality, z
2.
left endpoint,
r
0
​,for fitting interval:
r
0
≤ r ≤ 17
0
Mitchell J. Feigenbaum's one-parameter scaling law for one-dimensional iterative maps with
z
-unimodality is given by

λ
r+1
-
λ
r
~c
r
(δ(z))
for any
z>1
,
where1.
r≥0
is the order of the period-doubling pitchfork bifurcation;2.
λ
is the control parameter of the iterative maps;3.
λ
r
is the super-stable parameter value [1] for each bifurcation order (e.g.
λ
0
for period 1,
λ
1
for period 2,
λ
2
for period 4,
λ
3
for period 8, and so on);4.
c
is a constant;5.
δ(z)
is the Feigenbaum constant as a function of
z
[2]. For
z=2
,
δ(2)=4.669201609102990671…
.
On the left is the plot of
log|
λ
r+1
-
λ
r
|
versus
r
. For any
z>1
, the filled-in blue circles within the fitting interval fall nicely on a straight line with the slope
Δz±u≈-logδ(z)±u
, where
u
is the uncertainty, indicating the above scaling law. The uncertainty on
Δz
can be found from a standard linear regression analysis.
On the right is the plot of
δ(z)≈
|Δ(z)|
e
versus
z
. The accuracy of the estimated value of
δ(z)
can be increased by decreasing the length of the fitting interval, which is displayed in the dropdown menu button.