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.