# Estimating the Feigenbaum Constant from a One-Parameter Scaling Law

Estimating the Feigenbaum Constant from a One-Parameter Scaling Law

Mitchell J. Feigenbaum's one-parameter scaling law for one-dimensional iterative maps with -unimodality is given by

z

-~c

λ

r+1

λ

r

r

(δ(z))

z>1

where1. is the order of the period-doubling pitchfork bifurcation;2. is the control parameter of the iterative maps;3. is the super-stable parameter value [1] for each bifurcation order (e.g. for period 1, for period 2, for period 4, for period 8, and so on);4. is a constant;5. is the Feigenbaum constant as a function of [2]. For , .

r≥0

λ

λ

r

λ

0

λ

1

λ

2

λ

3

c

δ(z)

z

z=2

δ(2)=4.669201609102990671…

On the left is the plot of versus . For any , the filled-in blue circles within the fitting interval fall nicely on a straight line with the slope , where is the uncertainty, indicating the above scaling law. The uncertainty on can be found from a standard linear regression analysis.

log|-|

λ

r+1

λ

r

r

z>1

Δz±u≈-logδ(z)±u

u

Δz

On the right is the plot of versus . The accuracy of the estimated value of can be increased by decreasing the length of the fitting interval, which is displayed in the dropdown menu button.

δ(z)≈

|Δ(z)|

e

z

δ(z)