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)