Feigenbaum's Scaling Law for the Logistic Map
Feigenbaum's Scaling Law for the Logistic Map
Mitchell Feigenbaum's one-term parameter scaling laws for the logistic map are
-∼
λ
r
λ
F
c
1
r
δ
-∼
λ
r+1
λ
r
c
2
r
δ
where (1) is the order of the period-doubling pitchfork bifurcation, (2) is the control parameter of the logistic map, (3) is the superstable parameter value for each bifurcation order, (4) =3.569945671870944901… is a special parameter value corresponding to the Feigenbaum point of the logistic map, (5) and are constants, (6) is the Feigenbaum constant.
r
λ
λ
r
λ
F
c
1
c
2
δ=4.669201609102990671…
Superstable parameter values can be obtained using the standard Newton iterative scheme. For more information, please see the references in the "Details" section.
λ
r