Stack Diagram for 1D Box-Counting Steps
Stack Diagram for 1D Box-Counting Steps
The box-counting dimension [1–4] can be defined as
lim
ϵ0
lnN(ϵ)
ln1/ϵ
lim
r∞
lnN
r
ϵ
0
ln1
r
ϵ
0
where () is an initial box (or grid) size, is a natural number representing the box-counting step, is the box size for the scaling step, and is the number of boxes with the same size . It can be applied to any fractal, including wild fractals such as the Brownian motion. This Demonstration visualizes one-dimensional (1D) box-counting steps as a stack diagram.
ϵ
0
0<<1
ϵ
0
r
th
r
ϵ=
r
ϵ
0
th
r
N(ϵ)
ϵ
The test map used in this Demonstration is the well-known logistic map [3–7] ()=≡λ(1-), where is an iteration number, is the iterate starting from an initial condition , and is a control parameter value; has been fixed at 5.00001, and for imitating attractors, 4000 iterates are selected from to .
f
LM
x
i
x
i+1
x
i
x
i
i
x
i
th
i
x
0
λ
x
0
i=1001
5000