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Stack Diagram for 1D Box-Counting Steps

​
box–counting step r
24
initial grid size
ϵ
0
1
3
1
2
2
3
3
4
4
5
parameter values 3.56994
1.
2.
3.23607
3.49856
3.55464
3.56666
3.56924
3.56994
3.58
3.83
3.84
3.8493
4.
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
ϵ
0
(
0<
ϵ
0
<1
) is an initial box (or grid) size,
r
is a natural number representing the
th
r
box-counting step,
ϵ=
r
ϵ
0
is the box size for the
th
r
scaling step, and
N(ϵ)
is the number of boxes with the same size
ϵ
. 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.
The test map used in this Demonstration is the well-known logistic map [3–7]
f
LM
(
x
i
)=
x
i+1
≡λ
x
i
(1-
x
i
)
, where
i
is an iteration number,
x
i
is the
th
i
iterate starting from an initial condition
x
0
, and
λ
is a control parameter value;
x
0
has been fixed at 5.00001, and for imitating attractors, 4000 iterates are selected from
i=1001
to
5000
.
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