Box-Counting Algorithm of the Hénon Map
Box-Counting Algorithm of the Hénon Map
This Demonstration shows the orbit diagram (OD), the box counting diagram (BC), and the dimension estimation plot (DE) of the Hénon map [1–4]. It is known that the chaotic attractors in the Hénon map are neither area filling (dimension 2) nor a simple curve (of dimension 1) [4]. Therefore the dimensions of these complicated geometries must be non-integer values between 1 and 2, and the chaotic attractors are then called fractals or strange attractors [4, 5]. The capacity or box-counting dimension is the simplest possible way to measure such pathologies. It can be defined by
(,)=1+b-a,
x
n+1
y
n+1
y
n
2
x
n
x
n
d
box
d
box
lim
ϵ0
logN(ϵ)
log(1/ϵ)
lim
k∞
logN(k)
log(k/Δ)
where is the number of boxes with size covering the attractor [5–7]. Here is the box-counting step and is the size of the box at the initial step .
N(ϵ)=N(k)
ϵ=ϵ(k)=Δ/k
k≥1
ϵ(1)=Δ
k=1
• Drag the blue locator to change the position of the center .
(,)
x
c
y
c
• Drag the red locator to change the initial condition .
(,)
x
0
y
0
The sliders let you control the following:
• opacity, the opacity of the plotted points.
• point size, the size of the plotted points.
• , the sizes of the horizontal and vertical plot range; .
Δ
Δ=-=-=ϵ(1)
x
right
x
left
y
top
y
bottom
• , the number of iterations.
n
• , the number of initial iterations to be dropped.
n
drop
• , the main control parameter value.
a
• , the dissipation parameter value.
b
• , the box-counting step; the size of the boxes at the box-counting step is given by .
k
ϵ
th
k
ϵ=Δ/k
• , the maximum box-counting step.
k
max
The dropdown menu lets you control:
• , the number of initial box-counting steps to be dropped.
k
drop
And, finally, the resets let you reset the position of the locators.