WOLFRAM|DEMONSTRATIONS PROJECT

Lyapunov Fractals

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zoom
1
plot points
50
string
AB
ABB
AABB
iterations = 2 ×
5
10
20
30
The algorithm for computing the fractal is summarized as follows:
1. Choose a string of A’s and B’s of any nontrivial length (e.g., AABAB).
2. Construct the sequence
S
formed by successive terms in the string, repeated as many times as necessary.
3. Choose a point
(a,b)∈[0,4]×[0,4]
.
4. Define the function
r
n
=a
if
S
n
=A
, and
r
n
=b
if
S
n
=B
.
5. Let
x
0
=0.5
, and compute the iterates
x
n+1
=
r
n
x
n
(1-
x
n
)
.
6. Compute the Lyapunov exponent:
λ=
lim
N∞
1
N
N
∑
n=1
log

x
n+1

x
n
=
lim
N∞
1
N
N
∑
n=1
log
r
n
(1-2
x
n
)
. In practice,
λ
is approximated by choosing a suitably large
N
(in the Manipulate code, the variable “iterations” corresponds to
N
).
7. Color the point
(a,b)
according to the value of
λ
obtained.
8. Repeat steps 3–7 for each point in the image plane.