# Lyapunov Fractals

Lyapunov Fractals

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 formed by successive terms in the string, repeated as many times as necessary.

S

3. Choose a point .

(a,b)∈[0,4]×[0,4]

4. Define the function =a if =A, and =b if =B.

r

n

S

n

r

n

S

n

5. Let =0.5, and compute the iterates =(1-).

x

0

x

n+1

r

n

x

n

x

n

6. Compute the Lyapunov exponent: . In practice, is approximated by choosing a suitably large (in the Manipulate code, the variable “iterations” corresponds to ).

λ=log=log(1-2)

lim

N∞

1

N

N

∑

n=1

x

n+1

x

n

lim

N∞

1

N

N

∑

n=1

r

n

x

n

λ

N

N

7. Color the point according to the value of obtained.

(a,b)

λ

8. Repeat steps 3–7 for each point in the image plane.