Sensitive Dependence in Iterated Maps

number of steps

mapping

shift map

3/2 map

λ=3.5 quadratic map

λ=4 quadratic map

initial condition

1

π

1

2

1

3

size of perturbation

perturbation0.00001

5

10

15

20

25

30

0.0

0.2

0.4

0.6

0.8

1.0

values

5

10

15

20

25

30

0.001

0.005

0.010

0.050

0.100

0.500

1

difference

A classic result in chaos theory is that small perturbations in "chaotic" iterated maps grow roughly exponentially. Notice that once the perturbations have magnitudes of order 1, there is no longer the same kind of growth, and instead there are often seemingly random fluctuations, more characteristic of intrinsic randomness generation.