Deterministic Chaos

6.Q. IFS Driven by the Tent Map and the Logistic Map

Recall the Driven IFS method for representing data. We take the tent and logistic maps as the generators of our examples.

Here are some examples with the logisitc and tent maps, for the indicated s.

Logistic

0.9, 0.99

1.5, 1.5

2.4, 2.9

3.1, 3.5

3.55, 3.57

3.6, 3.7

3.8, 3.825

3.826, 3.827

3.828, 3.829

3.846, 3.999

Tent

1.2, 1.32

1.4, 1.5

1.7, 1.8

1.9, 1.999

s = 2.4. The nonzero fixed point is stable; from x = 0.5 the iterates spiral in to the fixed point, first in bin 4, then in bin 1, and the remaining values are in bin 2. s = 2.9. Here the spiral is even more pronounced, and long-lived, and now the fixed point is in bin 3.

Here are some animations of how the driven IFS varies with the s parameter of the logistic and tent maps.