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 = 3.1. The 2-cycle is stable. Note the alternating pattern of points in bins 1 and 4. s = 3.5. This is a 4 cycle, but the two largest cycle points are so close together, compared to the range from smallest to largest cycle point, that both lie in bin 4. Similarly, both of the smaller cycle points lie in bin 1.

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